CHAPTER I PROJECTIVE MODULES AND VECTOR BUNDLES

The basic objects studied in algebraic K-theory are projective modules over a ring, and vector bundles over schemes. In this first chapter we introduce the cast of characters. Much of this information is standard, but collected here for ease of reference in later chapters.

Here are a few running conventions we will use. The word ring will always mean an associative ring with 1. If R is a ring, the word R-module will mean right Rmodule unless explicitly stated otherwise. We will often use ‘f.g.’ to denote ‘finitely generated.’

§1. Free modules, GL_{n} and stably free modules

If R is a field, or a division ring, then R-modules are called vector spaces. Classical results in linear algebra state that every vector space has a basis, and that the rank (or dimension) of a vector space is independent of the choice of basis. However, much of this fails for arbitrary rings.

As with vector spaces, a basis of an R-module M is a subset {e_{i}}_{i∈I} such that every element of M can be expressed in a unique way as a finite sum ∑ e_{i}r_{i} with r_{i} ∈ R. If M has a fixed basis we call M a based free module, and define the rank of the based free module M to be the cardinality of its given basis. Finally, we say that a module M is free if there is a basis making it into a based free module.

The canonical example of a based free module is R^{n}, which consists of n-tuples of elements of R, or “column vectors” of length n.

Unfortunately, there are rings for which R^{n} ≅ R^{n+t}, t ≠ 0. We make the following definition to avoid this pathology, referring the curious reader to the exercises for more details. (If κ is an infinite cardinal number, then every basis of R^{κ} has cardinality κ. In particular R^{κ} cannot be isomorphic to R^{n} for finite n. See ch.2, 5.5 of [Cohn65].)

Definition 1.1 (IBP). We say that a ring R satisfies the (right) invariant basis property (or IBP) if R^{m} and R^{n} are not isomorphic for m ≠ n. In this case, the rank of a free R-module M is an invariant, independent of the choice of basis of M.

Most of the rings we will consider satisfy the invariant basis property. For example, commutative rings satisfy the invariant basis property, and so do group rings ℤ[G]. This is because a ring R must satisfy the IBP if there exists a ring map f: R → F from R to a field or division ring F. (If R is commutative we may take F = R/𝔪, where 𝔪 is any maximal ideal of R.) To see this, note that any basis of M maps to a basis of the vector space V = M ⊗_{R} F; since dim V is independent of the choice of basis, any two bases of M must have the same cardinality.

Typeset by AMS-TEX

Our choice of right modules dictates that we write R-module homomorphisms on the left. In particular, homomorphisms R^{n} → R^{m} may be thought of as m × n matrices with entries in R, acting on the column vectors in R^{n} by matrix multiplication. We write M_{n}(R) for the ring of n × n matrices, and write GL_{n}(R) for the group of invertible n × n matrices, i.e., the automorphisms of R^{n}. We will usually write R^{×} for the group GL_{1}(R) of units in R.

Example 1.1.1. Any finite-dimensional algebra R over a field (or division ring) F must satisfy the IBP, because the rank of a free R-module M is an invariant:

rank(M) = dim_{F} (M)/ dim_{F} (R).

For a simple artinian ring R we can say even more. Classical Artin-Wedderburn theory states that R = M_{n}(F) for some n and F, and that every right R-module M is a direct sum of copies of V = (F^{n})^{t}, the R-module consisting of row vectors over F of length n. Moreover, the number of copies of V is an invariant of M, called its length; the length is also dim_{F} (M)/n since dim_{F} (V) = n. In this case we also have rank(M) = length(M)/n = dim_{F} (M)/n^{2}.

There are noncommutative rings which do not satisfy the IBP, i.e., which have R^{m} ≅ R^{n} for some m ≠ n. Rank is not an invariant of a free module over these rings. One example is the infinite matrix ring End_{F} (F^{∞}) of endomorphisms of an infinite-dimensional vector space over a field F. Another is the cone ring C(R) associated to a ring R. (See the exercises.)

Unimodular Rows and Stably Free Modules

Definition 1.2. An R-module P is called stably free (of rank n−m) if P ⊕R^{m} ≅ R^{n} for some m and n. (If R satisfies the IBP then the rank of a stably free module is easily seen to be independent of the choice of m and n.) Every stably free module is the kernel of a surjective m × n matrix σ: R^{n} → R^{m}, because a lift of a basis for R^{m} yields a decomposition P ⊕ R^{m} ≅ R^{n}.

This raises a question: when are stably free modules free? Over some rings every stably free module is free (fields, ℤ and the matrix rings M_{n}(F) of Example 1.1.1 are classical cases), but in general this is not so even if R is commutative; see example 1.2.2 below.

1.2.1. The most important special case, at least for inductive purposes, is when m = 1, i.e., P ⊕ R ≅ R^{n}. In this case σ is a row vector, and we call σ a unimodular row. It is not hard to see that the following conditions on a sequence σ = (r_{1}, ..., r_{n}) of elements in R are equivalent for each n:

• σ is a unimodular row;

• R^{n} ≅ P ⊕ R, where P = ker(σ);

• R = r_{1}R + ⋯ + r_{n}R;

• 1 = r_{1}s_{1} + ⋯ + r_{n}s_{n} for some s_{i} ∈ R.

If R^{n} ≅ P ⊕ R with P free, then a basis of P would yield a new basis for R^{n} and hence an invertible matrix g whose first row is the unimodular row σ: R^{n} → R corresponding to P. This gives us a general criterion: P is a free module iff the corresponding unimodular row may be completed to an invertible matrix. (The invertible matrix is in GL_{n}(R) if R satisfies the IBP).

When R is commutative, every unimodular row of length 2 may be completed. Indeed, if r_{1}s_{1} + r_{2}s_{2} = 1, then the desired matrix is:

( \begin{matrix}r_{1} & r_{2} \\ −s_{2} & s_{1} \end{matrix} )

Hence R^{2} ≅ R ⊕ P implies that P ≅ R. In §3 we will obtain a stronger result: every stably free module of rank 1 is free. The fact that R is commutative is crucial; in Ex. 1.6 we give an example of a unimodular row of length 2 which cannot be completed over D[x, y], D a division ring.

Example 1.2.2. Here is an example of a unimodular row σ of length 3 which cannot be completed to an element of GL_{3}(R). Hence P = ker(σ) is a rank 2 stably free module P which is not free, yet P ⊕ R ≅ R^{3}. Let σ be the unimodular row σ = (x, y, z) over the commutative ring R = ℝ[x, y, z]/(x^{2} + y^{2} + z^{2} = 1). Every element (f, g, h) of R^{3} yields a vector field in 3-space (ℝ^{3}), and σ is the vector field pointing radially outward. Therefore an element in P yields a vector field in 3-space tangent to the 2-sphere S^{2}. If P were free, a basis of P would yield two tangent vector fields on S^{2} which are linearly independent at every point of S^{2} (because together with σ they span the tangent space of 3-space at every point). It is well known that this is impossible: you can’t comb the hair on a coconut. Hence P cannot be free.

The following theorem describes a “stable range” in which stably free modules are free (see 2.3 for a stronger version). A proof may be found in [Bass, V.3.5], using the “stable range” condition (S_{n}) of Ex. 1.5 below. Example 1.2.2 shows that this range is sharp.

Bass Cancellation Theorem for Stably Free Modules 1.3. Let R be a commutative noetherian ring of Krull dimension d. Then every stably free Rmodule of rank > d is a free module. Equivalently, every unimodular row of length n ≥ d + 2 may be completed to an invertible matrix.

The study of stably free modules has a rich history, and we cannot do it justice here. An excellent source for further information is the book [Lam].

EXERCISES

1.1. Semisimple rings. An R-module M is called simple if it has no submodules other than 0 and M, and semisimple if it is the direct sum of simple modules. A ring R is called semisimple if R is a semisimple R-module. If R is semisimple, show that R is a direct sum of a finite (say n) number of simple modules. Then use the Jordan-Hölder Theorem, which states that the length of a semisimple module is an invariant, to show that every stably free module is free. In particular, this shows that semisimple rings satisfy the IBP. Hint: Show that length= n · rank is an invariant of free R-modules. 1.2. (P.M.Cohn) Consider the following conditions on a ring R:

(I) R satisfies the invariant basis property (IBP);

(II) For all m and n, if R^{m} ≅ R^{n} ⊕ P then m ≥ n;

(III) For all n, if R^{n} ≅ R^{n} ⊕ P then P = 0.

Show that (III) ⇒ (II) ⇒ (I). For examples of rings satisfying (III) but not (II), resp. (II) but not (I), see [Cohn66]. 1.3. Show that (III) and the following matrix conditions are equivalent:

(a) For all n, every surjection R^{n} → R^{n} is an isomorphism;

(b) For all n, and f, g ∈ M_{n}(R), if fg = 1_{n}, then gf = 1_{n} and g ∈ GL_{n}(R). Then show that commutative rings satisfy (b), hence (III). 1.4. Show that right noetherian rings satisfy condition (b) of the previous exercise. Hence they satisfy (III), and have the right invariant basis property. 1.5. Stable Range Conditions. We say that a ring R satisfies condition (S_{n}) if for every unimodular row (r_{0}, r_{1}, ..., r_{n}) in R^{n+1} there is a unimodular row (r′_{1}, ..., r′_{n}) in R^{n} with r′_{i} = r_{i} − r_{0}t_{i} for some t_{1}, ..., t_{n} in R. The stable range of R, sr(R), is defined to be the smallest n such that R satisfies condition (S_{n}). (Warning: our (S_{n}) is the stable range condition SR_{n+1} of [Bass].)

(a) (Vaserstein) Show that (S_{n}) holds for all n ≥ sr(R).

(b) If sr(R) = n, show that all stably free projective modules of rank ≥ n are free. Bass’ Cancellation Theorem [Bass, V.3.5], which is used to prove 1.3 and 2.3 below, actually states that sr(R) ≤ d + 1 if R is a d-dimensional commutative noetherian ring, or more generally if Max(R) is a finite union of spaces of dimension ≤ d.

(c) Show that sr(R) = 1 for every artinian ring R. Conclude that all stably free projective R-modules are free over artinian rings.

(d) Show that if I is an ideal of R then sr(R) ≥ sr(R/I).

(e) (Veldkamp) If sr(R) = n for some n, show that R satisfies the invariant basis property (IBP). Hint: Consider an isomorphism B: R^{N} ≅ R^{N+n}, and apply (S_{n}) to convert B into a matrix of the form ({C \atop 0} ).

1.6. (Ojanguren-Sridharan) Let D be a division ring which is not a field. Choose α, β ∈ D such that αβ − βα ≠ 0, and show that σ = (x + α, y + β) is a unimodular row over R = D[x, y]. Let P = ker(σ) be the associated rank 1 stably free module; P ⊕ R ≅ R^{2}. Prove that P is not a free D[x, y]-module, using these steps:

(i) If P ≅ R^{n}, show that n = 1. Thus we may suppose that P ≅ R with 1 ∈ R corresponding to a vector \left[{r \atop s}\right] with r, s ∈ R.

(ii) Show that P contains a vector \left[{f \atop g}\right] with f = c_{1}x + c_{2}y + c_{3}xy + c_{4}y^{2} and

g = d_{1}x + d_{2}y + d_{3}xy + d_{4}x^{2}, (c_{i}, d_{i} ∈ D). (iii) Show that P cannot contain any vector \left[{f \atop g}\right] with f and g linear polynomials

in x and y. Conclude that the vector in (i) must be quadratic, and may be taken to be of the form given in (ii).

(iv) Show that P contains a vector \left[{f \atop g}\right] with f = γ_{0}+γ_{1}y+y^{2}, g = δ_{0}+δ_{1}x−αy−xy

and γ_{0} = βu^{−1}βu ≠ 0. This contradicts (iii), so we cannot have P ≅ R. 1.7. For any ring R, let R^{∞} be a fixed free R-module on a countably infinite basis. Then R^{∞} is naturally a left module over the endomorphism ring E = End_{R}(R^{∞}), and we can identify E with the ring of infinite column-finite matrices. If V is any R-module summand of R^{∞}, show that the right ideal I = {f ∈ E : f(R^{∞}) ⊆ V } is a projective E-module. Conclude that E ≅ E^{2} ≅ E^{3} ≅ ⋯ as right E-modules, and that P ⊕ E ≅ E for every f.g. projective E-module P. Is E ≅ E^{∞}? 1.8. Cone Ring. For any ring R, the endomorphism ring End_{R}(R^{∞}) of the previous exercise contains a smaller ring, namely the subring C(R) consisting of row-and-column finite matrices. The ring C(R) is called the cone ring of R. Show that C(R) ≅ C(R) ⊕ C(R) as right C(R)-modules. 1.9. To see why our notion of stably free module involves only finitely generated free modules, let R^{∞} be the infinitely generated free module of exercise 1.7. Prove that if P ⊕ R^{m} ≅ R^{∞} then P ≅ R^{∞}. Hint: The image of R^{m} is contained in some R^{n} ⊆ R^{∞}. Writing R^{∞} ≅ R^{n} ⊕ F and Q = P ∩ R^{n}, show that P ≅ Q ⊕ F and F ≅ F ⊕ R^{m}. This trick is a version of the Eilenberg Swindle of §2. 1.10. Excision for GL_{n}. If I is a ring without unit, let ℤ ⊕ I be the canonical augmented ring with underlying abelian group ℤ⊕I. Let GL_{n}(I) denote the kernel of the map GL_{n}(ℤ ⊕ I) → GL_{n}(ℤ), and let M_{n}(I) denote the matrices with entries in I. If g ∈ GL_{n}(I) then clearly g − 1_{n} ∈ M_{n}(I).

(i) Characterize the set of all x ∈ M_{n}(I) such that 1 + x ∈ GL_{n}(I). (ii) If I is an ideal in a ring R, show that GL_{n}(I) is the kernel of GL_{n}(R) →

GL_{n}(R/I), and so is independent of the choice of R. (iii) If x = (x_{ij}) is any nilpotent matrix in M_{n}(I), such as a strictly upper trian-

gular matrix, show that 1_{n} + x ∈ GL_{n}(I). 1.11. (Whitehead) If g ∈ GL_{n}(R), verify the following identity in GL_{2n}(R) :

( \begin{matrix}g & 0 \\ 0 & g^{−1} \end{matrix} ) = ( \begin{matrix}1 & g \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ −g^{−1} & 1 \end{matrix} ) ( \begin{matrix}1 & g \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

Conclude that if S → R is a ring surjection then there is a matrix h ∈ GL_{2n}(S) mapping to the block diagonal matrix with entries g, g^{−1} displayed above. 1.12. Radical Ideals. A 2-sided ideal I in R is called a radical ideal if 1 + x is a unit of R for every x ∈ I, i.e., if (∀x ∈ I)(∃y ∈ I)(x + y + xy = 0). Every ring has a unique largest radical ideal, called the Jacobson radical of R; it is the intersection of the maximal left ideals of R.

(i) Show that every nil ideal is a radical ideal. (A nil ideal is an ideal in which every element is nilpotent.)

(ii) A ring R is local if it has a unique maximal 2-sided ideal 𝔪, and R/𝔪 is a field

or division ring. Show that 𝔪 is the Jacobson radical of R. (iii) If I is a radical ideal, show that M_{n}(I) is a radical ideal of M_{n}(R) for every

n. Hint: Use elementary row operations to diagonalize any matrix which is congruent to 1_{n} modulo I.

(iv) If I is a radical ideal, show that GL_{n}(R) → GL_{n}(R/I) is surjective for each n. That is, there is a short exact sequence of groups:

1 → GL_{n}(I) → GL_{n}(R) → GL_{n}(R/I) → 1.

(v) If I is a radical ideal, show that sr(R) = sr(R/I), where sr is the stable range of Exercise 1.5. Conclude that sr(R) = 1 for every local ring R.

1.13. A von Neumann regular ring is a ring R such that for every r ∈ R there is an x ∈ R such that r = rxr. It is called unit-regular if for every r ∈ R there is a unit x ∈ R such that r = rxr. If R is von Neumann regular, show that:

(a) R is unit-regular ⟺ R has stable range 1 (in the sense of Exercise 1.5);

(b) If R is unit-regular then R satisfies condition (III) of Exercise 1.2. (The converse does not hold; see Example 5.10 of [Gdearl].)

A rank function on R is a set map ρ: R → [0, 1] such that: (i) ρ(0) = 0 and ρ(1) = 1; (ii) ρ(x) > 0 if x ≠ 0; (iii) ρ(xy) ≤ ρ(x), ρ(y); and (iv) ρ(e + f) = ρ(e) + ρ(f) if e, f are orthogonal idempotents in A. Goodearl and Handelman proved (18.4 of [Gdearl]) that if R is a simple von Neumann ring then:

(III) holds ⟺ R has a rank function.

(c) Let F be a field or division ring. Show that the matrix ring M_{n}(F) is unitregular, and that ρ_{n}(g) = rank(g)/n is a rank function on M_{n}(F). Then show that the ring End_{F} (F^{∞}) is von Neumann regular but not unit-regular.

(d) Consider the union R of the matrix rings M_{n!}(F), where we embed M_{n!}(F) in M_{(n+1)!}(F) ≅ M_{n!}(F) ⊗ M_{n+1}(F) as M_{n!} ⊗ 1. Show that R is a simple von Neumann regular ring, and that the union of the ρ_{n} of (c) gives a rank function ρ: R → [0, 1] with image ℚ ∩ [0, 1].

(e) Show that a commutative ring R is von Neumann regular if and only if it is reduced and has Krull dimension 0. These rings are called absolutely flat rings by Bourbaki, because every R-module is flat. Use Exercise 1.12 to conclude that every commutative 0-dimensional ring R has stable range 1 (and is unitregular).

§2. Projective Modules

Definition 2.1. An R-module P is called projective if there exists a module Q so that the direct sum P ⊕ Q is free. This is equivalent to saying that P satisfies the projective lifting property: For every surjection s: M → N of R-modules and every map g: P → N there exists a map f: P → M so that g = sf.

P
∃f↙ ↓ g
M ⟶^{s} N → 0

To see that these are equivalent, first observe that free modules satisfy this lifting property; in this case f is determined by lifting the image of a basis. To see that all projective modules satisfy the lifting property, extend g to a map from a free module P ⊕Q to N and lift that. Conversely, suppose that P satisfies the projective lifting property. Choose a surjection π: F → P with F a free module; the lifting property splits π, yielding F ≅ P ⊕ ker(π).

If P is a projective module, then P is generated by n elements iff there is a decomposition P ⊕ Q ≅ R^{n}. Indeed, the generators give a surjection π: R^{n} → P, and the lifting property yields the decomposition.

We will focus most of our attention on the category P(R) of finitely generated projective R-modules (f.g. projective modules for short); the morphisms are the Rmodule maps. Since the direct sum of projectives is projective, P(R) is an additive category. We may regard P as a covariant functor on rings, since if R → S is a ring map then there is an additive functor P(R) → P(S) sending P to P ⊗_{R} S.

Hom and ⊗. If P is a projective R-module, then it is well-known that P ⊗_{R} − is an exact functor on the category of (left) R-modules, and that Hom_{R}(P, −) is an exact functors on the category of (right) R-modules. (See [WHomo] for example.)

That is, any exact sequence 0 → L → M → N → 0 of R-modules yields exact sequences

0 → P ⊗ L → P ⊗ M → P ⊗ N → 0

and

0 → Hom(P, L) → Hom(P, M) → Hom(P, N) → 0.

Examples 2.1.1. Of course free modules and stably free modules are projective.

(1) If F is a field (or a division ring) then every F -module (vector space) is free, but this is not so for all rings.

(2) Consider the matrix ring R = M_{n}(F). The R-module V of Example 1.1.1 is projective but not free, because length(V) = 1 < n = length(R).

(3) Componentwise free modules. Another type of projective module arises for rings of the form R = R_{1} × R_{2}; both P = R_{1} × 0 and Q = 0 × R_{2} are projective but cannot be free because the element e = (0, 1) ∈ R satisfies P e = 0 yet R^{n}e ≠ 0. We say that a module M is componentwise free if there is a decomposition R = R_{1} × ⋯ × R_{c} and integers n_{i} such that M ≅ R_{1}^{n_{1}} × ⋯ × R_{c}^{n_{c}}. It is easy to see that all componentwise free modules are projective.

(4) Topological Examples. Other examples of nonfree projective modules come from topology, and will be discussed more in section 4 below. Consider the ring R = C^{0}(X) of continuous functions X → ℝ on a paracompact topological space X. If η: E → X is a vector bundle then by Ex. 4.8 the set Γ(E) = {s: X → E : ηs = 1_{X}} of continuous sections of η forms a projective R-module. For example, if T^{n} is the trivial bundle ℝ^{n} × X → X then Γ(T^{n}) = R^{n}. I claim that if E is a nontrivial vector bundle then Γ(E) cannot be a free R-module. To see this, observe that if Γ(E) were free then the sections {s_{1}, ..., s_{n}} in a basis would define a bundle map f: T^{n} → E such that Γ(T^{n}) = Γ(E). Since the kernel and cokernel bundles of f have no nonzero sections they must vanish, and f is an isomorphism.

When X is compact, the category P(R) of f.g. projective C^{0}(X)-modules is actually equivalent to the category of vector bundles over X; this result is called Swan’s Theorem. (See Ex. 4.9 for a proof.)

Idempotents 2.1.2. An element e of a ring R is called idempotent if e^{2} = e. If e ∈ R is idempotent then P = eR is projective because R = eR ⊕ (1 − e)R. Conversely, given any decomposition R ≅ P ⊕ Q, there are unique elements e ∈ P, f ∈ Q such that 1 = e + f in R. By inspection, e and f = 1 − e are idempotent. Thus idempotent elements of R are in 1-1 correspondence with decompositions R ≅ P ⊕ Q.

If e ≠ 0, 1 and R is commutative then P = eR cannot be free, because P (1−e) = 0 but R(1 − e) ≠ 0. The same is true for noetherian rings by Ex.1.4, but obviously cannot be true for rings such that R ≅ R ⊕ R; see Ex.1.2 (III).

Every finitely generated projective R-module arises from an idempotent element in a matrix ring M_{n}(R). To see this, note that if P ⊕ Q = R^{n} then the projectioninclusion R^{n} → P → R^{n} is an idempotent element e of M_{n}(R). By inspection, the image e(R^{n}) of e is P. The study of projective modules via idempotent elements can be useful, especially for rings of operators on a Banach space.

If R is a Principal Ideal Domain (PID), such as ℤ or F [x], then all projective R-modules are free. This follows from the Structure Theorem for modules over a PID (even for infinitely generated projectives).

Not many other classes of rings have all (f.g.) projective modules free. A famous theorem of Quillen and Suslin states that if R is a polynomial ring (or a Laurent polynomial ring) over a field or a PID then all projective R-modules are free; a good reference for this is the book [Lam]. In particular, if G is a free abelian group then the group ring ℤ[G] is the Laurent polynomial ring ℤ[x, x^{−1}, ..., z, z^{−1}], and has all projectives free. In contrast, if G is a nonabelian torsion-free nilpotent group, Artamanov proved in [Art] that there are always projective ℤ[G]-modules P which are stably free but not free: P ⊕ ℤ[G] ≅ (ℤ[G])^{2}.

It is an open problem to determine whether or not all projective ℤ[G]-modules are stably free when G is a finitely presented torsion-free group. Some partial results and other examples are given in [Lam].

For our purposes, local rings form the most important class of rings with all projectives free. A ring R is called a local ring if R has a unique maximal (2-sided) ideal 𝔪, and R/𝔪 is either a field or a division ring.

Lemma 2.2. If R is a local ring then every finitely generated projective R-module P is free. In fact P ≅ R^{p}, where p = dim_{R/𝔪}(P/𝔪P).

Proof. We first observe that every element u ∈ R − 𝔪 is a unit of R, i.e., uv = vu = 1 for some v. Indeed, by multiplying by a representative for the inverse of ū ∈ R/𝔪 we may assume that u ∈ 1 + 𝔪. Since 𝔪 is the Jacobson radical of R, any element of 1 + 𝔪 must be a unit of R.

Suppose that P ⊕ Q ≅ R^{n}. As vector spaces over F = R/𝔪, P/𝔪P ≅ F^{p} and Q/𝔪Q ≅ F^{q} for some p and q. Since F^{p} ⊕ F^{q} ≅ F^{n}, p + q = n. Choose elements {e_{1}, ..., e_{p}} of P and {e′_{1}, ..., e′_{q}} of Q mapping to bases of P/𝔪P and Q/𝔪Q. The e_{i} and e′_{j} determine a homomorphism R^{p} ⊕ R^{q} → P ⊕ Q ≅ R^{n}, which may be represented by a square matrix (r_{ij}) ∈ M_{n}(R) whose reduction (r̄_{ij}) ∈ M_{n}(F) is invertible. But every such matrix (r_{ij}) is invertible over R by Exercise 1.12. Therefore {e_{1}, ..., e_{p}, e′_{1}, ..., e′_{q}} is a basis for P ⊕ Q, and from this it follows that P is free on basis {e_{1}, ..., e_{p}}.

Remark 2.2.1. Even infinitely generated projective R-modules are free when R is local. See [Kap58].

Corollary 2.2.2. If 𝔭 is a prime ideal of a commutative ring R and P is a f.g. projective R-module, then the localization P_{𝔭} is isomorphic to (R_{𝔭})^{n} for some n ≥ 0. Moreover, there is an s ∈ R − 𝔭 such that the localization of P away from s is free:

(P [\frac{1}{s}]) ≅ (R[\frac{1}{s}])^{n}.

In particular, P_{𝔭′} ≅ (R_{𝔭′})^{n} for every other prime ideal 𝔭′ of R not containing s.

Proof. If P ⊕ Q = R^{m} then P_{𝔭} ⊕ Q_{𝔭} ≅ (R_{𝔭})^{m}, so P_{𝔭} is a f.g. projective R_{𝔭}- module. Since R_{𝔭} is a local ring, P_{𝔭} is free by 2.2. Now every element of P_{𝔭} is of the form p/s for some p ∈ P and s ∈ R−𝔭. By clearing denominators, we may find an Rmodule homomorphism f: R^{n} → P which becomes an isomorphism upon localizing at 𝔭. As coker(f) is a finitely generated R-module which vanishes upon localization, it is annihilated by some s ∈ R−𝔭. For this s, the map f[\frac{1}{s}]: (R[\frac{1}{s}])^{n} → P [\frac{1}{s}] is onto. Since P [\frac{1}{s}] is projective, (R[\frac{1}{s}])^{n} is the direct sum of P [\frac{1}{s}] and a f.g. R[\frac{1}{s}]-module M with M_{𝔭} = 0. Since M is annihilated by some t ∈ R − 𝔭 we have

f[\frac{1}{st}]: (R[\frac{1}{st}])^{n} ⟶^{≅} P [\frac{1}{st}].

Suppose that there is a ring homomorphism f: R → F from R to a field or a division ring F. If M is any R-module (projective or not) then the rank of M at f is the integer dim_{F} (M ⊗_{R} F). However, the rank depends upon f, as the example R = F ×F, M = F ×0 shows. When R is commutative, every such homomorphism factors through the field k(𝔭) = R_{𝔭}/𝔭R_{𝔭} for some prime ideal 𝔭 of R, so we may consider rank(M) as a function on the set Spec(R) of prime ideals in R.

Recall that the set Spec(R) of prime ideals of R has the natural structure of a topological space in which the basic open sets are

D(s) = \{𝔭 ∈ Spec(R) : s ∉ 𝔭\} ≅ Spec(R[\frac{1}{s}]) for s ∈ R.

Definition 2.2.3 (Rank). Let R be a commutative ring. The rank of a f.g. R-module M at a prime ideal 𝔭 of R is rank_{𝔭}(M) = dim_{k(𝔭)} M ⊗_{R} k(𝔭). Since M_{𝔭}/𝔭M_{𝔭} ≅ k(𝔭)^{rank_{𝔭}(M)}, rank_{𝔭}(M) is the minimal number of generators of M_{𝔭}.

If P is a f.g. projective R-module then rank(P): 𝔭 ↦ rank_{𝔭}(P) is a continuous function from the topological space Spec(R) to the discrete topological space ℕ ⊂ ℤ, as we see from Corollary 2.2.2. In this way, we shall view rank(P) as an element of the two sets [Spec(R), ℕ] and [Spec(R), ℤ] of continuous maps from Spec(R) to ℕ and to ℤ, respectively.

We say that P has constant rank n if n = rank_{𝔭}(P) is independent of 𝔭. If Spec(R) is topologically connected, every continuous function Spec(R) → ℕ must be constant, so every f.g. projective R-module has constant rank. For example, suppose that R is an integral domain with field of fractions F; then Spec(R) is connected, and every f.g. projective R-module P has constant rank: rank(P) = dim_{F} (P ⊗_{R} F).

If a module M is not projective, rank(M) need not be a continuous function on Spec(R), as the example R = ℤ, M = ℤ/p shows.

Componentwise free modules 2.2.4. Every continuous f: Spec(R) → ℕ induces a decomposition of Spec(R) into the disjoint union of closed subspaces f^{−1}(n). In fact, f takes only finitely many values (say n_{1}, ..., n_{c}), and it is possible to write R as R_{1} × ... × R_{c} such that f^{−1}(n_{i}) is homeomorphic to Spec(R_{i}). (See Ex. 2.4.) Given such a function f, form the componentwise free R-module:

R^{f} = R_{1}^{n_{1}} × ... × R_{c}^{n_{c}}.

Clearly R^{f} has constant rank n_{i} at every prime in Spec(R_{i}) and rank(R^{f}) = f. For n ≥ max{n_{i}}, R^{f} ⊕ R^{n−f} = R^{n}, so R^{f} is a f.g. projective R-module. Hence continuous functions Spec(R) → ℕ are in 1-1 correspondence with componentwise free modules.

The following variation allows us to focus on projective modules of constant rank in many arguments. Suppose that P is a f.g. projective R-module, so that rank(P) is a continuous function. Let R ≅ R_{1} ×⋯×R_{c} be the corresponding decomposition of R. Then each component P_{i} = P ⊗_{R}R_{i} of P is a projective R_{i}-module of constant rank and there is an R-module isomorphism P ≅ P_{1} × ⋯ × P_{c}.

The next theorem allows us to further restrict our attention to projective modules of rank ≤ dim(R). Its proof may be found in [Bass, IV]. We say that two R-modules M, M′ are stably isomorphic if M ⊕ R^{m} ≅ M′ ⊕ R^{m} for some m ≥ 0.

Bass-Serre Cancellation Theorem 2.3. Let R be a commutative noetherian ring of Krull dimension d, and let P be a projective R-module of constant rank n > d.

(a) (Serre) P ≅ P_{0} ⊕ R^{n−d} for some projective R-module P_{0} of constant rank d.

(b) (Bass) If P is stably isomorphic to P′ then P ≅ P′.

(c) (Bass) For all M, M′, if P ⊕ M is stably isomorphic to M′ then P ⊕ M ≅ M′.

Remark 2.3.1. If P is a projective module whose rank is not constant, then P ≅ P_{1} × ⋯ × P_{c} for some decomposition R ≅ R_{1} × ⋯ × R_{c}. (See Ex. 2.4.) In this case, we can apply the results in 2.3 to each P_{i} individually. The reader is invited to phrase 2.3 in this generality.

Locally Free Modules 2.4. Let R be commutative. An R-module M is called locally free if for every prime ideal 𝔭 of R there is an s ∈ R − 𝔭 so that M[\frac{1}{s}] is a free module. We saw in Corollary 2.2.2 that f.g. projective R-modules are locally free. In fact, the following are equivalent:

(1) M is a finitely generated projective R-module;

(2) M is a locally free R-module of finite rank (i.e., rank_{𝔭}(M) < ∞ for all 𝔭);

(3) M is a finitely presented R-module, and for every prime 𝔭 of R:

M_{𝔭} is a free R_{𝔭}-module.

Proof. The implication (2) ⇒ (3) follows from the theory of descent; nowadays we would say that M is coherent (locally finitely presented), hence finitely presented. (See [Hart, II].) To see that (3) ⇒ (1), note that finite presentation gives an exact sequence

R^{m} → R^{n} ⟶^{ε} M → 0.

We claim that the map ε^{∗}: Hom_{R}(M, R^{n}) → Hom_{R}(M, M) is onto. To see this, recall that being onto is a local property; locally ε_{𝔭}^{∗} is Hom(M_{𝔭}, R_{𝔭}^{n}) → Hom(M_{𝔭}, M_{𝔭}). This is a split surjection because M_{𝔭} is projective and ε_{𝔭}: R_{𝔭}^{n} → M_{𝔭} is a split surjection. If s: M → R^{n} is such that ε^{∗}(s) = εs is id_{M}, then s makes M a direct summand of R^{n}, and M is a f.g. projective module.

Open Patching Data 2.5. It is sometimes useful to be able to build projective modules by patching free modules. The following data suffices Suppose that s_{1}, ..., s_{c} ∈ R form a unimodular row, i.e., s_{1}R + ⋯ + s_{c}R = R. Then Spec(R) is covered by the open sets D(s_{i}) ≅ Spec(R[\frac{1}{s_{i}}]). Suppose we are given g_{ij} ∈ GL_{n}(R[\frac{1}{s_{i}s_{j}}]) with g_{ii} = 1 and g_{ij}g_{jk} = g_{ik} in GL_{n}(R[\frac{1}{s_{i}s_{j}s_{k}}]) for every i, j, k. Then

P = \{(x_{1}, ..., x_{c}) ∈ ∏_{i=1}^{c}(R[\frac{1}{s_{i}}])^{n} : g_{ij}(x_{j}) = x_{i} for all i, j\}

is a f.g. projective R-module by 2.4, because each P [\frac{1}{s_{i}}] is isomorphic to R[\frac{1}{s_{i}}]^{n}.

Milnor Squares 2.6. Another type of patching arises from an ideal I in R and a ring map f: R → S such that I is mapped isomorphically onto an ideal of S, which we also call I. In this case R is the “pullback” of S and R/I :

R = {(r̄, s) ∈ (R/I) × S : f̄(r̄) ≡ s modulo I};

the square

R ⟶^{f} S
↓  ↓
R/I ⟶^{f̄} S/I

is called a Milnor square, because their importance for in patching was emphasized by J. Milnor in [Milnor].

One special kind of Milnor square is the conductor square. This arises when R is commutative and S is a finite extension of R (S is often the integral closure of R). The ideal I is chosen to be the conductor ideal, i.e., the largest ideal of S contained in R, which is just I = {x ∈ R : xS ⊂ R} = ann_{R}(S/R). If S is reduced then I cannot lie in any minimal prime of R or S, so the rings R/I and S/I have lower Krull dimension.

Given a Milnor square, we can construct an R-module M = (M_{1}, g, M_{2}) from the following “descent data”: an S-module M_{1}, an R/I-module M_{2} and a S/I-module isomorphism g: M_{2} ⊗ S/I ≅ M_{1}/IM_{1}. In fact M is the kernel of the map

M_{1} × M_{2} → M_{1}/IM_{1}, (m_{1}, m_{2}) ↦ m̄_{1} − g(f̄(m_{2})).

We call M the R-module obtained by patching M_{1} and M_{2} together along g.

An important special case is when we patch S^{n} and (R/I)^{n} together along a matrix g ∈ GL_{n}(R/I). For example, R is obtained by patching R and R/I together along g=1. We will return to this point when we study K_{1}(R) and K_{0}(R).

Here is Milnor’s result.

Milnor Patching Theorem 2.7. In a Milnor square,

(1) If P is obtained by patching together a f.g. projective S-module P_{1} and a f.g. projective R/I-module P_{2}, then P is a f.g. projective R-module;

(2) P ⊗_{R} S ≅ P_{1} and P/IP ≅ P_{2};

(3) Every f.g. projective R-module arises in this way;

(4) If P is obtained by patching free modules along g ∈ GL_{n}(S/I), and Q is obtained by patching free modules along g^{−1}, then P ⊕ Q ≅ R^{2n}.

We shall prove part (3) here; the rest of the proof will be described in Exercise 2.8. If M is any R-module, the Milnor square gives a natural map from M to the R-module M′ obtained by patching M_{1} = M ⊗_{R}S and M_{2} = M ⊗_{R}(R/I) = M/IM along the canonical isomorphism

(M/IM) ⊗_{R/I} (S/I) ≅ M ⊗_{R} (S/I) ≅ (M ⊗_{R} S)/I(M ⊗_{R} S).

Tensoring M with 0 → R → (R/I) ⊕ S → S/I → 0 yields an exact sequence

Tor_{1}^{R}(M, S/I) → M → M′ → 0,

so in general M′ is just a quotient of M. However, if M is projective, the Torterm is zero and M ≅ M′. Thus every projective R-module may be obtained by patching, as (3) asserts.

Remark 2.7.1. Other examples of patching may be found in [Landsbg].

Eilenberg Swindle 2.8. The following “swindle,” discovered by Eilenberg, explains why we restrict our attention to finitely generated projective modules. Let R^{∞} be an infinitely generated free module. If P ⊕ Q = R^{n}, then

P ⊕ R^{∞} ≅ P ⊕ (Q ⊕ P) ⊕ (Q ⊕ P) ⊕ ⋯ ≅ (P ⊕ Q) ⊕ (P ⊕ Q) ⊕ ⋯ ≅ R^{∞}.

Moreover R^{∞} ≅ R^{∞} ⊕ R^{∞}, and if P ⊕ R^{m} ≅ R^{∞} then P ≅ R^{∞} (see Ex. 1.9). Here are a few more facts about infinitely generated projective modules:

• (Bass) If R is noetherian, every infinitely generated projective module P is free, unless there is an ideal I such that P/IP has fewer generators than P;

• (Kaplansky) Every infinitely generated projective module is the direct sum of countably generated projective modules;

• (Kaplansky) There are infinitely generated projectives P whose rank is finite but rank(P) is not continuous on Spec(R). (See Ex. 2.13)

EXERCISES

2.1 Radical ideals. Let I be a radical ideal in R (Exercise 1.12). If P_{1}, P_{2} are f.g. projective R-modules such that P_{1}/IP_{1} ≅ P_{2}/IP_{2}, show that P_{1} ≅ P_{2}. Hint: Modify the proof of 2.2, observing that Hom(P, Q) → Hom(P/I, Q/I) is onto. 2.2 Idempotent lifting. Let I be a nilpotent ideal, or more generally an ideal that is complete in the sense that every Cauchy sequence ∑_{n=1}^{∞} x_{n} with x_{n} ∈ I^{n} converges to a unique element of I. Show that there is a bijection between the isomorphism classes of f.g. projective R-modules and the isomorphism classes of f.g. projective R/I-modules. To do this, use Ex. 2.1 and proceed in two stages:

(i) Show that every idempotent ē ∈ R/I is the image of an idempotent e ∈ R, and that any other idempotent lift is ueu^{−1} for some u ∈ 1 + I. Hint: it suffices to suppose that I^{2} = 0 (consider the tower of rings R/I^{n}). If r is a lift of ē, consider elements of the form e = r + rxr + (1 − r)y(1 − r) and (1 + xe)e(1 + xe)^{−1}.

(ii) By applying (i) to M_{n}(R), show that every f.g. projective R/I-module is of the form P/IP for some f.g. projective R-module P.

2.3 Let e, e_{1} be idempotents in M_{n}(R) defining projective modules P and P_{1}. If e_{1} = geg^{−1} for some g ∈ GL_{n}(R), show that P ≅ P_{1}. Conversely, if P ≅ P_{1} show that for some g ∈ GL_{2n}(R):

( \begin{matrix}e_{1} & 0 \\ 0 & 0 \end{matrix} ) = g ( \begin{matrix}e & 0 \\ 0 & 0 \end{matrix} ) g^{−1}.

2.4 Rank. If R is a commutative ring and f: Spec(R) → ℤ is a continuous function, show that we can write R = R_{1} ×⋯×R_{c} in such a way that Spec(R) is the disjoint union of the Spec(R_{i}), and f is constant on each of the components Spec(R_{i}) of R. To do this, proceed as follows.

(i) Show that Spec(R) is quasi-compact and conclude that f takes on only finitely many values, say n_{1}, ..., n_{c}. Each V_{i} = f^{−1}(n_{i}) is a closed and open subset of Spec(R) because ℤ is discrete.

(ii) It suffices to suppose that R is reduced, i.e., has no non-zero nilpotent elements. To see this, let 𝔑 be the ideal of all nilpotent elements in R, so R/𝔑 is reduced. Since Spec(R) ≅ Spec(R/𝔑), we may apply idempotent lifting (Ex. 2.2).

(iii) Let I_{i} be the ideal defining V_{i}, i.e., I_{i} = ∩{𝔭 : 𝔭 ∈ V_{i}}. If R is reduced, show that I_{1} + ⋯ + I_{c} = R and that for every i ≠ j I_{i} ∩ I_{j} = ∅. Conclude using the Chinese Remainder Theorem, which says that R ≅ ∏ R_{i}. 2.5 Show that the following are equivalent for every commutative ring R:

(1) Spec(R) is topologically connected

(2) Every f.g. projective R-module has constant rank

(3) R has no idempotent elements except 0 and 1.

2.6 Dual Module. If P is a f.g. projective R-module, the dual module is P̌ = Hom_{R}(P, R). Show that P̌ is a f.g. projective R^{op}-module, where R^{op} denotes R with multiplication reversed. Now suppose that R is commutative, so that R = R^{op}. Show that rank(P) = rank(P̌) as functions from Spec(R) to ℤ. 2.7 Tensor Product. Let P and Q be projective modules over a commutative ring R. Show that the tensor product P ⊗_{R} Q is also a projective R-module, and is finitely generated if P and Q are. Finally, show that

rank(P ⊗_{R} Q) = rank(P) · rank(Q).

2.8 Milnor Patching. In this exercise we prove the Milnor Patching Theorem 2.7, that any R-module obtained by patching f.g. projective modules over S and R/I in a Milnor square is a f.g. projective R-module. Prove the following:

(i) If g ∈ GL_{n}(S/I) is the image of a matrix in either GL_{n}(S) or GL_{n}(R/I), the the patched module P = (S^{n}, g, (R/I)^{n}) is a free R-module.

(ii) Show that (P_{1}, g, P_{2}) ⊕ (Q_{1}, h, Q_{2}) ≅ (P_{1} ⊕ Q_{1}, \binom{g 0}{0 h} , P_{2} ⊕ Q_{2}).

(iii) If g ∈ GL_{n}(S/I), let M be the module obtained by patching S^{2n} and (R/I)^{2n} together along the matrix

( \begin{matrix}g & 0 \\ 0 & g^{−1} \end{matrix} ) ∈ GL_{2n}(S/I).

Use Ex. 1.11 to prove that M ≅ R^{2n}. This establishes Theorem 2.7, part (4). (iv) Given P_{1}⊕Q_{1} ≅ S^{n}, P_{2}⊕Q_{2} ≅ (R/I)^{n} and isomorphisms P_{1}/IP_{1} ≅ P_{2}⊗S/I,

Q_{1}/IQ_{1} ≅ Q_{2} ⊗ S/I, let P and Q be the R-modules obtained by patching the P_{i} and Q_{i} together. By (ii), P ⊕ Q is obtained by patching S^{n} and (R/I)^{n} together along some g ∈ GL_{n}(S/I). Use (iii) to show that P and Q are f.g. projective.

(v) If P_{1} ⊕Q_{1} ≅ S^{m} and P_{2} ⊕Q_{2} ≅ (R/I)^{n}, and g: P_{1}/IP_{1} ≅ P_{2} ⊗S/I, show that (Q_{1} ⊕ S^{n}) ⊗ S/I is isomorphic to (R/I^{m} ⊕ Q_{2}) ⊗ S/I. By (iv), this proves that (P_{1}, g, P_{2}) is f.g. projective, establishing part (1) of Theorem 2.7.

(vi) Prove part (2) of Theorem 2.7 by analyzing the above steps. 2.9 Consider a Milnor square (2.6). Let P_{1}, Q_{1} be f.g. projective S-modules, and P_{2}, Q_{2} be f.g. projective R/I-modules such that there are isomorphisms g: P_{2} ⊗ S/I ≅ P_{1}/IP_{1} and h: Q_{2} ⊗ S/I ≅ Q_{1}/IQ_{1}.

(i) If f: Q_{2} ⊗ S/I ≅ P_{1}/IP_{1}, show that (P_{1}, g, P_{2}) ⊕ (Q_{1}, h, Q_{2}) is isomorphic to (Q_{1}, gf^{−1}h, P_{2}) ⊕ (P_{1}, f, Q_{2}). Hint: Use Ex. 2.8 and the decomposition

( \begin{matrix}g & 0 \\ 0 & h \end{matrix} ) = ( \begin{matrix}gf^{−1}h & 0 \\ 0 & f \end{matrix} ) ( \begin{matrix}h^{−1}f & 0 \\ 0 & f^{−1}h \end{matrix} ) .

(ii) Conclude that (S^{n}, g, R/I^{n}) ⊕ (S^{n}, h, R/I^{n}) ≅ (S^{n}, gh, R/I^{n}) ⊕ R^{n}. 2.10 Suppose P, Q are modules over a commutative ring R such that P ⊗ Q ≅ R^{n} for some n ≠ 0. Show that P and Q are finitely generated projective R-modules. Hint: Find a generating set {p_{i} ⊗q_{i}|i = 1, ..., m} for P ⊗Q; the p_{i} ⊗q_{j} ⊗p_{k} generate P ⊗ Q ⊗ P. Show that {p_{i}} define a split surjection R^{m} → P. 2.11 Let M be a finitely generated module over a commutative ring R. Show that the following are equivalent for every n:

(1) M is a f.g. projective module of constant rank n

(2) M_{𝔭} ≅ R_{𝔭}^{n} for every prime ideal 𝔭 of R. Conclude that in 2.4 we may add:

(4) M is finitely generated, M_{𝔭} is free for every prime ideal 𝔭 of R, and rank(M) is a continuous function on Spec(R).

2.12 If f: R → S is a homomorphism of commutative rings, there is a continuous map f^{∗}: Spec(S) → Spec(R) sending 𝔭 to f^{−1}(𝔭). If P is a f.g. projective R-module, show that rank(P ⊗_{R} S) is the composition of f^{∗} and rank(P). In particular, show that if P has constant rank n, then so does P ⊗_{R} S. 2.13 (Kaplansky) Here is an example of an infinitely generated projective module whose rank is not continuous. Let R be the ring of continuous functions f: [0, 1] → ℝ on the unit interval and I the ideal of all functions f which vanish on some neighborhood [0, ε) of 0. Show that I is a projective R-module, yet rank(I): Spec(R) → {0, 1} is not continuous, so I is neither finitely generated nor free. We remark that every f.g. projective R-module is free; this follows from Swan’s Theorem, since every vector bundle on [0, 1] is trivial (by 4.6.1 below).

Hint: Show that the functions f_{n} = max\{0, t − \frac{1}{n}\} generate I, and construct a splitting to the map R^{∞} → I. To see that rank(I) is not continuous, consider the rank of I at the primes 𝔪_{t} = {f ∈ R : f(t) = 0}, 0 ≤ t ≤ 1.

§3. The Picard Group of a commutative ring

An algebraic line bundle L over a commutative ring R is just a f.g. projective R-module of constant rank 1. The name comes from the fact that if R is the ring of continuous functions on a compact space X, then a topological line bundle (vector bundle which is locally ℝ × X → X) corresponds to an algebraic line bundle by Swan’s Theorem (see example 2.1.1(4) or Ex. 4.9 below).

The tensor product L ⊗_{R} M ≅ M ⊗_{R} L of line bundles is again a line bundle (by Ex. 2.7), and L ⊗_{R} R ≅ L for all L. Thus up to isomorphism the tensor product is a commutative associative operation on line bundles, with identity element R.

Lemma 3.1. If L is a line bundle, then the dual module Ľ = Hom_{R}(L, R) is also a line bundle, and Ľ ⊗_{R} L ≅ R.

Proof. Since rank(Ľ) = rank(L) = 1 by Ex. 2.6, Ľ is a line bundle. Consider the evaluation map Ľ ⊗_{R} L → R sending f ⊗ x to f(x). If L ≅ R, this map is clearly an isomorphism. Therefore for every prime ideal 𝔭 the localization

(Ľ ⊗_{R} L)_{𝔭} = (L_{𝔭})ˇ ⊗_{R_{𝔭}} L_{𝔭} → R_{𝔭}

is an isomorphism. Since being an isomorphism is a local property of an R-module homomorphism, the evaluation map must be an isomorphism.

Definition. the Picard group Pic(R) of a commutative ring R is the set of isomorphism classes of line bundles over R. As we have seen, the tensor product ⊗_{R} endows Pic(R) with the structure of an abelian group, the identity element being [R] and the inverse being L^{−1} = Ľ.

Proposition 3.2. Pic is a functor from commutative rings to abelian groups. That is, if R → S is a ring homomorphism then Pic(R) → Pic(S) is a homomorphism sending L to L ⊗_{R} S.

Proof. If L is a line bundle over R, then L ⊗_{R} S is a line bundle over S (see Ex. 2.12), so ⊗_{R}S maps Pic(R) to Pic(S). The natural isomorphism (L ⊗_{R} M) ⊗_{R} S ≅ (L ⊗_{R} S) ⊗_{S} (M ⊗_{R} S), valid for all R-modules L and M, shows that ⊗_{R}S is a group homomorphism.

Lemma 3.3. If L is a line bundle, then End_{R}(L) ≅ R.

Proof. Multiplication by elements in R yields a map from R to End_{R}(L). As it is locally an isomorphism, it must be an isomorphism.

Determinant line bundle of a projective module

If M is any module over a commutative ring R and k ≥ 0, the k^{th} exterior product ∧^{k}M is the quotient of the k-fold tensor M ⊗ ⋯ ⊗ M by the submodule generated by terms m_{1} ⊗ ⋯ ⊗ m_{k} with m_{i} = m_{j} for some i ≠ j. By convention, ∧^{0}M = R and ∧^{1}M = M. The following facts are classical; see [B-AC, ch.2].

(i) ∧^{k}(R^{n}) is the free module of rank (_{k}^{n}) generated by terms e_{i_{1}} ∧ ⋯ ∧ e_{i_{k}} with 1 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ n. In particular, ∧^{n}(R^{n}) ≅ R on e_{1} ∧ ⋯ ∧ e_{n}.

(ii) If R → S is a ring map, there is a natural isomorphism (∧^{k}M) ⊗_{R} S ≅ ∧^{k}(M ⊗_{R} S), the first ∧^{k} being taken over R and the second being taken over S. In particular, rank(∧^{k}M) = \binom{rank M}{k} as functions from Spec(R) to ℕ.

(iii) (Sum Formula) If M = P ⊕ Q, there is a natural isomorphism

∧^{k}(P ⊕ Q) ≅ ⨁_{i=0}^{k}(∧^{i}P) ⊗ (∧^{k−i}Q).

If P is a projective module of constant rank n, then ∧^{k}P is a f.g. projective module of constant rank \binom{n}{k}, because ∧^{k}P is locally free: if P [\frac{1}{s}] ≅ (R[\frac{1}{s}])^{n} then (∧^{k}P)[\frac{1}{s}] ≅ (R[\frac{1}{s}])^{\binom{n}{k}}. In particular, ∧^{n}P is a line bundle, and ∧^{k}P = 0 for k > n. We write det(P) for ∧^{n}P, and call it the determinant line bundle of P.

If the rank of a projective module P is not constant, we define the determinant line bundle det(P) componentwise, using the following recipe. From §2 we find a decomposition R ≅ R_{1} ×⋯×R_{c} so that P ≅ P_{1} ×⋯×P_{c} and each P_{i} has constant rank n_{i} as an R_{i}-module. We then define det(P) to be (∧^{n_{1}}P_{1}) × ⋯ × (∧^{n_{c}}P^{c}); clearly det(P) is a line bundle on R. If P has constant rank n, this agrees with our above definition: det(P) = ∧^{n}P.

As the name suggests, the determinant line bundle is related to the usual determinant of a matrix. An n × n matrix g is just an endomorphism of R^{n}, so it induces an endomorphism ∧^{n}g of ∧^{n}R^{n} ≅ R. By inspection, ∧^{n}g is multiplication by det(g).

Using the determinant line bundle, we can also take the determinant of an endomorphism g of a f.g. projective R-module P. By the naturality of ∧^{n}, g induces an endomorphism det(g) of det(P). By Lemma 3.3, det(g) is an element of R, acting by multiplication; we call det(g) the determinant of the endomorphism g.

Here is an application of the det construction. Let L, L′ be stably isomorphic line bundles. That is, P = L ⊕ R^{n} ≅ L′ ⊕ R^{n} for some n. The Sum Formula (iii) shows that det(P) = L, and det(P) = L′, so L ≅ L′. Taking L′ = R, this shows that R is the only stably free line bundle. It also gives the following slight improvement upon the Cancellation Theorem 2.3 for 1-dimensional rings:

Proposition 3.4. Let R be a commutative noetherian 1-dimensional ring. Then all f.g. projective R-modules are completely classified by their rank and determinant. In particular, every f.g. projective R-module P of rank ≥ 1 is isomorphic to L⊕R^{f}, where L = det(P) and f = rank(P) − 1.

Invertible Ideals

When R is a commutative integral domain (=domain), we can give a particularly nice interpretation of Pic(R), using the following concepts. Let F be the field of fractions of R; a fractional ideal is a nonzero R-submodule I of F such that I ⊆ fR for some f ∈ F. If I and J are fractional ideals then their product IJ = {∑ x_{i}y_{i} : x_{i} ∈ I, y_{i} ∈ J} is also a fractional ideal, and the set Frac(R) of fractional ideals becomes an abelian monoid with identity element R. A fractional ideal I is called invertible if IJ = R for some other fractional ideal J; invertible ideals are sometimes called Cartier divisors. The set of invertible ideals is therefore an abelian group, and one writes Cart(R) or Pic(R, F) for this group.

If f ∈ F^{×}, the fractional ideal div(f) = fR is invertible because (fR)(f^{−1}R) = R; invertible ideals of this form are called principal divisors. Since (fR)(gR) = (fg)R, the function div: F^{×} → Cart(R) is a group homomorphism.

This all fits into the following overall picture (see Ex. 3.7 for a generalization).

Proposition 3.5. If R is a commutative integral domain, every invertible ideal is a line bundle, and every line bundle is isomorphic to an invertible ideal. If I and J are fractional ideals, and I is invertible, then I ⊗_{R} J ≅ IJ. Finally, there is an exact sequence of abelian groups:

1 → R^{×} → F^{×} ⟶^{div} Cart(R) → Pic(R) → 0.

Proof. If I and J are invertible ideals such that IJ ⊆ R, then we can interpret elements of J as homomorphisms I → R. If IJ = R then we can find x_{i} ∈ I and y_{i} ∈ J so that x_{1}y_{1} + ... + x_{n}y_{n} = 1. The {x_{i}} assemble to give a map R^{n} → I and the {y_{i}} assemble to give a map I → R^{n}. The composite I → R^{n} → I is the identity, because it sends r ∈ I to ∑ x_{i}y_{i}r = r. Thus I is a summand of R^{n}, i.e., I is a f.g. projective module. As R is an integral domain and I ⊆ F, rank(I) is the constant dim_{F} (I ⊗_{R} F) = dim_{F} (F) = 1. Hence I is a line bundle.

This construction gives a set map Cart(R) → Pic(R); to show that it is a group homomorphism, it suffices to show that I ⊗_{R} J ≅ IJ for invertible ideals. Suppose that I is a submodule of F which is also a line bundle over R. As I is projective, I ⊗_{R} − is an exact functor. Thus if J is an R-submodule of F then I ⊗_{R} J is a submodule of I ⊗_{R} F. The map I ⊗_{R} F → F given by multiplication in F is an isomorphism because I is locally free and F is a field. Therefore the composite

I ⊗_{R} J ⊆ I ⊗_{R} F ⟶^{multiply} F

sends ∑ x_{i} ⊗ y_{i} to ∑ x_{i}y_{i}. Hence I ⊗_{R} J is isomorphic to its image IJ ⊆ F. This proves the third assertion.

The kernel of Cart(R) → Pic(R) is the set of invertible ideals I having an isomorphism I ≅ R. If f ∈ I corresponds to 1 ∈ R under such an isomorphism then I ≅ fR = div(f). This proves exactness of the sequence at Cart(R).

Clearly the units R^{×} of R inject into F^{×}. If f ∈ F^{×} then fR = R iff f ∈ R and f is in no proper ideal, i.e., iff f ∈ R^{×}. This proves exactness at R^{×} and F^{×}.

Finally, we have to show that every line bundle L is isomorphic to an invertible ideal of R. Since rank(L) = 1, there is an isomorphism L ⊗_{R} F ≅ F. This gives an injection L ≅ L ⊗_{R} R ⊂ L ⊗_{R} F ≅ F, i.e., an isomorphism of L with a fractional ideal I. Choosing an isomorphism Ľ ≅ J, Lemma 3.1 yields

R ≅ L ⊗_{R} Ľ ≅ I ⊗_{R} J ≅ IJ.

Hence IJ = fR for some f ∈ F^{×}, and I(f^{−1}J) = R, so I is invertible.

Dedekind domains

Historically, the most important applications of the Picard group have been for Dedekind domains. A Dedekind domain is a commutative integral domain which is noetherian, integrally closed and has Krull dimension 1.

There are many equivalent definitions of Dedekind domain in the literature. Here is another: an integral domain R is Dedekind iff every fractional ideal of R is invertible. In a Dedekind domain every nonzero ideal (and fractional ideal) can be written uniquely as a product of prime ideals 𝔭_{1}^{n_{1}} ⋯ 𝔭_{r}^{n_{r}}. Therefore Cart(R) is the free abelian group on the set of (nonzero) prime ideals of R, and Pic(R) is the set of isomorphism classes of (actual) ideals of R.

Another property of Dedekind domains is that every f.g. torsionfree R-module M is projective. To prove this fact we use induction on rank_{0}(M) = dim_{F} (M ⊗ F), the case rank_{0}(M) = 0 being trivial. Set rank_{0}(M) = n + 1. As M is torsionfree, it is a submodule of M ⊗F ≅ F^{n+1}. The image of M under any coordinate projection F^{n+1} → F is a fractional ideal I_{0}. As I_{0} is invertible, the projective lifting property for I_{0} shows that M ≅ M′⊕I_{0} with rank_{0}(M′) = n. By induction, M ≅ I_{0}⊕⋯⊕I_{n} is a sum of ideals. By Propositions 3.4 and 3.5, M ≅ I ⊕ R^{n} for the invertible ideal I = det(M) = I_{0} ⋯ I_{n}.

Examples. Here are some particularly interesting classes of Dedekind domains.

• A principal ideal domain (or PID) is a domain R in which every ideal is rR for some r ∈ R. Clearly, these are just the Dedekind domains with Pic(R) = 0. Examples of PID’s include ℤ and polynomial rings k[x] over a field k.

• A discrete valuation domain (or DVR) is a local Dedekind domain. By Lemma 2.2, a DVR is a PID R with a unique maximal ideal M = πR. Fixing π, it isn’t hard to see that every ideal of R is of the form π^{i}R for some i ≥ 0. Consequently every fractional ideal of R can be written as π^{i}R for a unique i ∈ ℤ. By Proposition 3.5, F^{×} ≅ R^{×} × {π^{i}}. There is a (discrete) valuation ν on the field of fractions F : ν(f) is that integer i such that fR ≅ π^{i}R.

Examples of DVR’s include the p-adic integers ℤ̂_{p}, the power series ring k[[x]] over a field k, and localizations ℤ_{p} of ℤ.

• Let F be a number field, i.e., a finite field extension of ℚ. An algebraic integer of F is an element which is integral over ℤ, i.e., a root of a monic polynomial x^{n} +a_{1}x^{n−1} +⋯+a_{n} with integer coefficients (a_{i} ∈ ℤ). The set O_{F} of all algebraic integers of F is a ring—it is the integral closure of ℤ in F. A famous result in ring theory asserts that O_{F} is a Dedekind domain with field of fractions F. It follows that O_{F} is a lattice in F, i.e., a free abelian group of rank dim_{ℚ}(F).

In Number Theory, Pic(O_{F}) is called the ideal class group of the number field F. A fundamental theorem states that Pic(O_{F}) is always a finite group, but the precise structure of the ideal class group is only known for special number fields of small dimension. For example, if ξ_{p} = e^{2πi/p} then ℤ[ξ_{p}] is the ring of algebraic integers of ℚ(ξ_{p}), and the class group is zero if and only if p ≤ 19; Pic(ℤ[ξ_{23}]) is ℤ/3. More details may be found in books on number theory, such as [BSh].

• If C is a smooth affine curve over a field k, then the coordinate ring R of C is a Dedekind domain. One way to construct a smooth affine curve is to start with a smooth projective curve C̄. If {p_{0}, ..., p_{n}} is any nonempty set of points on C̄, the Riemann-Roch theorem implies that C = C̄ − {p_{0}, ..., p_{n}} is a smooth affine curve.

If k is algebraically closed, Pic(R) is a divisible abelian group. Indeed, the points of the Jacobian variety J(C̄) form a divisible abelian group, and Pic(R) is the quotient of J(C̄) by the subgroup generated by the classes of the prime ideals of R corresponding to p_{1}, ..., p_{n}.

This is best seen when k = ℂ, because smooth projective curves over ℂ are the same as compact Riemann surfaces. If C̄ is a compact Riemann surface of genus g, then as an abelian group the points of the Jacobian J(C̄) form the divisible group (ℝ/ℤ)^{2g}. In particular, when C = C̄ − {p_{0}} then Pic(R) ≅ J(C̄) ≅ (ℝ/ℤ)^{2g}.

For example, R = ℂ[x, y]/(y^{2} − x(x − 1)(x − β)) is a Dedekind domain with Pic(R) ≅ (ℝ/ℤ)^{2} if β ≠ 0, 1. Indeed, R is the coordinate ring of a smooth affine curve C obtained by removing one point from an elliptic curve (= a projective curve of genus g = 1).

The Weil Divisor Class group

Let R be an integrally closed domain (= normal domain) with field of fractions F. If R is a noetherian normal domain, it is well-known that:

(i) R_{𝔭} is a discrete valuation ring (DVR) for all height 1 prime ideals 𝔭; (ii) R = ∩R_{𝔭}, the intersection being over all height 1 primes 𝔭 of R, each R_{𝔭} being

a subring of F; (iii) Every r ≠ 0 in R is contained in only finitely many height 1 primes 𝔭. An integral domain R satisfying (i), (ii) and (iii) is called a Krull domain.

Krull domains are integrally closed because every DVR R_{𝔭} is integrally closed. For a Krull domain R, the group D(R) of Weil divisors is the free abelian group on the height 1 prime ideals of R. An effective Weil divisor is a divisor D = ∑ n_{i}[𝔭_{i}] with all the n_{i} ≥ 0.

We remark that effective divisors correspond to “divisorial” ideals of R, D corresponding to the intersection ∩𝔭_{i}^{(n_{i})} of the symbolic powers of the 𝔭_{i}.

If 𝔭 is a height 1 prime of R, the 𝔭-adic valuation ν_{𝔭}(I) of an invertible ideal I is defined to be that integer ν such that I_{𝔭} = 𝔭^{ν}R_{𝔭}. By (iii), v_{𝔭}(I) ≠ 0 for only finitely many 𝔭, so ν(I) = ∑ ν_{𝔭}(I)[𝔭] is a well-defined element of D(R). By 3.5, this gives a group homomorphism:

ν: Cart(R) → D(R).

If I is invertible, ν(I) is effective iff I ⊆ R. To see this, observe that ν(I) is effective ⟺ I_{𝔭} ⊆ R_{𝔭} for all 𝔭 ⟺ I ⊆ ∩I_{𝔭} ⊆ ∩R_{𝔭} = R. It follows that ν is an injection, because if both ν(I) and ν(I^{−1}) are effective then I and I^{−1} are ideals with product R; this can only happen if I = R.

The divisor class group Cl(R) of R is defined to be the quotient of D(R) by the subgroup of all ν(fR), f ∈ F^{×}. This yields a map Pic(R) → Cl(R) which is evidently an injection. Summarizing, we have proven:

Proposition 3.6. Let R be a Krull domain. Then Pic(R) is a subgroup of the class group Cl(R), and there is a commutative diagram with exact rows:

1 → R^{∗} → F^{∗} →^{div} Cart(R) → Pic(R) → 0
↓= ↓= ⋂ ν ⋂
1 → R^{∗} → F^{∗} →^{div} D(R) → Cl(R) → 0.

Remark 3.6.1. Both the Picard group and the divisor class group of a Krull domain R are invariant under polynomial and Laurent polynomial extensions. That is, Pic(R) = Pic(R[t]) = Pic(R[t, t^{−1}]) and Cl(R) = Cl(R[t]) = Cl(R[t, t^{−1}]). Most of this assertion is proven in [B-AC, ch.7,§1]; the Pic[t, t^{−1}] assertion is proven in [BM, 5.10].

Recall that an integral domain R is called factorial, or a Unique Factorization Domain (UFD) if every nonzero element r ∈ R is either a unit or a product of prime elements. (This implies that the product is unique up to order and primes differing by a unit). It is not hard to see that UFD’s are Krull domains; the following interpretation in terms of the class group is taken from [Matsu, §20].

Theorem 3.7. Let R be a Krull domain. Then R is a UFD ⟺ Cl(R) = 0.

Definition. A noetherian ring R is called regular if every R-module M has a finite resolution 0 → P_{n} → ⋯ → P_{0} → M → 0 with the P_{i} projective. Every localization S^{−1}R of a regular ring R is also a regular ring, because S^{−1}R-modules are also R-modules, and a localization of an R-resolution is an S^{−1}R-resolution.

Now suppose that (R, 𝔪) is a regular local ring. It is well-known [Matsu, §14,19] that R is a noetherian, integrally closed domain (hence Krull), and that if s ∈ 𝔪−𝔪^{2} then sR is a prime ideal.

Theorem 3.8. Every regular local ring is a UFD.

Proof. We proceed by induction on dim(R). If dim(R) = 0 then R is a field; if dim(R) = 1 then R is a DVR, hence a UFD. Otherwise, choose s ∈ 𝔪 − 𝔪^{2}. Since sR is prime, Ex. 3.8(b) yields Cl(R) ≅ Cl(R[\frac{1}{s}]). Hence it suffices to show that S = R[\frac{1}{s}] is a UFD. Let 𝔓 be a height 1 prime of S; we have to show that 𝔓 is a principal ideal. For every prime ideal 𝔔 of S, S_{𝔔} is a regular local ring of smaller dimension than R, so by induction S_{𝔔} is a UFD. Hence 𝔓_{𝔔} is principal: xS_{𝔔} for some x ∈ S. By 2.4, 𝔓 is projective, hence invertible. Let 𝔭 be the prime ideal of R such that 𝔓 = 𝔭[\frac{1}{s}] and choose an R-resolution 0 → P_{n} → ⋯ → P_{0} → 𝔭 → 0 of 𝔭 by f.g. projective R-modules P_{i}. Since R is local, the P_{i} are free. Since 𝔓 is projective, the localized sequence 0 → P_{n}[\frac{1}{s}] → ⋯ → P_{0}[\frac{1}{s}] → 𝔓 → 0 splits. Letting E (resp. F) denote the direct sum of the odd (resp. even) P_{i}[\frac{1}{s}], we have 𝔓 ⊕ E ≅ F. Since stably free line bundles are free, 𝔓 is free. That is, 𝔓 = xS for some x ∈ 𝔓, as desired.

Corollary 3.8.1. If R is a regular domain, then Cart(R) = D(R), and hence

Pic(R) = Cl(R).

Proof. We have to show that every height 1 prime ideal 𝔓 of R is invertible. For every prime ideal 𝔭 of R we have 𝔓_{𝔭} ≅ R_{𝔭} in the UFD R_{𝔭}. By 2.4 and 3.5, 𝔓 is an invertible ideal.

Remark 3.8.2. A ring is called locally factorial if R_{𝔭} is factorial for every prime ideal 𝔭 of R. For example, regular rings are locally factorial by 3.8. The proof of Cor. 3.8.1 shows that if R is a locally factorial Krull domain then Pic(R) = Cl(R).

Non-normal rings

The above discussion should make it clear that the Picard group of a normal domain is a classical object, even if it is hard to compute in practice. If R isn’t normal, we can get a handle on Pic(R) using the techniques of the rest of this section. For example, the next lemma allows us to restrict attention to reduced noetherian rings with finite normalization, because the quotient R_{red} of any commutative ring R by its nilradical (the ideal of nilpotent elements) is a reduced ring, and every commutative ring is the filtered union of its finitely generated subrings–rings having these properties.

Lemma 3.9. (1) Pic(R) = Pic(R_{red}).

(2) Pic commutes with filtered direct limits of rings. In particular, if R is the filtered union of subrings R_{α}, then Pic(R) ≅ \underrightarrow{lim} Pic(R_{α}).

Proof. Part (1) is an instance of idempotent lifting (Ex. 2.2). To prove (2), recall from 2.5 that a line bundle L over R may be given by patching data: a unimodular row (s_{1}, ..., s_{c}) and units g_{ij} over the R[\frac{1}{s_{i}s_{j}}]. If R is the filtered direct limit of rings R_{α}, this finite amount of data defines a line bundle L_{α} over one of the R_{α}, and we have L = L_{α} ⊗_{R_{α}} R. If L_{α} and L′_{α} become isomorphic over R, the isomorphism is defined over some R_{β}, i.e., L and L′ become isomorphic over R_{β}.

If R is reduced noetherian, its normalization S is a finite product of normal domains S_{i}. We would like to describe Pic(R) in terms of the more classical group Pic(S) = ∏ Pic(S_{i}), using the conductor square of 2.6. For this it is convenient to assume that S is finite over R, an assumption which is always true for rings of finite type over a field.

More generally, suppose that we are given a Milnor square (2.6):

R →^{f} S
↓  ↓
R/I →^{f̄} S/I.

Given a unit β of S/I, the Milnor Patching Theorem 2.7 constructs a f.g. projective R-module L_{β} = (S, β, R/I) with L_{β} ⊗_{R} S ≅ S and L_{β}/IL_{β} ≅ R/I. In fact L_{β} is a line bundle, because rank(L_{β}) = 1; every map from R to a field factors through either R/I or S (for every prime ideal 𝔭 of R either I ⊆ 𝔭 or R_{𝔭} ≅ S_{𝔭}). By Ex. 2.9, L_{α} ⊕ L_{β} ≅ L_{αβ} ⊕ R; applying ∧^{2} yields L_{α} ⊗_{R} L_{β} ≅ L_{αβ}. Hence the formula ∂(β) = [L_{β}] yields a group homomorphism

∂: (S/I)^{×} → Pic(R).

Theorem 3.10 (Units-Pic sequence). Given a Milnor square, the following sequence is exact. Here Δ denotes the diagonal map and ± denotes the difference map sending (s, r̄) to s̄f(r̄)^{−1}, resp. (L′, L) to L′ ⊗_{S} S/I ⊗_{R/I} L^{−1}.

1 → R^{×} →^{Δ} S^{×}× (R/I)^{×} →^{±} (S/I)^{×} →^{∂} Pic(R) →^{Δ} Pic(S) × Pic(R/I) →^{±} Pic(S/I)

Proof. Since R is the pullback of S and R/I, exactness at the first two places is clear. Milnor Patching 2.7 immediately yields exactness at the last two places, leaving only the question of exactness at (S/I)^{×}. Given s ∈ S^{×} and r̄ ∈ (R/I)^{×}, set β = ±(s, r̄) = s̄f(r̄)^{−1}, where s̄ denotes the reduction of s modulo I. By inspection, λ = (s, r̄) ∈ L_{β} ⊂ S × R/I, and every element of L_{β} is a multiple of λ. It follows that L_{β} ≅ R. Conversely, suppose given β ∈ (S/I)^{×} with L_{β} ≅ R. If λ = (s, r̄) is a generator of L_{β} we claim that s and r̄ are units, which implies that β = s̄f(r̄)^{−1} and finishes the proof. If s′ ∈ S maps to β ∈ S/I then (s′, 1) ∈ L_{β}; since (s′, 1) = (xs, xr̄) for some x ∈ R this implies that r̄ ∈ (R/I)^{×}. If t ∈ S maps to f(r̄)^{−1}β^{−1} ∈ S/I then st ≡ 1 modulo I. Now I ⊂ sR because I × 0 ⊂ L_{β}, so st = 1 + sx for some x ∈ R. But then s(t − x) = 1, so s ∈ S^{×} as claimed.

Example 3.10.1. (Cusp). Let k be a field and let R be k[x, y]/(x^{3} = y^{2}), the coordinate ring of the cusp in the plane. Setting x = t^{2}, y = t^{3} makes R isomorphic to the subring k[t^{2}, t^{3}] of S = k[t]. The conductor ideal from S to R is I = t^{2}S, so we get a conductor square with R/I = k and S/I = k[t]/(t^{2}). Now Pic(k[t]) = 0 and (S/I)^{×} ≅ k^{×}×k with α ∈ k corresponding to (1+αt) ∈ (S/I)^{×}. Hence Pic(R) ≅ k. A little algebra shows that a nonzero α ∈ k corresponds to the invertible prime ideal 𝔭 = (1 − α^{2}x, x − αy)R corresponding to the point (x, y) = (α^{−2}, α^{−3}) on the cusp.

Example 3.10.2. (Node). Let R be k[x, y]/(y^{2} = x^{2} + x^{3}), the coordinate ring of the node in the plane over a field k with char(k) ≠ 2. Setting x = t^{2} − 1 and y = tx makes R isomorphic to a subring of S = k[t] with conductor ideal I = xS. We get a conductor square with R/I = k and S/I ≅ k ×k. Since (S/I)^{×} ≅ k^{×} ×k^{×} we see that Pic(R) ≅ k^{×}. A little algebra shows that α ∈ k^{×} corresponds to the invertible prime ideal 𝔭 corresponding to the point (x, y) = (\frac{4α}{(α−1)^{2}} , \frac{4α(α+1)}{(α−1)^{3}}) on

the node corresponding to t = (\frac{1+α}{1−α}).

Seminormal rings

A reduced commutative ring R is called seminormal if whenever x, y ∈ R satisfy x^{3} = y^{2} there is an s ∈ R with s^{2} = x, s^{3} = y. If R is an integral domain, there is an equivalent definition: R is seminormal if every s in the field of fractions satisfying s^{2}, s^{3} ∈ R belongs to R. Normal domains are clearly seminormal; the node (3.10.2) is not normal (t^{2} = 1 + x), but it is seminormal (see Ex. 3.13). Arbitrary products of seminormal rings are also seminormal, because s may be found slotwise. The cusp (3.10.1) is the universal example of a reduced ring which is not seminormal.

Our interest in seminormal rings lies in the following theorem, first proven by C. Traverso for geometric rings and extended by several authors. For normal domains, it follows from Remark 3.6.1 above. Our formulation is taken from [Swan80].

Theorem 3.11. (Traverso). The following are equivalent for a commutative ring R:

(1) R_{red} is seminormal;

(2) Pic(R) = Pic(R[t]);

(3) Pic(R) = Pic(R[t_{1}, ..., t_{n}]) for all n.

Remark 3.11.1. If R is seminormal, it follows that R[t] is also seminormal. By Ex. 3.11, R[t, t^{−1}] and the local rings R_{𝔭} are also seminormal. However, the Pic[t, t^{−1}] analogue of Theorem 3.11 fails. For example, if R is the node (3.10.2) then Pic(R[t, t^{−1}]) ≅ Pic(R) × ℤ. For more details, see [Weib91].

To prove Traverso’s theorem, we shall need the following standard result about units of polynomial rings.

Lemma 3.12. Let R be a commutative ring with nilradical 𝔑. If r_{0} + r_{1}t + ⋯ + r_{n}t^{n} is a unit of R[t] then r_{0} ∈ R^{×} and r_{1}, ..., r_{n} are nilpotent. Consequently, if NU(R) denotes the subgroup 1 + t𝔑[t] of R[t]^{×} then:

(1) R[t]^{×} = R^{×} × NU(R);

(2) If R is reduced then R^{×} = R[t]^{×};

(3) Suppose that R is an algebra over a field k. If char(k) = p, NU(R) is a p-group. If char(k) = 0, NU(R) is a uniquely divisible abelian group (= a ℚ-module).

Proof of Traverso’s Theorem. We refer the reader to Swan’s paper for the proof that (1) implies (2) and (3). By Lemma 3.9, we may suppose that R is reduced but not seminormal. Choose x, y ∈ R with x^{3} = y^{2} such that no s ∈ R satisfies s^{2} = x, s^{3} = y. Then the reduced ring S = R[s]/(s^{2} − x, s^{3} − y)_{red} is strictly larger than R. Since I = xS is an ideal of both R and S, we have Milnor squares

\begin{matrix}R & →^{f} & S \\ ↓ & & ↓ \\ R/I & →^{f̄} & S/I \end{matrix}  and  \begin{matrix}R[t] & →^{f} & S[t] \\ ↓ & & ↓ \\ R/I[t] & →^{f̄} & S/I[t]. \end{matrix}

The Units-Pic sequence 3.10 of the first square is a direct summand of the Units-Pic sequence for the second square. Using Lemma 3.12, we obtain the exact quotient sequence

0 → NU(R/I) → NU(S/I) ⟶^{∂} \frac{Pic(R[t])}{Pic(R)} .

By construction, s ∉ R and s̄ ∉ R/I. Hence ∂(1 + s̄t) is a nonzero element of the quotient Pic(R[t])/ Pic(R). Therefore if R isn’t seminormal we have Pic(R) ≠ Pic(R[t]), which is the (3) ⇒ (2) ⇒ (1) half of Traverso’s theorem.

EXERCISES

In these exercises, R is always a commutative ring. 3.1 Show that the following are equivalent for every R-module L:

(a) There is a R-module M such that L ⊗ M ≅ R.

(b) L is an algebraic line bundle.

(c) L is a finitely generated R-module and L_{𝔭} ≅ R_{𝔭} for every prime ideal 𝔭 of R. Hint: Use Exercises 2.10 and 2.11. 3.2 Show that the tensor product P ⊗_{R} Q of two line bundles may be described using “Open Patching” 2.5 as follows. Find s_{1}, ..., s_{r} ∈ R forming a unimodular row, such that P (resp. Q) is obtained by patching the R[\frac{1}{s_{i}}] by units g_{ij} (resp. h_{ij}) in R[\frac{1}{s_{i}s_{j}}]^{×}. Then P ⊗_{R} Q is obtained by patching the R[\frac{1}{s_{i}}] using the units f_{ij} = g_{ij}h_{ij}). 3.3 Let P be a locally free R-module, obtained by patching free modules of rank n by g_{ij} ∈ GL_{n}(R[\frac{1}{s_{i}s_{j}}]). Show that det(P) is the line bundle obtained by patching free modules of rank 1 by the units det(g_{ij}) ∈ (R[\frac{1}{s_{i}s_{j}}])^{×}.

3.4 Let P and Q be f.g. projective modules of constant ranks m and n respectively. Show that there is a natural isomorphism (det P)^{⊗n} ⊗ (det Q)^{⊗m} → det(P ⊗ Q).

Hint: Send (p_{11}∧⋯⊗⋯∧p_{mn})⊗(q_{11}∧⋯⊗⋯∧q_{mn}) to (p_{11}∧q_{11})∧⋯∧(p_{mn}∧q_{mn}). Then show that this map is locally an isomorphism. 3.5If an ideal I ⊆ R is a projective R-module and J ⊆ R is any other ideal, show that I ⊗_{R} J is isomorphic to the ideal IJ of R. 3.6 Excision for Pic. If I is a commutative ring without unit, let Pic(I) denote the kernel of the canonical map Pic(ℤ ⊕ I) → Pic(ℤ). Write I^{×} for the group GL_{1}(I) of Ex. 1.10. Show that if I is an ideal of R then there is an exact sequence:

1 → I^{×} → R^{×} → (R/I)^{×} →^{∂} Pic(I) → Pic(R) → Pic(R/I).

3.7 (Roberts-Singh) This exercise generalizes Proposition 3.5. Let R ⊆ S be an inclusion of commutative rings. An R-submodule I of S is called an invertible R-ideal of S if IJ = R for some other R-submodule J of S.

(i) If I ⊆ S is an invertible R-ideal of S, show that I is finitely generated over R, and that IS = S.

(ii) Show that the invertible R-ideals of S form an abelian group Pic(R, S) under

multiplication. (iii) Show that every invertible R-ideal of S is a line bundle over R. Hint: use

Ex. 3.5 to compute its rank. Conversely, if I is a line bundle over R contained in S and IS = S, then I is an R-ideal.

(iv) Show that there is a natural exact sequence:

1 → R^{×} → S^{×} ⟶^{div} Pic(R, S) → Pic(R) → Pic(S).

3.8 Relative Class groups. Suppose that R is a Krull domain and that R_{S} = S^{−1}R for some multiplicatively closed set S in R. Let D(R, R_{S}) denote the free abelian group on the height 1 primes 𝔭 of R such that 𝔭 ∩ S ≠ ϕ. Since D(R_{S}) is free on the remaining height 1 primes of R, D(R) = D(R, R_{S}) ⊕ D(R_{S}).

(a) Show that the group Pic(R, R_{S}) of Ex. 3.7 is a subgroup of D(R, R_{S}), and that there is an exact sequence compatible with Ex. 3.7

1 → R^{×} → R_{S}^{×} → D(R, R_{S}) → Cl(R) → Cl(R_{S}) → 0.

(b) Suppose that sR is a prime ideal of R. Prove that (R[\frac{1}{s}])^{×} ≅ R^{×} × ℤ^{n} and that Cl(R) ≅ Cl(R[\frac{1}{s}]).

(c) Suppose that every height 1 prime 𝔭 of R with 𝔭∩S ≠ ϕ is an invertible ideal. Show that Pic(R, R_{S}) = D(R, R_{S}) and that Pic(R) → Pic(R_{S}) is onto. (This always happens if R is a regular ring, or if the local rings R_{M} are unique factorization domains for every maximal ideal M of R with M ∩ S ≠ ∅.)

3.9 Suppose that we are given a Milnor square with R ⊆ S. If s̄ ∈ (S/I)^{×} is the image of a nonzerodivisor s ∈ S, show that ∂(s̄) ∈ Pic(R) is the class of the ideal (sS) ∩ R. 3.10 Let R be a 1-dimensional noetherian ring with finite normalization S, and let I be the conductor ideal from S to R. Show that for every maximal ideal 𝔭 of R, 𝔭 is a line bundle ⟺ I ⊈ 𝔭. Using Ex. 3.9, show that these 𝔭 generate Pic(R). 3.11 If R is seminormal, show that every localization S^{−1}R is seminormal.

3.12 Seminormality is a local property. Show that the following are equivalent:

(a) R is seminormal;

(b) R_{𝔪} is seminormal for every maximal ideal 𝔪 of R;

(c) R_{𝔭} is seminormal for every prime ideal 𝔭 of R. 3.13 If R is a pullback of a diagram of seminormal rings, show that R is seminormal. This shows that the node (3.10.2) is seminormal. 3.14 Normal rings. A ring R is called normal if each local ring R_{𝔭} is an integrally closed domain. If R and R′ are normal rings, so is the product R × R′. Show that normal domains are normal rings, and that every reduced 0-dimensional ring is normal. Then show that every normal ring is seminormal. 3.15 Seminormalization. Show that every reduced commutative ring R has an extension R ⊆ ^{+}R with ^{+}R seminormal, satisfying the following universal property: if S is seminormal, then every ring map R → S has a unique extension ^{+}R → S. The extension ^{+}R is unique up to isomorphism, and is called the seminormalization of R. Hint: First show that it suffices to construct the seminormalization of a noetherian ring R whose normalization S is finite. In that case, construct the seminormalization as a subring of S, using the observation that if x^{3} = y^{2} for x, y ∈ R, there is an s ∈ S with s^{2} = x, s^{3} = y. 3.16 An extension R ⊂ R′ is called subintegral if Spec(R′) → Spec(R) is a bijection, and the residue fields k(𝔭) = R_{𝔭}/𝔭R_{𝔭} and R′_{𝔭}/𝔭R′_{𝔭} are isomorphic. Show that the seminormalization R ⊂ ^{+}R of the previous exercise is a subintegral extension. 3.17 Let R be a commutative ring with nilradical 𝔑.

(a) Show that the subgroup 1 + 𝔑[t, t^{−1}] of R[t, t^{−1}]^{×} is the product of the three groups 1 + 𝔑, N_{t}U(R) = 1 + t𝔑[t], and N_{t^{−1}}U(R) = 1 + t^{−1}𝔑[t^{−1}].

(b) Show that there is a homomorphism t: [Spec(R), ℤ] → R[t, t^{−1}]^{×} sending f to the unit t^{f} of R[t, t^{−1}] which is t^{n} on the factor R_{i} of R where f = n. Here R_{i} is given by 2.2.4 and Ex. 2.4.

(c) Show that there is a natural decomposition

R[t, t^{−1}]^{×} ≅ R^{×} × N_{t}U(R) × N_{t^{−1}}U(R) × [Spec(R), ℤ],

or equivalently, that there is a split exact sequence:

1 → R^{×} → R[t]^{×} × R[t^{−1}]^{×} → R[t, t^{−1}]^{×} → [Spec(R), ℤ] → 0.

3.18 Show that the following sequence is exact:

1 → Pic(R) → Pic(R[t]) × Pic(R[t^{−1}]) → Pic(R[t, t^{−1}]).

Hint: If R is finitely generated, construct a diagram whose rows are Units-Pic sequences 3.10, and whose first column is the naturally split sequence of Ex. 3.17. 3.19 (NPic) Let NPic(R) denote the cokernel of the natural map Pic(R)→Pic(R[t]). Show that Pic(R[t]) ≅ Pic(R) × NPic(R), and that NPic(R) = 0 iff R_{red} is a seminormal ring. If R is an algebra over a field k, prove that:

(a) If char(k) = p > 0 then NPic(R) is a p-group;

(b) If char(k) = 0 then NPic(R) is a uniquely divisible abelian group. To do this, first reduce to the case when R is finitely generated, and proceed by induction on dim(R) using conductor squares.

§4. Topological Vector Bundles and Chern Classes

Because so much of the theory of projective modules is based on analogy with the theory of topological vector bundles, it is instructive to review the main aspects of the structure of vector bundles. Details and further information may be found in [CC], [Atiyah] or [Huse]. We will work with vector spaces over ℝ or ℂ.

Let X be a topological space. A family of vector spaces over X is a topological space E, together with a continuous map η: E → X and a finite dimensional vector space structure (over ℝ or ℂ) on each fiber E_{x} = η^{−1}(x), x ∈ X. We require the vector space structure on E_{x} to be compatible with the topology on E. By a homomorphism from one family η: E → X to another family φ: F → X we mean a continuous map f: E → F with η = φf, such that each induced map f_{x}: E_{x} → F_{x} is a linear map of vector spaces. There is an evident category of families of vector spaces over X and their homomorphisms.

For example, if V is an n-dimensional vector space, the projection from T^{n} = X × V to X forms a “constant” family of vector spaces. We call such a family, and any family isomorphic to it, a trivial vector bundle over X.

If Y ⊆ X, we write E|Y for the restriction η^{−1}(Y) of E to Y; the restriction η|Y: E|Y → Y of η makes E|Y into a family of vector spaces over Y. More generally, if f: Y → X is any continuous map then we can construct an induced family f^{∗}(ξ): f^{∗}E → Y as follows. The space f^{∗}E is the subspace of Y × E consisting of all pairs (y, e) such that f(y) = ξ(e), and f^{∗}E → Y is the restriction of the projection map. Since the fiber of f^{∗}E at y ∈ Y is E_{f(y)}, f^{∗}E is a family of vector spaces over Y.

A vector bundle over X is a family of vector spaces η: E → X such that every point x ∈ X has a neighborhood U such that η|U: E|U → U is trivial. A vector bundle is also called a locally trivial family of vector spaces.

The most historically important example of a vector bundle is the tangent bundle T X → X of a smooth manifold X. Another famous example is the Möbius bundle E over S^{1}; E is the open Möbius strip and E_{x} ≅ ℝ for each x ∈ S^{1}.

Suppose that f: X → Y is continuous. If E → X is a vector bundle, then the induced family f^{∗}E → Y is a vector bundle on Y. To see this, note that if E is trivial over a neighborhood U of f(y) then f^{∗}E is trivial over f^{−1}(U).

The symbol VB(X) denotes the category of vector bundles and homomorphisms. If emphasis on the field is needed, we write VB_{ℝ}(X) or VB_{ℂ}(X). The induced bundle construction gives rise to an additive functor f^{∗} from VB(X) to VB(Y).

The Whitney sum E ⊕ F of two vector bundles η: E → X and φ: F → X is the family of all the vector spaces E_{x} ⊕ F_{x}, topologized as a subspace of E × F. Since E and F are locally trivial, so is E ⊕ F; hence E ⊕ F is a vector bundle. By inspection, the Whitney sum is the product in the category VB(X). Since there is a natural notion of the sum f + g of two homomorphisms f, g: E → F, this makes VB(X) into an additive category with Whitney sum the direct sum operation.

A sub-bundle of a vector bundle η: E → X is a subspace F of E which is a vector bundle under the induced structure. That is, each fiber F_{x} is a vector subspace of E_{x} and the family F → X is locally trivial. The quotient bundle E/F is the union of all the vector spaces E_{x}/F_{x}, given the quotient topology. Since F is locally a Whitney direct summand in E, we see that E/F is locally trivial, hence a vector bundle. This gives a “short exact sequence” of vector bundles in VB(X):

0 → F → E → E/F → 0.

A vector bundle E → X is said to be of finite type if there is a finite covering U_{1}, …, U_{n} of X such that each each E|U_{i} is a trivial bundle. Every bundle over a compact space X must be of finite type; the same is true if X is a finite-dimensional connected CW complex [Huse, §3.5], or more generally if there is an integer n such that every open cover of X has a refinement V such that no point of X is contained in more that n elements of V. We will see in Exercise 4.15 that the canonical line bundle on infinite dimensional projective space ℙ^{∞} is an example of a vector bundle which is not of finite type.

Riemannian Metrics. Let E → X be a real vector bundle. A Riemannian metric on E is a family of inner products β_{x}: E_{x} × E_{x} → ℝ, x ∈ X, which varies continuously with x (in the sense that β is a continuous function on the Whitney sum E ⊕E). The notion of Hermitian metric on a complex vector bundle is defined similarly. A fundamental result [Huse, 3.5.5 and 3.9.5] states that every vector bundle over a paracompact space X has a Riemannian (or Hermitian) metric.

Dimension of vector bundles

If E is a vector bundle over X then dim(E_{x}) is a locally constant function on X with values in ℕ = {0, 1, ...}. Hence dim(E) is a continuous function from X to the discrete topological space ℕ; it is the analogue of the rank of a projective module. An n-dimensional vector bundle is a bundle E such that dim(E) = n is constant; 1-dimensional vector bundles are frequently called line bundles. The Möbius bundle is an example of a nontrivial line bundle.

A vector bundle E is called componentwise trivial if we can write X as a disjoint union of components X_{i} in such a way that each E|X_{i} is trivial. Isomorphism classes of componentwise trivial bundles are in 1-1 correspondence with the set [X, ℕ] of all continuous maps from X to ℕ. To see this, note that any continuous map f: X → ℕ induces a decomposition of X into components X_{i} = f^{−1}(i). Given such an f, let T^{f} denote the disjoint union

T^{f} = ∐_{i∈ℕ} X_{i} × F^{i}, F = ℝ or ℂ.

The projection T^{f} → ⨿X_{i} = X makes T^{f} into a componentwise trivial vector bundle with dim(T^{f}) = f. Conversely, if E is componentwise trivial, then E ≅ T^{dim(E)}. Note that T^{f} ⊕ T^{g} ≅ T^{f+g}. Thus if f is bounded then by choosing g = n − f we can make T^{f} into a summand of the trivial bundle T^{n}.

The following result, which we cite from [Huse, 3.5.8 and 3.9.6], illustrates some of the similarities between VB(X) and the category of f.g. projective modules. It is proven using the Riemannian (or Hermitian) metric on E: F_{x}^{⊥} is the subspace of E_{x} perpendicular to F_{x}.

Subbundle Theorem 4.1. Let E → X be a vector bundle on a paracompact topological space X. Then:

(1) If F is a sub-bundle of E, there is a sub-bundle F^{⊥} such that E ≅ F ⊕ F^{⊥}.

(2) E has finite type if and only if E is isomorphic to a sub-bundle of a trivial bundle. That is, if and only if there is another bundle F such that E ⊕ F is trivial.

Corollary 4.1.1. Suppose that X is compact, or that X is a finite-dimensional CW complex with finitely many components. Then every vector bundle over X is a Whitney direct summand of a trivial bundle.

Example 4.1.2. If X is a smooth d-dimensional manifold, its tangent bundle T X → X is a d-dimensional real vector bundle. Embedding X in ℝ^{n} allows us to form the normal bundle NX → X; N_{x}X is the orthogonal complement of T_{x}X in ℝ^{n}. Clearly T X ⊕ NX is the trivial n-dimensional vector bundle X × ℝ^{n} → X over X.

Example 4.1.3. Consider the canonical line bundle E_{1} on projective n-space; a point x of ℙ^{n} corresponds to a line L_{x} in n + 1-space, and the fiber of E_{1} at x is just L_{x}. In fact, E_{1} is a subbundle of the trivial bundle T^{n+1}. Letting F_{x} be the n-dimensional hyperplane perpendicular to L_{x}, the family of vector spaces F forms a vector bundle such that E_{1} ⊕ F = T^{n+1}.

Example 4.1.4. (Global sections) A global section of a vector bundle η: E → X is a continuous map s: X → E such that ηs = 1_{X}. It is nowhere zero if s(x) ≠ 0 for all x ∈ X. Every global section s determines a map from the trivial line bundle T^{1} to E; if s is nowhere zero then the image is a line subbundle of E. If X is paracompact the Subbundle Theorem determines a splitting E ≅ F ⊕ T^{1}.

Patching vector bundles

4.2 One technique for creating vector bundles uses transition functions. The idea is to patch together a collection of vector bundles which are defined on subspaces of X. A related technique is the clutching construction discussed in 4.7 below.

Let η: E → X be an n-dimensional vector bundle on X over the field F (F is ℝ or ℂ). Since E is locally trivial, we can find an open covering {U_{i}} of X, and isomorphisms h_{i} : U_{i} × F^{n} ≅ E|U_{i}. If U_{i} ∩ U_{j} ≠ ϕ, the isomorphism

h_{i}^{−1}h_{j} : (U_{i} ∩ U_{j}) × F^{n} ≅ η|U_{i} ∩ U_{j} ≅ (U_{i} ∩ U_{j}) × F^{n}

sends (x, v) ∈ (U_{i} ∩ U_{j}) × F^{n} to (x, g_{ij}(x)(v)) for some g_{ij}(x) ∈ GL_{n}(F).

Conversely, suppose we are given maps g_{ij}: U_{i} ∩ U_{j} → GL_{n}(F) such that g_{ii} = 1 and g_{ij}g_{jk} = g_{ik} on U_{i} ∩ U_{j} ∩ U_{k}. On the disjoint union of the U_{i} × F^{n}, form the equivalence relation ∼ which is generated by the relation that (x, v) ∈ U_{j} × F^{n} and (x, g_{ij}(x)(v)) ∈ U_{i} × F^{n} are equivalent for every x ∈ U_{i} ∩ U_{j}. Let E denote the quotient space (∐ U_{i} × F^{n})/ ∼. It is not hard to see that there is an induced map η: E → X making E into a vector bundle over X.

We call E the vector bundle obtained by patching via the transition functions g_{ij}; this patching construction is the geometric motivation for open patching of projective modules in 2.5.

Construction 4.2.1. (Tensor product). Let E and F be vector bundles over X. There is a vector bundle E ⊗ F over X whose fiber over x ∈ X is the vector space tensor product E_{x} ⊗ F_{x}, and dim(E ⊗ F) = dim(E) dim(F).

To construct E ⊗ F, we first suppose that E and F are trivial bundles, i.e., E = X ×V and F = X ×W for vector spaces V, W. In this case we let E ⊗F be the trivial bundle X × (V ⊗ W). In the general case, we proceed as follows. Restricting to a component of X on which dim(E) and dim(F) are constant, we may assume that E and F have constant ranks m and n respectively. Choose a covering {U_{i}} and transition maps g_{ij}, g′_{ij} defining E and F by patching. Identifying M_{m}(F)⊗M_{n}(F) with M_{mn}(F) gives a map GL_{m}(F) × GL_{n}(F) → GL_{mn}(F), and the elements g_{ij} ⊗ g′_{ij} give transition maps from U_{i} ∩ U_{j} to GL_{mn}(F). The last assertion comes from the classical vector space formula dim(E_{x} ⊗ F_{x}) = dim(E_{x}) dim(F_{x}).

Construction 4.2.2. (Determinant bundle). For every n-dimensional vector bundle E, there is an associated “determinant” line bundle det(E) = ∧^{n}E whose fibers are the 1-dimensional vector spaces ∧^{n}(E_{x}). In fact, det(E) is a line bundle obtained by patching, the transition functions for det(E) being the determinants det(g_{ij}) of the transition functions g_{ij} of E. More generally, if E is any vector bundle then this construction may be performed componentwise to form a line bundle det(E) = ∧^{dim(E)}E. As in §3, if L is a line bundle and E = L ⊕ T^{f}, then det(E) = L, so E uniquely determines L. Taking E trivial, this shows that line bundles cannot be stably trivial.

Orthogonal and unitary structure groups 4.2.3. We say that an ndimensional vector bundle E → X has structure group O_{n} or U_{n} if the transition functions g_{ij} map U_{i} ∩ U_{j} to the subgroup O_{n} of GL_{n}(ℝ) or the subgroup U_{n} of GL_{n}(ℂ). If X is paracompact, this can always be arranged, because then E has a (Riemannian or Hermitian) metric. Indeed, it is easy to continuously modify the isomorphisms h_{i}: U_{i} × F^{n} → E|U_{i} so that on each fiber the map F^{n} ≅ E_{x} is an isometry. But then the fiber isomorphisms g_{ij}(x) are isometries, and so belong to O_{n} or U_{n}. Using the same continuous modification trick, any vector bundle isomorphism between vector bundles with a metric gives rise to a metric-preserving isomorphism. If X is paracompact, this implies that VB_{n}(X) is also the set of equivalence classes of vector bundles with structure group O_{n} or U_{n}.

The following pair of results forms the historical motivation for the Bass-Serre Cancellation Theorem 2.3. Their proofs may be found in [Huse, 8.1].

Real Cancellation Theorem 4.3. Suppose X is a d-dimensional CW complex, and that η: E → X is an n-dimensional real vector bundle with n > d. Then (i) E ≅ E_{0} ⊕ T^{n−d} for some d-dimensional vector bundle E_{0} (ii) If F is another bundle and E ⊕ T^{k} ≅ F ⊕ T^{k}, then E ≅ F.

Corollary 4.3.1. Over a 1-dimensional CW complex, every real vector bundle E of rank ≥ 1 is isomorphic to L ⊕ T^{f}, where L = det(E) and f = dim(E) − 1.

Complex Cancellation Theorem 4.4. Suppose X is a d-dimensional CW complex, and that η: E → X is a complex vector bundle with dim(E) ≥ d/2.

(i) E ≅ E_{0} ⊕ T^{k} for some vector bundle E_{0} of dimension ≤ d/2 (ii) If F is another bundle and E ⊕ T^{k} ≅ F ⊕ T^{k}, then E ≅ F.

Corollary 4.4.1. Let X be a CW complex of dimension ≤ 3. Every complex vector bundle E of rank ≥ 1 is isomorphic to L ⊕ T^{f}, where L = det(E) and f = dim(E) − 1.

Vector bundles are somewhat more tractable than projective modules, as the following result shows. Its proof may be found in [Huse, 3.4.7].

Homotopy Invariance Theorem 4.5. If f, g: Y → X are homotopic maps and Y is paracompact, then f^{∗}E ≅ g^{∗}E for every vector bundle E over X.

Corollary 4.6. If X and Y are homotopy equivalent paracompact spaces, there is a 1-1 correspondence between isomorphism classes of vector bundles on X and Y.

Application 4.6.1. If Y is a contractible paracompact space then every vector bundle over Y is trivial.

Clutching Construction 4.7. Here is an analogue for vector bundles of Milnor Patching 2.7 for projective modules. Suppose that X is a paracompact space, expressed as the union of two closed subspaces X_{1} and X_{2}, with X_{1} ∩ X_{2} = A. Given vector bundles E_{i} → X_{i} and an isomorphism g: E_{1}|A → E_{2}|A, we form a vector bundle E = E_{1} ∪_{g} E_{2} over X as follows. As a topological space E is the quotient of the disjoint union (E_{1} ∐ E_{2}) by the equivalence relation indentifying e_{1} ∈ E_{1}|A with g(e_{1}) ∈ E_{2}|A. Clearly the natural projection η: E → X makes E a family of vector spaces, and E|X_{i} ≅ E_{i}. Moreover, E is locally trivial over X (see [Atiyah, p.21]; paracompactness is needed to extend g off of A). The isomorphism g: E_{1}|A ≅ E_{2}|A is called the clutching map of the construction. As with Milnor patching, every vector bundle over X arises by this clutching construction. A new feature, however, is homotopy invariance: if f, g are homotopic clutching isomorphisms E_{1}|A ≅ E_{2}|A, then E_{1} ∪_{f} E_{2} and E_{1} ∪_{g} E_{2} are isomorphic vector bundles over X.

Proposition 4.8. Let SX denote the suspension of a paracompact space X. A choice of basepoint for X yields a 1-1 correspondence between the set VB_{n}(SX) of isomorphism classes of n-dimensional (real, resp. complex) vector bundles over SX and the set of based homotopy classes of maps

[X, O_{n}]_{∗}, resp. [X, U_{n}]_{∗},

from X to the orthogonal group O_{n}, resp. to the unitary group U_{n}.

Sketch of Proof. SX is the union of two contractible cones C_{1} and C_{2} whose intersection is X. As every vector bundle on the cones C_{i} is trivial, every vector bundle on SX is obtained from an isomorphism of trivial bundles over X via the clutching construction. Such an isomorphism is given by a homotopy class of maps from X to GL_{n}, or equivalently to the deformation retract O_{n}, resp. U_{n} of GL_{n}. The indeterminacy in the resulting map from [X, GL_{n}] to classes of vector bundles is eliminated by insisting that the basepoint of X map to 1 ∈ GL_{n}.

Vector Bundles on Spheres

Proposition 4.8 allows us to use homotopy theory to determine the vector bundles on the sphere S^{d}, because S^{d} is the suspension of S^{d−1}. Hence n-dimensional (real, resp. complex) bundles on S^{d} are in 1-1 correspondence with elements of π_{d−1}(O_{n}) and π_{d−1}(U_{n}), respectively. For example, every line bundle over S^{d} is trivial if d ≥ 3, because the appropriate homotopy groups of O_{1} ≅ S^{0} and U_{1} ≅ S^{1} vanish. The classical calculation of the homotopy groups of O_{n} and U_{n} (see [Huse, 7.12]) yields the following facts: (4.9.1) On S^{1}, there are |π_{0}(O_{n})| = 2 real vector bundles of dimension n for all n ≥ 1. The nontrivial line bundle on S^{1} is the Möbius bundle. The Whitney sum of the Möbius bundle with trivial bundles yields all the other nontrivial bundles. Since |π_{0}(U_{n})| = 1 for all n, every complex vector bundle on S^{1} is trivial. (4.9.2) On S^{2}, the situation is more complicated. Since π_{1}(O_{1}) = 0 there are no nontrivial real line bundles on S^{2}. There are infinitely many 2-dimensional real vector bundles on S^{2} (indexed by the degree d of their clutching functions), because π_{1}(O_{2}) = ℤ. However, there is only one nontrivial n-dimensional real vector bundle for each n ≥ 3, because π_{1}(O_{n}) = ℤ/2. A real 2-dimensional bundle E is stably trivial (and E ⊕ T ≅ T^{3}) iff the degree d is even. The tangent bundle of S^{2} has degree d = 2.

There are infinitely many complex line bundles L_{d} on S^{2}, indexed by the degree d (in π_{1}(U_{1}) = ℤ) of their clutching function. The Complex Cancellation theorem (4.4) states that every other complex vector bundle on S^{2} is isomorphic to a Whitney sum L_{d} ⊕ T^{n}, and that all the L_{d} ⊕ T^{n} are distinct. (4.9.3) Every vector bundle on S^{3} is trivial. This is a consequence of the classical result that π_{2}(G) = 0 for every compact Lie group G, such as G = O_{n} or U_{n}. (4.9.4) As noted above, every line bundle on S^{4} is trivial. S^{4} carries infinitely many distinct n-dimensional vector bundles for n ≥ 5 over ℝ, and for n ≥ 2 over ℂ, because π_{3}(O_{n}) = ℤ for n ≥ 5 and π_{3}(U_{n}) = ℤ for n ≥ 2. In the intermediate range, we have π_{3}(O_{2}) = 0, π_{3}(O_{3}) = ℤ and π_{3}(O_{4}) = ℤ ⊕ ℤ. Every 5-dimensional real bundle comes from a unique 3-dimensional bundle but every 4-dimensional real bundle on S^{4} is stably isomorphic to infinitely many other distinct 4-dimensional vector bundles. (4.9.5) For d ≥ 3 there are no 2-dimensional real vector bundles on S^{d}, because the appropriate homotopy groups of O_{2} ≅ S^{1}×ℤ/2 vanish. This vanishing phenomenon doesn’t persist though; if d ≥ 5 the 2-dimensional complex bundles, as well as the 3dimensional real bundles on S^{d}, correspond to elements of π_{d−1}(O_{3}) ≅ π_{d−1}(U_{2}) ≅ π_{d−1}(S^{3}). This is a finite group which is rarely trivial.

Classifying Vector Bundles

One feature present in the theory of vector bundles, yet absent in the theory of projective modules, is the classification of vector bundles using Grassmannians.

If V is any finite-dimensional vector space, the set G_{n}(V) of all n-dimensional linear subspaces of V is a smooth manifold, called the Grassmann manifold of nplanes in V. If V ⊂ W, then G_{n}(V) is naturally a submanifold of G_{n}(W). The infinite Grassmannian G_{n} is the union of the G_{n}(V) as V ranges over all finitedimensional subspaces of a fixed infinite-dimensional vector space (ℝ^{∞} or ℂ^{∞}); thus G_{n} is an infinite-dimensional CW complex. For example, if n = 1 then G_{1} is either ℝℙ^{∞} or ℂℙ^{∞}, depending on whether the vector spaces are over ℝ or ℂ.

There is a canonical n-dimensional vector bundle E_{n}(V) over each G_{n}(V), whose fibre over each x ∈ G_{n}(V) is the linear subspace of V corresponding to x. To topologize this family of vector spaces, and see that it is a vector bundle, we define E_{n}(V) to be the sub-bundle of the trivial bundle G_{n}(V) × V → G_{n}(V) having the prescribed fibers. For n = 1 this is just the canonical line bundle on projective space described in Example 4.1.3.

The union (as V varies) of the E_{n}(V) yields an n-dimensional vector bundle E_{n} → G_{n}, called the n-dimensional classifying bundle because of the following theorem (see [Huse, 3.7.2]).

Classification Theorem 4.10. Let X be a paracompact space. Then the set VB_{n}(X) of isomorphism classes of n-dimensional vector bundles over X is in 1-1 correspondence with the set [X, G_{n}] of homotopy classes of maps from X into G_{n}:

VB_{n}(X) ≅ [X, G_{n}].

In more detail, every n-dimensional vector bundle η: E → X is isomorphic to f^{∗}(E_{n}) for some map f: X → G_{n}, and E determines f up to homotopy.

Remark 4.10.1. (Classifying Spaces) The Classification Theorem 4.10 states that the contravariant functor VB_{n} is “representable” by the infinite Grassmannian G_{n}. Because X is paracompact we may assume (by 4.2.3) that all vector bundles have structure group O_{n}, resp. U_{n}. For this reason, the infinite Grassmannian G_{n} is called the classifying space of O_{n}, resp. U_{n} (depending on the choice of ℝ or ℂ). It is the custom to write BO_{n}, resp. BU_{n}, for the Grassmannian G_{n} (or any space homotopy equivalent to it) over ℝ, resp. over ℂ.

In fact, there are homotopy equivalences Ω(BO_{n}) ≃ O_{n} and Ω(BU_{n}) ≃ U_{n}. We can deduce this from 4.8 and 4.10: for any paracompact space X we have [X, O_{n}]_{∗} ≅ VB_{n}(SX) ≅ [SX, BO_{n}] ≅ [X, Ω(BO_{n})]_{∗}, and [X, U_{n}]_{∗} ≅ [X, Ω(BU_{n})]_{∗} similarly. Taking X to be O_{n} and Ω(BO_{n}), resp. U_{n} and Ω(BU_{n}), yields the homotopy equivalences.

It is well-known that there are canonical isomorphisms [X, ℝℙ^{∞}] ≅ H^{1}(X; ℤ/2) and [X, ℂℙ^{∞}] ≅ H^{2}(X; ℤ) respectively. Therefore the case n = 1 may be reformulated as follows.

Classification Theorem for Line Bundles 4.11. If X is paracompact, there are natural isomorphisms:

w_{1} : VB_{1,ℝ}(X) = {real line bundles on X} ≅ H^{1}(X; ℤ/2)
c_{1} : VB_{1,ℂ}(X) = {complex line bundles on X} ≅ H^{2}(X; ℤ).

Remark 4.11.1. Since H^{1}(X) and H^{2}(X) are abelian groups, it follows that the set VB_{1}(X) of isomorphism classes of line bundles is an abelian group. We can understand this group structure in a more elementary way, as follows. The tensor product E ⊗ F of line bundles is again a line bundle by 4.2.1, and ⊗ is the product in the group VB_{1}(X). The inverse of E in this group is the dual bundle Ě of Ex. 4.3, because Ě ⊗ E is a trivial line bundle (see Ex. 4.4).

Riemann Surfaces 4.11.2. Here is a complete classification of complex vector bundles on a Riemann surface X. Recall that a Riemann surface is a compact 2-dimensional oriented manifold; the orientation gives a canonical isomorphism H^{2}(X; ℤ) = ℤ. If L is a complex line bundle, the degree of L is that integer d such that c_{1}(L) = d. By Theorem 4.11, there is a unique complex line bundle O(d) of each degree on X. By Corollary 4.4.1, every complex vector bundle of rank r on X is isomorphic to O(d) ⊕ T^{r−1} for some d. Therefore complex vector bundles on a Riemann surface are completely classified by their rank and degree.

For example, the tangent bundle T_{X} of a Riemann surface X has the structure of a complex line bundle, because every Riemann surface has the structure of a 1-dimensional complex manifold. The Riemann-Roch Theorem states that T_{X} has degree 2−2g, where g is the genus of X. (Riemann surfaces are completely classified by their genus g ≥ 0, a Riemann surface of genus g being a surface with g “handles.”)

In contrast, there are 2^{2g} distinct real line bundles on X, because H^{1}(X; ℤ/2) ≅ (ℤ/2)^{2g}. The Real Cancellation Theorem 4.3 shows that every real vector bundle is the sum of a trivial bundle and a bundle of dimension ≤ 2, but there are infinitely many 2-dimensional bundles over X. For example, the complex line bundles O(d) all give distinct 2-dimensional real vector bundles on X; they are distinguished by an invariant called the Euler class (see [CC]).

Characteristic Classes

By Theorem 4.11, the determinant line bundle det(E) of a vector bundle E yields a cohomology class: if E is a real vector bundle, it is the first Stiefel-Whitney class w_{1}(E) in H^{1}(X; ℤ/2); if E is a complex vector bundle, it is the first Chern class c_{1}(E) in H^{2}(X, ℤ). These classes fit into a more general theory of characteristic classes, which are constructed and described in the book [CC]. Here is an axiomatic description of these classes.

Axioms for Stiefel-Whitney classes 4.12. The Stiefel–Whitney classes of a real vector bundle E over X are elements w_{i}(E) ∈ H^{i}(X; ℤ/2), which satisfy the following axioms. By convention w_{0}(E) = 1. (SW1) (Dimension) If i > dim(E) then w_{i}(E) = 0. (SW2) (Naturality) If f: Y → X is continuous then f^{∗}: H^{i}(X; ℤ/2) → H^{i}(Y; ℤ/2) sends w_{i}(E) to w_{i}(f^{∗}E). If E and E′ are isomorphic bundles then w_{i}(E) = w_{i}(E′). (SW3) (Whitney sum formula) If E and F are bundles, then in the graded cohomology ring H^{∗}(X; ℤ/2) we have:

w_{n}(E ⊕ F) = ∑ w_{i}(E)w_{n−i}(F) = w_{n}(E) + w_{n−1}(E)w_{1}(F) + ⋯ + w_{n}(F).

(SW4) (Normalization) For the canonical line bundle E_{1} over ℝℙ^{∞}, w_{1}(E_{1}) is the unique nonzero element of H^{1}(ℝℙ^{∞}; ℤ/2) ≅ ℤ/2.

The axioms (SW2) and (SW4), together with the Classification Theorem 4.10, show that w_{1} classifies real line bundles in the sense that it gives the isomorphism VB_{1}(X) ≅ H^{1}(X; ℤ/2) of Theorem 4.11. The fact that w_{1}(E) = w_{1}(det E) is a consequence of the “Splitting Principle” for vector bundles, and is left to the exercises.

Since trivial bundles are induced from the map X → {∗}, it follows from (SW1) and (SW2) that w_{i}(T^{n}) = 0 for every trivial bundle T^{n} (and i ≠ 0). The same is true for componentwise trivial bundles; see Ex. 4.2. From (SW3) it follows that w_{i}(E ⊕ T^{n}) = w_{i}(E) for every bundle E and every trivial bundle T^{n}.

The total Stiefel-Whitney class w(E) of E is defined to be the formal sum

w(E) = 1 + w_{1}(E) + ⋯ + w_{i}(E) + ...

in the complete cohomology ring Ĥ^{∗}(X; ℤ/2) = ∏_{i} H^{i}(X; ℤ/2), which consists of all formal infinite series a_{0}+a_{1}+… with a_{i} ∈ H^{i}(X; ℤ/2). With this formalism, the Whitney sum formula becomes a product formula: w(E ⊕ F) = w(E)w(F). Now the collection U of all formal sums 1 + a_{1} + ⋯ in Ĥ^{∗}(X; ℤ/2) forms an abelian group under multiplication (the group of units of Ĥ^{∗}(X; ℤ/2) if X is connected). Therefore if E ⊕F is trivial we can compute w(F) via the formula w(F) = w(E)^{−1}.

For example, consider the canonical line bundle E_{1}(ℝ^{n}) over ℝℙ^{n}. By axiom (SW4) we have w(E_{1}) = 1+x in the ring H^{∗}(ℝℙ^{n}; ℤ/2) ≅ 𝔽_{2}[x]/(x^{n+1}). We saw in Example 4.1.3 that there is an n-dimensional vector bundle F with F ⊕E_{1} = T^{n+1}. Using the Whitney Sum formula (SW3), we compute that w(F) = 1 + x + ⋯ + x^{n}. Thus w_{i}(F) = x^{i} for i ≤ n and w_{i}(F) = 0 for i > n.

Stiefel-Whitney classes were named for E. Stiefel and H. Whitney, who discovered the w_{i} independently in 1935, and used them to study the tangent bundle of a smooth manifold.

Axioms for Chern classes 4.13. If E is a complex vector bundle over X, the Chern classes of E are certain elements c_{i}(E) ∈ H^{2i}(X; ℤ), with c_{0}(E) = 1. They satisfy the following axioms. Note that the natural inclusion of S^{2} ≅ ℂℙ^{1} in ℂℙ^{∞} induces a canonical isomorphism H^{2}(ℂℙ^{∞}; ℤ) ≅ H^{2}(S^{2}; ℤ) ≅ ℤ. (C1) (Dimension) If i > dim(E) then c_{i}(E) = 0 (C2) (Naturality) If f: Y → X is continuous then f^{∗}: H^{2i}(X; ℤ) → H^{2i}(Y; ℤ) sends c_{i}(E) to c_{i}(f^{∗}E). If E ≅ E′ then c_{i}(E) = c_{i}(E′). (C3) (Whitney sum formula) If E and F are bundles then

c_{n}(E ⊕ F) = ∑ c_{i}(E)c_{n−i}(F) = c_{n}(E) + c_{n−1}(E)c_{1}(F) + ⋯ + c_{n}(F).

(C4) (Normalization) For the canonical line bundle E_{1} over ℂℙ^{∞}, c_{1}(E_{1}) is the canonical generator x of H^{2}(ℂℙ^{∞}; ℤ) ≅ ℤ.

Axioms (C2) and (C4) and the Classification Theorem 4.10 imply that the first Chern class c_{1} classifies complex line bundles; it gives the isomorphism VB_{1}(X) ≅ H^{2}(X; ℤ) of Theorem 4.11. The identity c_{1}(E) = c_{1}(det E) is left to the exercises.

The total Chern class c(E) of E is defined to be the formal sum

c(E) = 1 + c_{1}(E) + ⋯ + c_{i}(E) + · · · .

in the complete cohomology ring Ĥ^{∗}(X; ℤ) = ∏_{i} H^{i}(X; ℤ). With this formalism, the Whitney sum formula becomes c(E ⊕ F) = c(E)c(F). As with Stiefel-Whitney classes, axioms (C1) and (C2) imply that for a trivial bundle T^{n} we have c_{i}(T^{n}) = 0 (i ≠ 0), and axiom (C3) implies that for all E

c_{i}(E ⊕ T^{n}) = c_{i}(E).

For example, consider the canonical line bundle E_{1}(ℂ^{n}) over ℂℙ^{n}. By axiom (C4), c(E_{1}) = 1 + x in the truncated polynomial ring H^{∗}(ℂℙ^{n}; ℤ) ≅ ℤ[x]/(x^{n+1}). We saw in Example 4.1.3 that there is a canonical n-dimensional vector bundle F with F ⊕ E_{1} = T^{n+1}. Using the Whitney Sum Formula (C3), we compute that c(F) = ∑(−1)^{i}x^{i}. Thus c_{i}(F) = (−1)^{i}x^{i} for all i ≤ n.

Chern classes are named for S.-S. Chern, who discovered them in 1946 while studying L. Pontrjagin’s 1942 construction of cohomology classes p_{i}(E) ∈ H^{4i}(X; ℤ) associated to a real vector bundle E. In fact, p_{i}(E) is (−1)^{i}c_{2i}(E ⊗ℂ), where E ⊗ℂ is the complexification of E (see Ex. 4.5). However, the Whitney sum formula for Pontrjagin classes only holds up to elements of order 2 in H^{4n}(X; ℤ); see Ex. 4.13.

EXERCISES

4.1 Let η: E → X and φ: F → X be two vector bundles, and form the induced bundle η^{∗}F over E. Show that the Whitney sum E ⊕ F → X is η^{∗}F, considered as a bundle over X by the map η^{∗}F → E → X. 4.2 Show that all of the uncountably many vector bundles on the discrete space X = ℕ are componentwise trivial. Let T^{ℕ} → ℕ be the bundle with dim(T_{n}^{ℕ}) = n for all n. Show that every componentwise trivial vector bundle T^{f} → Y over every space Y is isomorphic to f^{∗}T^{ℕ}. Use this to show that the Stiefel-Whitney and Chern classes vanish for componentwise trivial vector bundles. 4.3 If E and F are vector bundles over X, show that there are vector bundles Hom(E, F), Ě and ∧^{k}E over X whose fibers are, respectively: Hom(E_{x}, F_{x}), the dual space (Ě_{x}) and the exterior product ∧^{k}(E_{x}). Then show that there are natural isomorphisms (E ⊕ F)ˇ ≅ Ě ⊕ F̌, Ě ⊗ F ≅ Hom(E, F), ∧^{1}E ≅ E and

∧^{k}(E ⊕ F) ≅ ∧^{k}E ⊕ (∧^{k−1}E ⊗ F) ⊕ ⋯ ⊕ (∧^{i}E ⊗ ∧^{k−i}F) ⊕ ⋯ ⊕ ∧^{k}F.

4.4 Show that the global sections of the bundle Hom(E, F) of Ex. 4.3 are in 1-1 correspondence with vector bundle maps E → F. (Cf. 4.1.4.) If E is a line bundle, show that the vector bundle Ě ⊗ E ≅ Hom(E, E) is trivial. 4.5 Complexification. Let E → X be a real vector bundle. Show that there is a complex vector bundle E_{ℂ} → X with fibers E_{x} ⊗_{ℝ} ℂ and that there is a natural isomorphism (E ⊕ F)_{ℂ} ≅ (E_{ℂ}) ⊕ (F_{ℂ}). Then show that E_{ℂ} → X, considered as a real vector bundle, is isomorphic to the Whitney sum E ⊕ E. 4.6 Complex conjugate bundle. If F → X is a complex vector bundle, given by transition functions g_{ij}, let F̄ denote the complex vector bundle obtained by using the complex conjugates ḡ_{ij} for transition functions; F̄ is called the complex conjugate bundle of F. Show that F and F̄ are isomorphic as real vector bundles, and that the complexification F_{ℂ} → X of Ex. 4.5 is isomorphic to the Whitney sum F ⊕ F̄. If F = E_{ℂ} for some real bundle E, show that F ≅ F̄. Finally, show that for every complex line bundle L on X we have L̄ ≅ Ľ. 4.7 Use the formula L̄ ≅ Ľ of Ex. 4.6 to show that c_{1}(Ē) = −c_{1}(E) in H^{2}(X; ℤ) for every complex vector bundle E. 4.8 Global sections. If η: E → X is a vector bundle, let Γ(E) denote the set of all global sections of E (see 4.1.4). Show that Γ(E) is a module over the ring C^{0}(X) of continuous functions on X (taking values in ℝ or ℂ). If E is an n-dimensional trivial bundle, show that Γ(E) is a free C^{0}(X)-module of rank n.

Conclude that if X is paracompact then Γ(E) is a locally free C^{0}(X)-module in the sense of 2.4, and that Γ(E) is a finitely generated projective module if X is compact or if E is of finite type. This is the easy half of Swan’s theorem; the rest is given in the next exercise. 4.9 Swan’s Theorem. Let X be a compact space, and write R for C^{0}(X). Show that the functor Γ of the previous exercise is a functor from VB(X) to the category P(R) of f.g. projective modules, and that the homomorphisms

Γ: Hom_{VB(X)}(E, F) → Hom_{P(R)}(Γ(E), Γ(F))  (∗)

are isomorphisms. This proves Swan’s Theorem, that Γ is an equivalence of categories VB(X) ≈ P(C^{0}(X)). Hint: First show that (∗) holds when E and F are trivial bundles, and then use Corollary 4.1.1. 4.10 Projective and Flag bundles. If E → X is a vector bundle, consider the subspace E_{0} = E − X of E, where X lies in E as the zero section. The units ℝ^{×} (or ℂ^{×}) act fiberwise on E_{0}, and the quotient space ℙ(E) obtained by dividing out by this action is called the projective bundle associated to E. If p: ℙ(E) → X is the projection, the fibers p^{−1}(x) are projective spaces.

(a) Show that there is a line sub-bundle L of p^{∗}E over ℙ(E). Use the Subbundle Theorem to conclude that p^{∗}E ≅ E′ ⊕ L.

Now suppose that E → X is an n-dimensional vector bundle, and let 𝔽(E) be the flag space f: 𝔽(E) → X obtained by iterating the construction

⋯ → ℙ(E′′) → ℙ(E′) → ℙ(E) → X.

(b) Show that the bundle f^{∗}E → 𝔽(E) is a direct sum L_{1} ⊕ ⋯ ⊕ L_{n} of line bundles.

4.11 If E is a direct sum L_{1} ⊕ ⋯ ⊕ L_{n} of line bundles, show that det(E) ≅ L_{1} ⊗ ⋯ ⊗ L_{n}. Then use the Whitney Sum formula to show that w_{1}(E) = w_{1}(det(E)), resp. c_{1}(E) = c_{1}(det(E)). Prove that every w_{i}(E), resp. c_{i}(E) is the i^{th} elementary symmetric function of the n cohomology classes {w_{1}(L_{i})}, resp. {c_{1}(L_{i})}. 4.12 Splitting Principle. Write H^{i}(X) for H^{i}(X; ℤ/2) or H^{2i}(X; ℤ), depending on whether our base field is ℝ or ℂ, and let p: 𝔽(E) → X be the flag bundle of a vector bundle E over X (see Ex. 4.10). Prove that p^{∗}: H^{i}(X) → H^{i}(𝔽(E)) is an injection. Then use Ex. 4.11 to show that the characteristic classes w_{i}(E) or c_{i}(E) in H^{i}(X) may be calculated inside H^{i}(𝔽(E)). Hint: for a trivial bundle this follows easily from the Künneth formula. 4.13 Pontrjagin classes. In this exercise we assume the results of Ex. 4.6 on the conjugate bundle F̄ of a complex bundle F. Use the Splitting Principle to show that c_{i}(F̄) = (−1)^{i}c_{i}(F). Then prove the following:

(i) The Pontrjagin classes p_{n}(F) of F (considered as a real bundle) are

p_{n}(F) = c_{n}(F)^{2} + 2 ∑_{i=1}^{n−1}(−1)^{i}c_{n−i}(F)c_{n+i}(F) + (−1)^{n}2c_{2n}(F).

(ii) If F = E ⊗ ℂ for some real bundle E, the odd Chern classes c_{1}(F), c_{3}(F), …

all have order 2 in H^{∗}(X; ℤ). (iii) The Whitney sum formula for Pontrjagin classes holds modulo 2:

p_{n}(E ⊕ E′) − ∑ p_{i}(E)p_{n−i}(E′) has order 2 in H^{4n}(X; ℤ).

4.14 Disk with double origin. This exercise shows that the classification theorems 4.10 and 4.11 fail for locally compact spaces which aren’t Hausdorff. Let D denote the closed unit disk in ℝ^{2}. The disk with double origin is the non-Hausdorff space X obtained from the disjoint union of two copies of D by identifying together the common subsets D − {0}. Show that [X, BO_{n}] = [X, BU_{n}] = 0 for all n, but that VB_{2,ℝ}(X) ≅ ℤ, VB_{n,ℝ}(X) ≅ ℤ/2 for n ≥ 3 and VB_{n,ℂ}(X) ≅ ℤ ≅ H^{2}(X; ℤ) for all n ≥ 1. 4.15 Show that the canonical line bundles E_{1} over ℝℙ^{∞} and ℂℙ^{∞} do not have finite type. Hint: Use characteristic classes and the Subbundle Theorem, or II.3.7.2. 4.16 Consider the suspension SX of a paracompact space X. Show that every vector bundle E over SX has finite type. Hint: If dim(E) = n, use 4.8 and Ex. 1.11 to construct a bundle E′ such that E ⊕ E′ ≅ T^{2n}.

§5. Algebraic Vector Bundles

Modern Algebraic Geometry studies sheaves of modules over schemes. This generalizes modules over commutative rings, and has many features in common with the topological vector bundles that we considered in the last section. In this section we discuss the main aspects of the structure of algebraic vector bundles.

We will assume the reader has some rudimentary knowledge of the language of schemes, in order to get to the main points quickly. Here is a glossary of the basic concepts; details for most things may be found in [Hart], but the ultimate source is [EGA].

A ringed space (X, O_{X}) is a topological space X equipped with a sheaf of rings O_{X}; it is a locally ringed space if each O_{X}(U) is a commutative ring, and if for every x ∈ X the stalk ring O_{X,x} = \underrightarrow{lim}_{x∈U} O_{X}(U) is a local ring. By definition, an affine

scheme is a locally ringed space isomorphic to (Spec(R), R̃) for some commutative ring R (where R̃ is the canonical structure sheaf), and a scheme is a ringed space (X, O_{X}) which can be covered by open sets U_{i} such that each (U_{i}, O_{X}|U_{i}) is an affine scheme.

An O_{X}-module is a sheaf F on X such that (i) for each open U ⊆ X the set F(U) is an O_{X}(U)-module, and (ii) if V ⊂ U then the restriction map F(U) → F(V) is compatible with the module structures. A morphism F → G of O_{X}-modules is a sheaf map such that each F(U) → G(U) is O_{X}(U)-linear. The category O_{X}-mod of all O_{X}-modules is an abelian category.

A global section of an O_{X}-module F is an element e_{i} of F(X). We say that F is generated by global sections if there is a set {e_{i}}_{i∈I} of global sections of F whose restrictions e_{i}|_{U} generate F(U) as an O_{X}(U)-module for every open U ⊆ X. We can reinterpret these definitions as follows. Giving a global section e of F is equivalent to giving a morphism O_{X} → F of O_{X}-modules, and to say that F is generated by the global sections {e_{i}} is equivalent to saying that the corresponding morphism ⊕_{i∈I} O_{X} → F is a surjection.

Free modules. We say that F is a free O_{X}-module if it is isomorphic to a direct sum of copies of O_{X}. A set {e_{i}} ⊂ F(X) is called a basis of F if the restrictions e_{i}|_{U} form a basis of each F(U), i.e., if the e_{i} provide an explicit isomorphism ⊕O_{X} ≅ F.

The rank of a free O_{X}-module F is not well-defined over all ringed spaces. For example, if X is a 1-point space then O_{X} is just a ring R and an O_{X}-module is just an R-module, so our remarks in §1 about the invariant basis property (IBP) apply. There is no difficulty in defining the rank of a free O_{X}-module when (X, O_{X}) is a scheme, or a locally ringed space, or even more generally when any of the rings O_{X}(U) satisfy the IBP. We shall avoid these difficulties by assuming henceforth that (X, O_{X}) is a locally ringed space.

We say that an O_{X}-module F is locally free if X can be covered by open sets U for which F|_{U} is a free O_{U}-module. The rank of a locally free module F is defined at each point x of X: rank_{x}(F) is the rank of the free O_{U}-module F|_{U}, where U is a neighborhood of x on which F|_{U} is free. Since the function x ↦ rank_{x}(F) is locally constant, rank(F) is a continuous function on X. In particular, if X is a connected space then every locally free module has constant rank.

Definition 5.1 (Vector Bundles). A vector bundle over a ringed space X is a locally free O_{X}-module whose rank is finite at every point. We will write VB(X) or VB(X, O_{X}) for the category of vector bundles on (X, O_{X}); the morphisms in VB(X) are just morphisms of O_{X}-modules. Since the direct sum of locally free modules is locally free, VB(X) is an additive category.

A line bundle L is a locally free module of constant rank 1. A line bundle is also called an invertible sheaf because as we shall see in 5.3 there is another sheaf L′ such that L ⊗ L′ = O_{X}.

These notions are the analogues for ringed spaces of f.g. projective modules and algebraic line bundles, as can be seen from the discussion in 2.4 and §3. However, the analogy breaks down if X is not locally ringed; in effect locally projective modules need not be locally free.

Example 5.1.1. (Topological spaces). Fix a topological space X. Then X_{top} = (X, O_{top}) is a locally ringed space, where O_{top} is the sheaf of (ℝ or ℂ-valued) continuous functions on X: O_{top}(U) = C^{0}(U) for all U ⊆ X. The following constructions give an equivalence between the category VB(X_{top}) of vector bundles over the ringed space X_{top} and the category VB(X) of (real or complex) topological vector bundles over X in the sense of §4. Thus our notation is consistent with the notation of §4.

If η: E → X is a topological vector bundle, then the sheaf E of continuous sections of E is defined by E(U) = {s: U → E : ηs = 1_{U}}. By Ex. 4.8 we know that E is a locally free O_{top}-module. Conversely, given a locally free O_{top}-module E, choose a cover {U_{i}} and bases for the free O_{top}-modules E|U_{i}; the base-change isomorphisms over the U_{i} ∩ U_{j} are elements g_{ij} of GL_{n}(C^{0}(U_{i} ∩ U_{j})). Interpreting the g_{ij} as maps U_{i} ∩ U_{j} → GL_{n}(ℂ), they become transition functions for a topological vector bundle E → X in the sense of 4.2.

Example 5.1.2 (Affine schemes). Suppose X = Spec(R). Every R-module M yields an O_{X}-module M̃, and R̃ = O_{X}. Hence every free O_{X}-module arises as M̃ for a free R-module M. The O_{X}-module F = P̃ associated to a f.g. projective R-module P is locally free by 2.4, and the two rank functions agree: rank(P) = rank(F). Conversely, if F is locally free O_{X}-module, it can be made trivial on a covering by open sets of the form U_{i} = D(s_{i}), i.e., there are free modules M_{i} such that F|_{U_{i}} = M̃_{i}. The isomorphisms between the restrictions of M̃_{i} and M̃_{j} to U_{i} ∩ U_{j} amount to open patching data defining a projective R-module P as in 2.5. In fact it is not hard to see that F ≅ P̃. Thus vector bundles on Spec(R) are in 1-1 correspondence with f.g. projective R-modules. And it is no accident that the notion of an algebraic line bundle over a ring R in §3 corresponds exactly to the notion of a line bundle over the ringed space (Spec(R), R̃).

More is true: the categories VB(X) and P(R) are equivalent when X = Spec(R). To see this, recall that an O_{X}-module is called quasicoherent if it is isomorphic to some M̃ ([Hart, II.5.4]). The above correspondence shows that every vector bundle is quasicoherent. It turns out that the category O_{X}-mod_{qcoh} of quasicoherent O_{X}- modules is equivalent to the category R-mod of all R-modules (see [Hart, II.5.5]). Since the subcategories VB(Spec R) and P(R) correspond, they are equivalent.

Definition (Coherent modules). Suppose that X is any scheme. We say that a sheaf of O_{X}-modules F is quasicoherent if X may be covered by affine opens U_{i} = Spec(R_{i}) such that each F|U_{i} is M̃_{i} for an R_{i}-module M_{i}. (If X is affine, this agrees with the definition of quasicoherent in Example 5.1.2 by [Hart, II.5.4].) We say that F is coherent if moreover each M_{i} is a finitely presented R_{i}-module.

If X is affine then F = M̃ is coherent iff M is a finitely presented R-module by [EGA, I(1.4.3)]. In particular, if R is noetherian then “coherent” is just a synonym for “finitely generated.” If X is a noetherian scheme, our definition of coherent module agrees with [Hart] and [EGA]. For general schemes, our definition is slightly stronger than in [Hart], and slightly weaker than in [EGA, 0_{I}(5.3.1)]; O_{X} is always coherent in our sense, but not in the sense of [EGA].

The equivalent conditions for locally free modules in 2.4 translate into:

Lemma 5.1.3. For every scheme X and O_{X}-module F, the following conditions are equivalent:

(1) F is a vector bundle (i.e., is locally free of finite rank);

(2) F is quasicoherent and the stalks F_{x} are free O_{X,x}-modules of finite rank;

(3) F is coherent and the stalks F_{x} are free O_{X,x}-modules;

(4) For every affine open U = Spec(R) in X, F|_{U} is a f.g. projective R-module.

Example 5.1.4. (Analytic spaces). Analytic spaces form another family of locally ringed spaces. To define them, one proceeds as follows. On the topological space ℂ^{n}, the subsheaf O_{an} of O_{top} consisting of analytic functions makes (ℂ^{n}, O_{an}) into a locally ringed space. A basic analytic set in an open subset U of ℂ^{n} is the zero locus V of a finite number of holomorphic functions, made into a locally ringed space (V, O_{V,an}) as follows. If I_{V} is the subsheaf of O_{U,an} consisting of functions vanishing on V, the quotient sheaf O_{V,an} = O_{U,an}/I_{V} is supported on V, and is a subsheaf of the sheaf O_{V,top}. By definition, an analytic space X_{an} = (X, O_{an}) is a ringed space which is locally isomorphic to a basic analytic set. A good reference for analytic spaces is [GA]; the original source is Serre’s [GAGA].

Let X_{an} be an analytic space. For clarity, a vector bundle over X_{an} (in the sense of Definition 5.1) is sometimes called an analytic vector bundle. There is also a notion of coherence on an analytic space: an O_{an}-module F is called coherent if it is locally finitely presented in the sense that in a neighborhood U of any point it is presented as a cokernel:

O_{U,an}^{n} → O_{U,an}^{m} → F|_{U} → 0.

One special class of analytic spaces is the class of Stein spaces. It is known that analytic vector bundles are the same as topological vector bundles over a Stein space. For example, any analytic subspace of ℂ^{n} is a Stein space. See [GR].

Morphisms of ringed spaces

Here are two basic ways to construct new ringed spaces and morphisms: (1) If A is a sheaf of O_{X}-algebras, (X, A) is a ringed space;

(2) If f: Y → X is a continuous map and (Y, O_{Y}) is a ringed space, the direct

image sheaf f_{∗}O_{Y} is a sheaf of rings on X, so (X, f_{∗}O_{Y}) is a ringed space. A morphism of ringed spaces f: (Y, O_{Y}) → (X, O_{X}) is a continuous map f: Y → X together with a map f^{#}: O_{X} → f_{∗}O_{Y} of sheaves of rings on X. In case (1) there is a morphism i: (X, A) → (X, O_{X}); in case (2) the morphism is (Y, O_{Y}) → (X, f_{∗}O_{Y}); in general, every morphism factors as (Y, O_{Y}) → (X, f_{∗}O_{Y}) → (X, O_{X}).

A morphism of ringed spaces f: X → Y between two locally ringed spaces is a morphism of locally ringed spaces if in addition for each point y ∈ Y the map of stalk rings O_{X,f(y)} → O_{Y,y} sends the maximal ideal 𝔪_{f(y)} into the maximal ideal 𝔪_{y}. A morphism of schemes is a morphism of locally ringed spaces f: Y → X between schemes.

If F is an O_{Y} -module, then the direct image sheaf f_{∗}F is an f_{∗}O_{Y} -module, and hence also an O_{X}-module. Thus f_{∗} is a functor from O_{Y} -modules to O_{X}-modules, making O_{X}-mod covariant in X. If F is a vector bundle over Y then f_{∗}F is a vector bundle over (X, f_{∗}O_{Y}). However, f_{∗}F will not be a vector bundle over (X, O_{X}) unless f_{∗}O_{Y} is a locally free O_{X}-module of finite rank, which rarely occurs.

If f: Y → X is a proper morphism between noetherian schemes then Serre’s “Theorem B” states that if F is a coherent O_{Y} -module then the direct image f_{∗}F is a coherent O_{X}-module. (See [EGA, III(3.2.2)] or [Hart, III.5.2 and II.5.19].)

Example 5.2.1 (Projective Schemes). When Y is a projective scheme over a field k, the structural map π: Y → Spec(k) is proper. In this case the direct image π_{∗}F = H^{0}(Y, F) is a finite-dimensional vector space over k. Indeed, every coherent k-module is finitely generated. Not surprisingly, dim_{k} H^{0}(Y, F) gives an important invariant for coherent modules (and vector bundles) over projective schemes.

The functor f_{∗} has a left adjoint f^{∗} (from O_{X}-modules to O_{Y} -modules):

Hom_{O_{Y}} (f^{∗}E, F) ≅ Hom_{O_{X}} (E, f_{∗}F)

for every O_{X}-module E and O_{Y} -module F. The explicit construction is given in [Hart, II.5], and shows that f^{∗} sends free O_{X}-modules to free O_{Y} -modules, with f^{∗}O_{X} ≅ O_{Y}. If i: U ⊂ X is the inclusion of an open subset then i^{∗}E is just E|U; it follows that if E|U is free then (f^{∗}E)|f^{−1}(U) is free. Thus f^{∗} sends locally free O_{X}- modules to locally free O_{Y} -modules, and yields a functor f^{∗}: VB(X) → VB(Y), making VB(X) contravariant in the ringed space X.

Example 5.2.2. If R and S are commutative rings then ring maps f^{#}: R → S are in 1-1 correspondence with morphisms f: Spec(S) → Spec(R) of ringed spaces. The direct image functor f_{∗} corresponds to the forgetful functor from S-modules to R-modules, and the functor f^{∗} corresponds to the functor ⊗_{R}S from R-modules to S-modules.

Associated analytic and topological bundles 5.2.3. Suppose that X is a scheme of finite type over ℂ, such as a subvariety of ℙ_{ℂ}^{n} or 𝔸_{ℂ}^{n} = Spec(ℂ[x_{1}, ..., x_{n}]). The closed points X(ℂ) of X have the natural structure of an analytic space; in particular it is a locally compact topological space. Indeed, X(ℂ) is covered by open sets U(ℂ) homeomorphic to analytic subspaces of 𝔸^{n}(ℂ), and 𝔸^{n}(ℂ) ≅ ℂ^{n}. Note that if X is a projective variety then X(ℂ) is compact, because it is a closed subspace of the compact space ℙ^{n}(ℂ) ≅ ℂℙ^{n}.

Considering X(ℂ) as topological and analytic ringed spaces as in Examples 5.1.1 and 5.1.4, the evident continuous map τ: X(ℂ) → X induces morphisms of ringed spaces X(ℂ)_{top} → X(ℂ)_{an} ⟶ X. This yields functors from VB(X, O_{X}) to VB(X(ℂ)_{an}), and from VB(X_{an}) to VB(X(ℂ)_{top}) ≅ VB_{ℂ}(X(ℂ)). Thus every vector bundle E over the scheme X has an associated analytic vector bundle E_{an}, as well as an associated complex vector bundle τ^{∗}E over X(ℂ). In particular, every vector bundle E on X has topological Chern classes c_{i}(E) = c_{i}(τ^{∗}E) in the group H^{2i}(X(ℂ); ℤ).

The main theorem of [GAGA] is that if X is a projective algebraic variety over ℂ then it has there is an equivalence between the categories of coherent modules over X and over X_{an}. In particular, the categories of vector bundles VB(X) and VB(X_{an}) are equivalent.

A similar situation arises if X is a scheme of finite type over ℝ. Let X(ℝ) denote the closed points of X with residue field ℝ; it too is a locally compact space. We consider X(ℝ) as a ringed space, using ℝ-valued functions as in Example 5.1.1. There is a morphism of ringed spaces τ: X(ℝ) → X, and the functor τ^{∗} sends VB(X) to VB_{ℝ}(X(ℝ)). That is, every vector bundle F over X has an associated real vector bundle τ^{∗}F over X(ℝ); in particular, every vector bundle F over X has Stiefel-Whitney classes w_{i}(F) = w_{i}(τ^{∗}F) ∈ H^{i}(X(ℝ); ℤ/2).

Patching and Operations 5.3. Just as we built up projective modules by patching in 2.5, we can obtain a locally free sheaf F by patching (or glueing) locally free sheaves F_{i} of O_{U_{i}}-modules via isomorphisms g_{ij} between F_{j}|U_{i} ∩ U_{j} and F_{i}|U_{i} ∩ U_{j}, as long as g_{ii} = 1 and g_{ij}g_{jk} = g_{ik} for all i, j, k.

The patching process allows us to take any natural operation on free modules and extend it to locally free modules. For example, if O_{X} is commutative we can construct tensor products F ⊗ G, Hom-modules Hom(F, G), dual modules F̌ and exterior products ∧^{i}F using P ⊗_{R} Q, Hom_{R}(P, Q), P̌ and ∧^{i}P. If F and G are vector bundles, then so are F ⊗ G, Hom(F, G), F̌ and ∧^{i}F. All of the natural isomorphisms such as F̌ ⊗ G ≅ Hom(F, G) hold for locally free modules, because a sheaf map is an isomorphism if it is locally an isomorphism.

The Picard group and determinant bundles

If (X, O_{X}) is a commutative ringed space, the set Pic(X) of isomorphism classes of line bundles forms a group, called the Picard group of X. To see this, we modify the proof in §3: the dual Ľ of a line bundle L is again a line bundle and Ľ⊗L ≅ O_{X} because by Lemma 3.1 this is true locally. Note that if X is Spec(R), we recover the definition of §3: Pic(Spec(R)) = Pic(R).

If F is locally free of rank n, then det(F) = ∧^{n}(F) is a line bundle. Operating componentwise as in §3, every locally free O_{X}-module F has an associated determinant line bundle det(F). The natural map det(F) ⊗ det(G) → det(F ⊕ G) is an isomorphism because this is true locally by the Sum Formula in §3 (see Ex. 5.4 for a generalization). Thus det is a useful invariant of a locally free O_{X}-module. We will discuss Pic(X) in terms of divisors at the end of this section.

Projective schemes

If X is a projective variety, maps between vector bundles are most easily described using graded modules. Following [Hart, II.2], this trick works more generally if X is Proj(S) for a commutative graded ring S = S_{0} ⊕S_{1} ⊕· · ·. By definition, the scheme Proj(S) is the union of the affine open sets D_{+}(f) = Spec S_{(f)}, where f ∈ S_{n} (n ≥ 1) and S_{(f)} is the degree 0 subring of the ℤ-graded ring S[\frac{1}{f}]. To cover Proj(S), it suffices to use D_{+}(f) for a family of f’s generating the ideal S_{+} = S_{1} ⊕ S_{2} ⊕ ⋯ of S. For example, projective n-space over R is ℙ_{R}^{n} = Proj(R[X_{0}, ..., X_{n}]); it is covered by the D_{+}(X_{i}) and if x_{j} = X_{j}/X_{i} then D_{+}(X_{i}) = Spec(R[x_{1}, ..., x_{n}]).

If M = ⊕_{i∈ℤ}M_{i} is a graded S-module, there is an associated O_{X}-module M̃ on X = Proj(S). The restriction of M̃ to D_{+}(f) is the sheaf associated to M_{(f)}, the S_{(f)}-module which constitutes the degree 0 submodule of M[\frac{1}{f}]; more details of the construction of M̃ are given in [Hart, II.5.11]. Clearly S̃ = O_{X}. The functor M ↦ M̃ is exact, and has the property that M̃ = 0 whenever M_{i} = 0 for large i.

Example 5.3.1 (Twisting Line Bundles). The most important example of this construction is when M is S(n), the module S regraded so that the degree i part is S_{n+i}; the associated sheaf S̃(n) is written as O_{X}(n). If f ∈ S_{1} then S(n)_{(f)} ≅ S_{(f)}, so if S is generated by S_{1} as an S_{0}-algebra then O_{X}(n) is a line bundle on X = Proj(S); it is called the n^{th} twisting line bundle. If F is any O_{X}-module, we write F(n) for F ⊗ O_{X}(n), and call it “F twisted n times.”

We will usually assume that S is generated by S_{1} as an S_{0}-algebra, so that the O_{X}(n) are line bundles. This hypothesis ensures that every quasicoherent O_{X}- module has the form M̃ for some M ([Hart, II.5.15]). It also ensures that the canonical maps M̃ ⊗_{O_{X}} Ñ → (M ⊗_{S} N)˜ are isomorphisms, so if F = M̃ then F(n) is the O_{X}-module associated to M(n) = M ⊗_{S} S(n). Since S(m)⊗S(n) ≅ S(m+n) we have the formula

O_{X}(m) ⊗ O_{X}(n) ≅ O_{X}(m + n).

Thus there is a homomorphism from ℤ to Pic(X) sending n to O_{X}(n). Operating componentwise, the same formula yields a homomorphism [X, ℤ] → Pic(X).

Here is another application of twisting line bundles. An element x ∈ M_{n} gives rise to a graded map S(−n) → M and hence a sheaf map O_{X}(−n) → M̃. Taking the direct sum over a generating set for M, we see that for every quasicoherent O_{X}- module F there is a surjection from a locally free module ⊕O_{X}(−n_{i}) onto F. In contrast, there is a surjection from a free O_{X}-module onto F iff F can be generated by global sections, which is not always the case.

If P is a graded f.g. projective S-module, the O_{X}-module P̃ is a vector bundle over Proj(S). To see this, suppose the generators of P lie in degrees n_{1}, ..., n_{r} and set F = S(−n_{1}) ⊕ ⋯ ⊕ S(−n_{r}). The kernel Q of the surjection F → P is a graded S-module, and that the projective lifting property implies that P ⊕ Q ≅ F. Hence P̃ ⊕ Q̃ is the direct sum F̃ of the line bundles O_{X}(−n_{i}), proving that P̃ is a vector bundle.

Example 5.4 (No projective vector bundles). Consider the projective line ℙ_{R}^{1} = Proj(S), S = R[x, y]. Associated to the “Koszul” exact sequence of graded S-modules

0 → S(−2) ⟶^{(y,−x)} S(−1) ⊕ S(−1) ⟶^{(x,y)} S → R → 0  (5.4.1)

is the short exact sequence of vector bundles over ℙ_{R}^{1}:

0 → O_{ℙ^{1}}(−2) → O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−1) → O_{ℙ^{1}} → 0.  (5.4.2)

The sequence (5.4.2) cannot split, because there are no nonzero maps from O_{ℙ^{1}} to O_{ℙ^{1}}(−1) (see Ex. 5.2). This shows that the projective lifting property of §2 fails for the free module O_{ℙ^{1}}. In fact, the projective lifting property fails for every vector bundle over ℙ_{R}^{1}; the category of O_{ℙ^{1}}-modules has no “projective objects.” This failure is the single biggest difference between the study of projective modules over rings and vector bundles over schemes.

The strict analogue of the Cancellation Theorem 2.3 does not hold for projective schemes. To see this, we cite the following result from [Atiy56]. A vector bundle is called indecomposable if it cannot be written as the sum of two proper sub-bundles. For example, every line bundle is indecomposable.

Krull-Schmidt Theorem 5.5 (Atiyah). Let X be a projective scheme over a field k. Then the Krull-Schmidt theorem holds for vector bundles over X. That is, every vector bundle over X can be written uniquely (up to reordering) as a direct sum of indecomposable vector bundles.

In particular, the direct sums of line bundles O_{X}(n_{1}) ⊕ ⋯ ⊕ O_{X}(n_{r}) are all distinct whenever dim(X) ≠ 0, because then all the O_{X}(n_{i}) are distinct.

Example 5.5.1. If X is a smooth projective curve over ℂ, then the associated topological space X(ℂ) is a Riemann surface. We saw in 4.11.2 that every topological line bundle on X(ℂ) is completely determined by its topological degree, and that every topological vector bundle is completely determined by its rank and degree. Now it is not hard to show that the twisting line bundle O_{X}(d) has degree d. Hence every topological vector bundle E of rank r and degree d is isomorphic to the direct sum O_{X}(d) ⊕ T^{r−1}. Moreover, the topological degree of a line bundle agrees with the usual algebraic degree one encounters in Algebraic Geometry.

The Krull-Schmidt Theorem shows that for each r ≥ 2 and d ∈ ℤ there are infinitely many vector bundles over X with rank r and degree d. Indeed, there are infinitely many ways to choose integers d_{1}, …, d_{r} so that ∑ d_{i} = d, and these choices yield the vector bundles O_{X}(d_{1}) ⊕ ⋯ ⊕ O_{X}(d_{r}), which are all distinct with rank r and degree d.

For X = ℙ_{k}^{1}, the only indecomposable vector bundles are the line bundles O(n). This is a theorem of A. Grothendieck, proven in [Groth57]. Using the Krull-Schmidt Theorem, we obtain the following classification.

Theorem 5.6 (Classification of Vector Bundles over ℙ_{k}^{1}). Let k be an algebraically closed field. Every vector bundle F over X = ℙ_{k}^{1} is a direct sum of the line bundles O_{X}(n) in a unique way. That is, F determines a finite decreasing family of integers n_{1} ≥ ⋯ ≥ n_{r} such that

F ≅ O_{X}(n_{1}) ⊕ ⋯ ⊕ O_{X}(n_{r}).

The classification over other spaces is much more complicated than it is for ℙ^{1}. The following example is taken from [Atiy57]. Atiyah’s result holds over any algebraically closed field k, but we shall state it for k = ℂ because we have not yet introduced the notion on the degree of a line bundle. (Using the Riemann-Roch theorem, we could define the degree of a line bundle L over an elliptic curve as the integer dim H^{0}(X, L(n)) − n for n >> 0.)

Classification of vector bundles over elliptic curves 5.7. Let X be a smooth elliptic curve over ℂ. Every vector bundle E over X has two integer invariants: its rank, and its degree, which we saw in 5.5.1 is just the Chern class c_{1}(E) ∈ H^{2}(X(ℂ); ℤ) ≅ ℤ of the associated topological vector bundle over the Riemann surface X(ℂ) of genus 1, defined in 5.2.3. Let VB_{r,d}^{ind}(X) denote the set of isomorphism classes of indecomposable vector bundles over X having rank r and degree d. Then for all r ≥ 1 and d ∈ ℤ:

(1) All the vector bundles in the set VB_{r,d}^{ind}(X) yield the same topological vector bundle E over X(ℂ). This follows from Example 5.5.1.

(2) There is a natural identification of each VB_{r,d}^{ind}(X) with the set X(ℂ); in particular, there are uncountably many indecomposable vector bundles of rank r and degree d.

(3) Tensoring with the twisting bundle O_{X}(d) induces a bijection between VB_{r,0}^{ind}(X) and VB_{r,d}^{ind}(X).

(4) The r^{th} exterior power ∧^{r} maps VB_{r,d}^{ind}(X) onto VB_{1,d}^{ind}(X). This map is a bijection iff r and d are relatively prime. If (r, d) = h then for each line bundle L of degree d there are h^{2} vector bundles E with rank r and determinant L.

Construction 5.8 (Projective bundles). If E is an vector bundle over a scheme X, we can form a projective space bundle ℙ(E), which is a scheme equipped with a map π: ℙ(E) → X and a canonical line bundle O(1). To do this, we first construct ℙ(E) when X is affine, and then glue the resulting schemes together.

If M is any module over a commutative ring R, the i^{th} symmetric product Sym^{i}M is the quotient of the i-fold tensor product M ⊗ ⋯ ⊗ M by the permutation action of the symmetric group, identifying m_{1} ⊗ ⋯ ⊗ m_{i} with m_{σ(1)} ⊗ ⋯ ⊗ m_{σ(i)} for every permutation σ. The obvious concatenation product (Sym^{i}M) ⊗_{R} (Sym^{j}M) → Sym^{i+j}M makes Sym(M) = ⊕Sym^{i}(M) into a graded commutative R-algebra, called the symmetric algebra of M. As an example, note that if M = R^{n} then Sym(M) is the polynomial ring R[x_{1}, ..., x_{n}]. This construction is natural in R: if R → S is a ring homomorphism, then Sym(M) ⊗_{R} S ≅ Sym(M ⊗_{R} S).

If E is a f.g. projective R-module, let ℙ(E) denote the scheme Proj(Sym(E)). This scheme comes equipped with a map π: ℙ(E) → Spec(R) and a canonical line bundle O(1); the scheme ℙ(E) with this data is called the projective space bundle associated to E. If E = R^{n}, then ℙ(E) is just the projective space ℙ_{R}^{n−1}. In general, the fact that E is locally free implies that Spec(R) is covered by open sets D(s) = Spec(R[\frac{1}{s}]) on which E is free. If E[\frac{1}{s}] is free of rank n then the restriction of ℙ(E) to D(s) is

ℙ(E[\frac{1}{s}]) ≅ Proj(R[\frac{1}{s}][x_{1}, ..., x_{n}]) = ℙ_{D(s)}^{n−1}.

Hence ℙ(E) is locally just a projective space over Spec(R). The vector bundles O(1) and π^{∗}Ẽ on ℙ(E) are the sheaves associated to the graded S-modules S(1) and E ⊗_{R} S, where S is Sym(E). The concatenation E ⊗ Sym^{j}(E) → Sym^{1+j}(E) yields an exact sequence of graded modules,

0 → E_{1} → E ⊗_{R} S → S(1) → R(−1) → 0  (5.8.1)

hence a natural short exact sequence of ℙ(E)-modules

0 → E_{1} → π^{∗}Ẽ → O(1) → 0.  (5.8.2)

Since π^{∗}Ẽ and O(1) are locally free, E_{1} is locally free and rank(E_{1}) = rank(E) − 1. For example, if E = R^{2} then ℙ(E) is ℙ_{R}^{1} and E_{1} is O(−1) because (5.8.1) is the sequence (5.4.1) tensored with S(1). That is, (5.8.2) is just:

0 → O(−1) → O ⊕ O → O(+1) → 0.

Having constructed ℙ(E) over affine schemes, we now suppose that E is a vector bundle over any scheme X. We can cover X by affine open sets U and construct the projective bundles ℙ(E|U) over each U. By naturality of the construction of ℙ(E|U), the restrictions of ℙ(E|U) and ℙ(E|V) to U ∩V may be identified with each other. Thus we can glue the ℙ(E|U) together to obtain a projective space bundle ℙ(E) over X; a patching process similar to that in 5.3 yields a canonical line bundle O(1) over ℙ(E).

By naturality of E ⊗_{R} Sym(E) → Sym(E)(1), we have a natural short exact sequence of vector bundles on ℙ(E), which is locally the sequence (5.8.2):

0 → E_{1} → π^{∗}E → O(1) → 0.  (5.8.3)

Let ρ denote the projective space bundle ℙ(E_{1}) → ℙ(E) and let E_{2} denote the kernel of ρ^{∗}E_{1} → O(1). Then (πρ)^{∗}E has a filtration E_{2} ⊂ ρ^{∗}E_{1} ⊂ (ρπ)^{∗}E with filtration quotients O(1) and ρ^{∗}O(1). This yields a projective space bundle ℙ(E_{2}) → ℙ(E_{1}). As long as E_{i} has rank ≥ 2 we can iterate this construction, forming a new projective space bundle ℙ(E_{i}) and a vector bundle E_{i+1}. If rank E = r, E_{r−1} will be a line bundle. We write 𝔽(E) for ℙ(E_{r−2}), and call it the flag bundle of E. We may summarize the results of this construction as follows.

Theorem 5.9 (Splitting Principle). Given a vector bundle E of rank r on a scheme X, there exists a morphism f: 𝔽(E) → X such that f^{∗}E has a filtration

f^{∗}E = E′_{0} ⊃ E′_{1} ⊃ ⋯ ⊃ E′_{r} = 0

by sub-vector bundles whose successive quotients E′_{i}/E′_{i+1} are all line bundles.

Cohomological classification of vector bundles

The formation of vector bundles via the patching process in 5.3 may be codified into a classification of rank n vector bundles via a Čech cohomology set Ȟ^{1}(X, GL_{n}(O_{X})) which is associated to the sheaf of groups G = GL_{n}(O_{X}). This cohomology set is defined more generally for any sheaf of groups G as follows. A Čech 1-cocycle for an open cover U = {U_{i}} of X is a family of elements g_{ij} in G(U_{i} ∩ U_{j}) such that g_{ij} = 1 and g_{ij}g_{jk} = g_{ik} for all i, j, k. Two 1-cocycles {g_{ij}} and {h_{ij}} are said to be equivalent if there are f_{i} ∈ G(U_{i}) such that h_{ij} = f_{i}g_{ij}f_{j}^{−1}. The equivalence classes of 1-cocycles form the set Ȟ^{1}(U, G). If V is a refinement of a cover U, there is a set map from Ȟ^{1}(U, G) to Ȟ^{1}(V, G). The cohomology set Ȟ^{1}(X, G) is defined to be the direct limit of the Ȟ^{1}(U, G) as U ranges over the system of all open covers of X.

We saw in 5.3 that every rank n vector bundle arises from patching, using a 1-cocycle for G = GL_{n}(O_{X}). It isn’t hard to see that equivalent cocycles give isomorphic vector bundles. From this, we deduce the following result.

Classification Theorem 5.10. For every ringed space X, the set VB_{n}(X) of isomorphism classes of vector bundles of rank n over X is in 1-1 correspondence with the cohomology set Ȟ^{1}(X, GL_{n}(O_{X})):

VB_{n}(X) ≅ Ȟ^{1}(X, GL_{n}(O_{X})).

When G is an abelian sheaf of groups, such as O_{X}^{∗} = GL_{1}(O_{X}), it is known that the Čech set Ȟ^{1}(X, G) agrees with the usual sheaf cohomology group H^{1}(X, G) (see Ex. III.4.4 of [Hart]). In particular, each Ȟ^{1}(X, G) is an abelian group. A little work, detailed in [EGA, 0_{I}(5.6.3)]) establishes:

Corollary 5.10.1. For every locally ringed space X the isomorphism of Theorem 5.10 is a group isomorphism:

Pic(X) ≅ H^{1}(X, O_{X}^{∗}).

As an application, suppose that X is the union of two open sets V_{1} and V_{2}. Write U(X) for the group H^{0}(X, O_{X}^{∗}) = O_{X}^{∗}(X) of global units on X. The cohomology Mayer-Vietoris sequence translates to the following exact sequence.

1 → U(X) → U(V_{1}) × U(V_{2}) → U(V_{1} ∩ V_{2}) ⟶^{∂}
⟶^{∂} Pic(X) → Pic(V_{1}) × Pic(V_{2}) → Pic(V_{1} ∩ V_{2}).

(5.10.2)

To illustrate how this sequence works, consider the standard covering of ℙ_{R}^{1} by Spec(R[t]) and Spec(R[t^{−1}]). Their intersection is Spec(R[t, t^{−1}]). Comparing (5.10.2) with the sequences of Ex. 3.17 and Ex. 3.18 yields

Theorem 5.11. For any commutative ring R,

U(ℙ_{R}^{1}) = U(R) = R^{∗} and Pic(ℙ_{R}^{1}) ≅ Pic(R) × [Spec(R), ℤ].

As in 5.3.1, the continuous function n: Spec(R) → ℤ corresponds to the line bundle O(n) on ℙ_{R}^{1} obtained by patching R[t] and R[t^{−1}] together via t^{n} ∈ R[t, t^{−1}]^{∗}.

Here is an application of Corollary 5.10.1 to nonreduced schemes. Suppose that I is a sheaf of nilpotent ideals, and let X_{0} denote the ringed space (X, O_{X}/I). Writing I^{∗} for the sheaf GL_{1}(I) of Ex. 1.10, we have an exact sequence of sheaves of abelian groups:

1 → I^{∗} → O_{X}^{∗} → O_{X_{0}}^{∗} → 1.

The resulting long exact cohomology sequence starts with global units:

U(X) → U(X_{0}) → H^{1}(X, I^{∗}) → Pic(X) → Pic(X_{0}) → H^{2}(X, I^{∗}) · · · . (5.11.1)

Thus Pic(X) → Pic(X_{0}) may not be an isomorphism, as it is in the affine case (Lemma 3.9).

Invertible ideal sheaves

Suppose that X is an integral scheme, i.e., that each O_{X}(U) is an integral domain. The function field K(X) of X is the common quotient field of the integral domains O_{X}(U). Following the discussion in §3, we use K to denote the constant sheaf U ↦ K(X) and consider O_{X}-submodules of K. Those that lie in some fO_{X} we call fractional; a fractional ideal I is called invertible if IJ = O_{X} for some J. As in Proposition 3.5, invertible ideals are line bundles and I ⊗ J ≅ IJ. The set Cart(X) of invertible ideals in K is therefore an abelian group.

Proposition 5.12. If X is an integral scheme, there is an exact sequence

1 → U(X) → K(X)^{∗} → Cart(X) → Pic(X) → 1.  (5.12.1)

Proof. The proof of 3.5 goes through to prove everything except that every line bundle L on X is isomorphic to an invertible ideal. On any affine open set U we have (L⊗K)|U ≅ K|U, a constant sheaf on U. This implies that L⊗K ≅ K, because over an irreducible scheme like X any locally constant sheaf must be constant. Thus the natural inclusion of L in L ⊗ K expresses L as an O_{X}-submodule of K, and the rest of the proof of 3.5 goes through.

Here is another way to understand Cart(X). Let K^{∗} denote the constant sheaf of units of K; it contains the sheaf O_{X}^{∗}. Associated to the exact sequence

1 → O_{X}^{∗} → K^{∗} → K^{∗}/O_{X}^{∗} → 1

is a long exact cohomology sequence. Since X is irreducible and K^{∗} is constant, we have H^{0}(X, K^{∗}) = K(X)^{∗} and H^{1}(X, K^{∗}) = 0. Since U(X) = H^{0}(X, O_{X}^{∗}) we get the exact sequence

1 → U(X) → K(X)^{∗} → H^{0}(X, K^{∗}/O_{X}^{∗}) → Pic(X) → 1.  (5.12.2)

Motivated by this sequence, we use the term Cartier divisor for a global section of the sheaf K^{∗}/O_{X}^{∗}. A Cartier divisor can be described by giving an open cover {U_{i}} of X and f_{i} ∈ K(X)^{∗} such that f_{i}/f_{j} is in O_{X}^{∗}(U_{i} ∩ U_{j}) for each i and j.

Lemma 5.12.3. Over every integral scheme X, there is a 1-1 correspondence between Cartier divisors on X and invertible ideal sheaves. Under this identification the sequences (5.12.1) and (5.12.2) are the same.

Proof. If I ⊂ K is an invertible ideal, there is a cover {U_{i}} on which it is trivial, i.e., I|U_{i} ≅ O_{U_{i}}. Choosing f_{i} ∈ I(U_{i}) ⊆ K(X) generating I|U_{i} gives a Cartier divisor. This gives a set map Cart(X) → H^{0}(X, K^{∗}/O_{X}^{∗}); it is easily seen to be a group homomorphism compatible with the map from K(X)^{∗}, and with the map to Pic(X) = H^{1}(X, O_{X}^{∗}). This gives a map between the sequences (5.12.1) and (5.12.2); the 5-lemma implies that Cart(X) ≅ H^{0}(X, K^{∗}/O_{X}^{∗}).

Variation 5.12.4. Let D be a Cartier divisor, represented by {(U_{i}, f_{i})}. Historically, the invertible ideal sheaf associated to D is the subsheaf L(D) of K defined by letting L(D)|U_{i} be the submodule of K(X) generated by f_{i}^{−1}. Since f_{i}/f_{j} is a unit on U_{i} ∩ U_{j}, these patch to yield an invertible ideal. If I is invertible and D is the Cartier divisor attached to I by (5.12.3), then L(D) is I^{−1}. Under the correspondence D ↔ L(D) the sequences (5.12.1) and (5.12.2) differ by a minus sign.

For example if X = ℙ_{R}^{1}, let D be the Cartier divisor given by t^{n} on Spec(R[t]) and 1 on Spec(R[t^{−1}]). The correspondence of Lemma 5.12.3 sends D to O(n), but L(D) ≅ O(−n).

Weil divisors

There is a notion of Weil divisor corresponding to that for rings (see 3.6). We say that a scheme X is normal if all the local rings O_{X,x} are normal domains (if X is affine this is the definition of Ex. 3.14), and Krull if it is integral, separated and has an affine cover {Spec(R_{i})} with the R_{i} Krull domains. For example, if X is noetherian, integral and separated, then X is Krull iff it is normal.

A prime divisor on X is a closed integral subscheme Y of codimension 1; this is the analogue of a height 1 prime ideal. A Weil divisor is an element of the free abelian group D(X) on the set of prime divisors of X; we call a Weil divisor D = ∑ n_{i}Y_{i} effective if all the n_{i} ≥ 0.

Let K(X) be the function field of X. Every prime divisor Y yields a discrete valuation on K(X), because the local ring O_{X,y} at the generic point y of Y is a DVR. Conversely, each discrete valuation on K(X) determines a unique prime divisor on X, because X is separated [Hart, Ex. II(4.5)]. Having made these observations, the discussion in §3 applies to yield group homomorphisms ν: K(X)^{∗} → D(X) and ν: Cart(X) → D(X). We define the divisor class group Cl(X) to be the quotient of D(X) by the subgroup of all Weil divisors ν(f), f ∈ K(X)^{∗}. The proof of Proposition 3.6 establishes the following result.

Proposition 5.13. Let X be Krull. Then Pic(X) is a subgroup of the divisor class group Cl(X), and there is a commutative diagram with exact rows:

1 → U(X) → K(X)^{∗} → Cart(X) → Pic(X) → 1
↓ ↓ ∩ ν ∩
1 → U(X) → K(X)^{∗} → D(X) → Cl(X) → 1.

A scheme X is called regular (resp. locally factorial) if the local rings O_{X,x} are all regular local rings (resp. UFD’s). By (3.8), regular schemes are locally factorial. Suppose that X is locally factorial and Krull. If I_{Y} is the ideal of a prime divisor Y and U = Spec(R) is an affine open subset of X, I_{Y} |U is invertible by Corollary 3.8.1. Since ν(I_{Y}) = Y, this proves that ν : Cart(X) → D(X) is onto. Inspecting the diagram of Proposition 5.13, we have:

Proposition 5.14. Let X be an integral, separated and locally factorial scheme. Then

Cart(X) ≅ D(X) and Pic(X) ≅ Cl(X).

Example 5.14.1. ([Hart, II(6.4)]). If X is the projective space ℙ_{k}^{n} over a field k, then Pic(ℙ_{k}^{n}) ≅ Cl(ℙ_{k}^{n}) ≅ ℤ. By Theorem 5.11, Pic(ℙ^{n}) is generated by O(1). The class group Cl(ℙ^{n}) is generated by the class of a hyperplane H, whose corresponding ideal sheaf I_{H} is isomorphic to O(1). If Y is a hypersurface defined by a homogeneous polynomial of degree d, we say deg(Y) = d; Y ∼ dH in D(ℙ^{n}).

The degree of a Weil divisor D = ∑ n_{i}Y_{i} is defined to be ∑ n_{i} deg(Y_{i}); the degree function D(ℙ^{n}) → ℤ induces the isomorphism Cl(ℙ^{n}) ≅ ℤ. We remark that when k = ℂ the degree of a Weil divisor agrees with the topological degree of the associated line bundle in H^{2}(ℂℙ^{n}; ℤ) = ℤ, defined by the first Chern class as in Example 4.11.2.

Blowing Up 5.14.2. Let X be a smooth variety over an algebraically closed field, and let Y be a smooth subvariety of codimension ≥ 2. If the ideal sheaf of Y is I, then I/I^{2} is a vector bundle on Y. The blowing up of X along Y is a nonsingular variety X̃, containing a prime divisor Ỹ ≅ ℙ(I/I^{2}), together with a map π: X̃ → X such that π^{−1}(Y) = Ỹ and X̃ − Ỹ ≅ X − Y (see [Hart, II.7]). For example, the blowing up of a smooth surface X at a point x is a smooth surface X̃, and the smooth curve Ỹ ≅ ℙ^{1} is called the exceptional divisor.

The maps π^{∗}: Pic(X) → Pic(X̃) and ℤ → Pic(X̃) sending n to n[Ỹ] give rise to an isomorphism (see [Hart, Ex. II.8.5 or V.3.2]):

Pic(X̃) ≅ Pic(X) ⊕ ℤ.

Example 5.14.3. Consider the rational ruled surface S in ℙ^{1} × ℙ^{2}, defined by X_{i}Y_{j} = X_{j}Y_{i} (i, j = 1, 2), and the smooth quadric surface Q in ℙ^{3}, defined by xy = zw. Now S is obtained by blowing up ℙ_{k}^{2} at a point [Hart, V.2.11.5], while Q is obtained from ℙ_{k}^{2} by first blowing up two points, and then blowing down the line between them [Hart, Ex. V.4.1]. Thus Pic(S) = Cl(S) and Pic(Q) = Cl(Q) are both isomorphic to ℤ × ℤ. For both surfaces, divisors are classified by a pair (a, b) of integers (see [Hart, II.6.6.1]).

EXERCISES

5.1 Give an example of a ringed space (X, O_{X}) such that the rank of O_{X}(X) is well-defined, but such that the rank of O_{X}(U) is not well-defined for any proper open U ⊆ X. 5.2 Show that the global sections of the vector bundle Hom(E, F) are in 1-1 correspondence with vector bundle maps E → F. Conclude that there is a non-zero map O(m) → O(n) over ℙ_{R}^{1} only if m ≤ n. 5.3 Projection Formula. If f: (X, O_{X}) → (Y, O_{Y}) is a morphism of ringed spaces, F is an O_{X}-module and E is a locally free O_{Y} -module of finite rank, show that there is a natural isomorphism f_{∗}(F ⊗_{O_{X}} f^{∗}E) ≅ f_{∗}(F) ⊗_{O_{Y}} E. 5.4 Let 0 → E → F → G → 0 be an exact sequence of locally free sheaves. Show that each ∧^{n}F has a finite filtration

∧^{n}F = F^{0} ⊇ F^{1} ⊇ ⋯ ⊇ F^{n+1} = 0

with successive quotients F^{i}/F^{i+1} ≅ (∧^{i}E) ⊗ (∧^{n−i}G). In particular, show that det(F) ≅ det(E) ⊗ det(G). 5.5 Let S be a graded ring generated by S_{1} and set X = Proj(S). Show that O_{X}(n)ˇ ≅ O_{X}(−n) and Hom(O_{X}(m), O_{X}(n)) ≅ O_{X}(n − m). 5.6 Serre’s “Theorem A.” Suppose that X is Proj(S) for a graded ring S which is finitely generated as an S_{0}-algebra by S_{1}. Recall from 5.3.1 (or [Hart, II.5.15]) that every quasicoherent O_{X}-module F is isomorphic to M̃ for some graded S-module M. In fact, we can take M_{n} to be H^{0}(X, F(n)).

(a) If M is generated by M_{0} and the M_{i} with i < 0, show that the sheaf M̃ is generated by global sections. Hint: consider M_{0} ⊕ M_{1} ⊕ · · ·.

(b) By (a), O_{X}(n) is generated by global sections if n ≥ 0. Is the converse true?

(c) If M is a finitely generated S-module, show that M̃(n) is generated by global sections for all large n (i.e., for all n ≥ n_{0} for some n_{0}).

(d) If F is a coherent O_{X}-module, show that F(n) is generated by global sections for all large n. This result is known as Serre’s “Theorem A,” and it implies that O_{X}(1) is an ample line bundle in the sense of [EGA, II(4.5.5)].

5.7 Let X be a d-dimensional quasi-projective variety, i.e., a locally closed integral subscheme of some ℙ_{k}^{n}, where k is an algebraically closed field.

(a) Suppose that E is a vector bundle generated by global sections. If rank(E) > d, Bertini’s Theorem implies that E has a global section s such that s_{x} ∉ 𝔪_{x}E_{x} for each x ∈ X. Establish the analogue of the Serre Cancellation Theorem 2.3(a), that there is a short exact sequence of vector bundles

0 → O_{X} →^{s} E → F → 0.

(b) Now suppose that X is a curve. Show that every vector bundle E is a successive extension of invertible sheaves in the sense that there is a filtration of E

E = E_{0} ⊃ E_{1} ⊃ ⋯ ⊃ E_{r} = 0.

by sub-bundles such that each E_{i}/E_{i+1} is a line bundle. Hint: by Ex. 5.6(d), E(n) is generated by global sections for large n.

5.8 Complex analytic spaces. Recall from Example 5.1.4 that a complex analytic space is a ringed space (X, O_{X}) which is locally isomorphic to a basic analytic subset of ℂ^{n}.

(a) Use Example 5.2.3 to show that every analytic vector bundle on ℂ^{n} is free, i.e., O_{an}^{r} for some r. What about ℂ^{n} − 0?

(b) Let X be the complex affine node defined by the equation y^{2} = x^{3} − x^{2}. We saw in 3.10.2 that Pic(X) ≅ ℂ^{×}. Use (4.9.1) to show that Pic(X(ℂ)_{an}) = 0.

(c) (Serre) Let X be the scheme Spec(ℂ[x, y])−{0}, 0 being the origin. Using the affine cover of X by D(x) and D(y), show that Pic(X) = 0 but Pic(X_{an}) ≠ 0.

5.9 Picard Variety. Let X be a scheme over ℂ and X_{an} = X(ℂ)_{an} the associated complex analytic space of Example 5.1.4. There is an exact sequence of sheaves of abelian groups on the topological space X(ℂ) underlying X_{an}:

0 → ℤ ⟶^{2πi} O_{X_{an}} ⟶^{exp} O_{X_{an}}^{∗} → 0,  (∗)

where ℤ is the constant sheaf on X(ℂ).

(a) Show that the Chern class c_{1}: Pic(X_{an}) → H^{2}(X(ℂ)_{top}; ℤ) of Example 5.2.3 is naturally isomorphic to the composite map

Pic(X_{an}) ≅ H^{1}(X_{an}, O_{X_{an}}^{∗}) ≅ H^{1}(X(ℂ)_{top}; O_{X_{an}}^{∗}) ⟶^{∂} H^{2}(X(ℂ)_{top}; ℤ)

coming from Corollary 5.10.1, the map X_{an} → X(ℂ)_{top} of Example 5.1.4, and boundary map of (∗).

Now suppose that X is projective. The image of Pic(X) ≅ Pic(X_{an}) in H^{2}(X(ℂ); ℤ) is called the Néron-Severi group NS(X) and the kernel of Pic(X) → NS(X) is written as Pic^{0}(X). Since H^{2}(X(ℂ); ℤ) is a finitely generated abelian group, so is NS(X). It turns out that H^{1}(X(ℂ), O_{X_{an}}) ≅ ℂ^{n} for some n, and that H^{1}(X(ℂ); ℤ) ≅ ℤ^{2n} is a lattice in H^{1}(X, O_{X_{an}}).

(b) Show that Pic^{0}(X) is isomorphic to H^{1}(X, O_{X})/H^{1}(X(ℂ); ℤ). Thus Pic^{0}(X) is a complex analytic torus; in fact it is the set of closed points of an abelian variety, called the Picard variety of X.

5.10 If E and F are f.g. projective R-modules, show that their projective bundles ℙ(E) and ℙ(F) are isomorphic as schemes over R if and only if E ≅ F ⊗_{R} L for some line bundle L on R. 5.11 Let X be a Krull scheme and Z an irreducible closed subset with complement U. Define a map ρ: Cl(X) → Cl(U) of class groups by sending the Weil divisor ∑ n_{i}Y_{i} to ∑ n_{i}(Y_{i} ∩ U), ignoring terms n_{i}Y_{i} for which Y_{i} ∩ U = ϕ. (Cf. Ex. 3.8.)

(a) If Z has codimension ≥ 2, show that ρ: Cl(X) ≅ Cl(U).

(b) If Z has codimension 1, show that there is an exact sequence

ℤ ⟶^{[Z]} Cl(X) →^{ρ} Cl(U) → 0.

(c) If X is a smooth projective curve over a field k and x ∈ X is a closed point, the

complement U = X−{x} is affine. Show that the map ℤ ⟶^{[x]} Pic(X) = Cl(X) in part (b) is injective. If k(x) = k, conclude that Pic(X) ≅ Pic(U)×ℤ. What happens if k(x) ≠ k?