2. Coherent Sheaves and Cohomology. Coherent sheaf cohomology encodes global properties of schemes. We will discuss some of the main results, particularly Serre’s Theorems A and B. Serre’s FAC probably remains the best reference, though it was written before the Grothendieck “revolution.”

Coherent Sheaves: Quasi-coherent sheaves over a scheme “globalize” modules over a commutative ring. Locally free sheaves are the analogues of vector bundles, and invertible sheaves are the analogues of line bundles.

A sheaf F of abelian groups on a scheme X is a sheaf of O_{X}-modules if each F(U) is a module over O_{X}(U) in such a way that the module structures are compatible with the restriction maps.

Example. If M is a module over A, then there is a natural associated sheaf of O_{Spec(A)}-modules, denoted by M̃, such that:

M̃(U_{f}) ≅ M_{f} on basic open sets and

M̃_{x} ≅ M_{P_{x}} for each x ∈ Spec(A).

A sheaf of O_{X}-modules on an arbitrary scheme is quasi-coherent if it is locally of the form M̃ on open subsets of the form Spec(A). It is coherent if, moreover, the modules M are each finitely generated A-modules. It is locally free if the stalks are each free O_{X,x}-modules.

The assignment M ↦ M̃ with the natural map on morphisms is an equivalence of categories between A-modules and quasi-coherent sheaves over Spec(A). If A is Noetherian, it is an equivalence between finitely generated A-modules and coherent sheaves over Spec(A). Over an arbitrary scheme X, the category of quasi-coherent sheaves is closed under kernels, cokernels and images, and an extension of quasi-coherent sheaves is again quasi-coherent. The same is true for coherent sheaves if X is Noetherian, meaning that X is covered by finitely many Noetherian open affine subschemes.

If i : X → Y is a closed immersion, the ideal sheaf I_{X/Y} is the kernel:

0 → I_{X/Y} → O_{Y} →^{i^{#}} i_{∗}O_{X} → 0

and is locally of the form Ĩ, where I ⊂ A are the ideals corresponding to the closed subschemes Spec(A) ∩ X of each open Spec(A) ⊂ Y. This is therefore quasi-coherent, and coherent if A is Noetherian.

Conversely, if I is a quasi-coherent sub-sheaf of O_{Y}, then there is a uniquely determined closed subscheme X ⊂ Y such that I = I_{X/Y}.

If f : X → Y is a morphism of schemes and F is a sheaf of O_{X}-modules, then f_{∗}F is a sheaf of O_{Y} -modules via the map f^{#} : O_{Y} → f_{∗}O_{X}. In the special case where X = Spec(A) and Y = Spec(B), then f_{∗}(M̃) = M̃, where the M on the right is viewed as a B-module via the associated ring homomorphism ϕ : B → A. It follows that the push-forward of a quasicoherent sheaf is again quasi-coherent.

Given a sheaf G on Y, recall that the inverse image sheaf f^{−1}G is the sheaf on X associated to the presheaf U ↦ lim_{V ⊇f(U)} G(V). If G is a sheaf of O_{Y} -modules, then f^{−1}G is not a sheaf of O_{X}-modules in any reasonable way, but f^{#} determines a map f^{−1}O_{Y} → O_{X} and the pull-back:

f^{∗}G := f^{−1}G ⊗_{f^{−1}O_{Y}} O_{X}

is rigged to be a sheaf of O_{X}-modules. In the special case where X = Spec(A) and Y = Spec(B), then f^{∗}(Ñ) = Ñ ⊗_{B} A, and it follows that f^{∗} commutes with tensor products (note that f_{∗} does not!), and (quasi)-coherent sheaves pull back to (quasi)-coherent sheaves.

An S-scheme X is separated if Δ : X → X ×_{S} X is a closed immersion. In that case, let I be the ideal sheaf of the diagonal. Then Ω_{X/S} := Δ^{∗}(I/I^{2}) is the sheaf of relative differentials of X over S. It is quasi-coherent by our discussion thus far, but this can be seen directly. Namely, for each Spec(B) ⊂ S and Spec(A) ⊂ X lying over Spec(B), the sheaf Ω_{X/S} restricts to Ω̃_{A/B} on Spec(A), where Ω_{A/B} is the module of Kähler differentials for the associated map ϕ : B → A. If in addition S is Noetherian and X is an Sscheme of finite type, meaning that each π^{−1}(Spec(B)) as above is covered by finitely many affines Spec(A) such that each A is a finitely generated B-algebra, then Ω_{X/S} is coherent because the Ω_{A/B} are finitely generated A-modules . A k-variety satisfies these assumptions, by definition, and if X is a non-singular k-variety, then Ω_{X} := Ω_{X/Spec(k)} is a locally free coherent sheaf of constant rank equal to the dimension of X, and vice-versa.

A morphism f : X → S is proper if X is separated and of finite type as an S-scheme and if in addition the projection maps π_{Y} : X ×_{S} Y → Y to all other S-schemes Y take closed sets to closed sets. Closed immersions and projective morphisms are proper, the latter by a theorem of Grothendieck.

A morphism f : X → S is finite if the inverse image of an affine open subscheme Spec(B) ⊂ S is always affine (call it Spec(A) ⊂ X) and A is a finitely generated B-module. Finite morphisms are proper and quasi-finite, meaning that the inverse image of a point in S is always a finite set.

If f : X → S is proper and S is Noetherian, then coherent sheaves on X push forward to coherent sheaves on S (This is non-trivial to prove!) In particular, if F is a coherent sheaf on a projective k-scheme X, then the space Γ(X, F) of global sections is a finite-dimensional vector space over k.

If F is a sheaf of O_{X}-modules on a scheme X, then Γ(X, F) is a Γ(X, O_{X})- module by definition. We say F is spanned by a collection {s_{i}} of sections of F if the images of the s_{i} generate every stalk of F. Alternatively, the s_{i} determine a map ⊕O_{X} → F, and F is spanned by the sections exactly when this map of sheaves of O_{X}-modules is surjective.

The fiber of F at x is the vector space:

F(x) := F_{x} ⊗_{O_{X,x}} k(x)

For example, the fiber Ω_{X}(x) of a proper k-scheme X at a k-rational point is the cotangent space m_{x}/m_{x}^{2}, so that the sheaf Ω_{X} can be seen as the cotangent sheaf, even if X is singular.

It follows from Nakayama’s lemma that if a map F → G of coherent sheaves induces a surjective map F(x) → G(x) at a point x ∈ X, then the sheaf map is surjective on stalks at x (hence in a neighborhood of x). In particular, the fiber dimension of a coherent sheaf F is an upper-semi-continuous function on X and F is generated by sections {s_{i}} if and only each fiber F(x) is generated by the sections.

A coherent sheaf L is invertible if it is locally free of rank one. The dual sheaf L^{∗} := Hom_{O_{X}}(L, O_{X}) inverts L in the sense that L ⊗ L^{∗} ≅ O_{X} hence the terminology. Isomorphism classes of invertible sheaves form an abelian group, the Picard group P ic(X), under tensor product.

Another Example. Suppose M is a graded module over R := A[x_{0}, ..., x_{n}]. There is a natural sheaf M̂ of O_{P_{A}^{n}}-modules on P_{A}^{n} with the property that M̂|_{U_{f}} ≅ M̃_{(f)} for each “basic” open affine. This exhibits M̂ is a quasi-coherent sheaf, which is coherent if A is Noetherian and M is finitely generated.

In particular, if M = ⊕M_{e} is a graded R-module, let M(d) denote the “shifted” graded R-module with M(d)_{e} = M_{d+e}. In particular, the sheaves O_{P_{A}^{n}}(d) := R̂(d) are invertible, and satisfy M̂(d) ≅ M̂ ⊗_{OPn_{A}} O_{P_{A}^{n}}(d) for all graded modules M.

This time, the assignment M ↦ M̂ does not give an equivalence between the categories of graded R-modules and of quasi-coherent sheaves on P_{A}^{n} because graded modules which are isomorphic in all sufficiently high degrees give rise to the same sheaves. This is the only ambiguity, however, and a sort of inverse is given by the twisted global sections:

M̂ ↦ Γ_{∗}M̂ := ⨁_{d∈Z} Γ(P_{A}^{n}, M̂(d))

If F is a quasi-coherent sheaf on P_{A}^{n}, then F ↦ Γ_{∗}F ↦ Γ̂_{∗}F yields a quasi-coherent sheaf which is isomorphic to F. It follows for example that closed subschemes of P_{A}^{n} correspond to homogeneous ideals in R.

If S is a scheme, then the invertible sheaves O_{P_{A}^{n}}(d) over open affines Spec(A) ⊂ S glue to an invertible sheaf O_{P_{S}^{n}}(d). An invertible sheaf L on an S-scheme X is ample (or S-ample) if there is a closed immersion f : X → P_{S}^{n} (so X is projective, in particular) and a positive integer d such that L^{⊗d} ≅ f^{∗}O_{P_{S}^{n}}(1). L is very ample if d may be chosen to be 1.

Any S-morphism f : X → P_{S}^{n} determines n + 1 global sections s_{0}, ..., s_{n} generating the invertible sheaf L := f^{∗}O_{P_{S}^{n}}(1) on X. These sections are pulled back from the “coordinate functions” x_{0}, ..., x_{n} on P_{S}^{n}. Conversely, an invertible sheaf L on X together with n + 1 generating sections s_{0}, ..., s_{n} uniquely determine an S-morphism to P_{S}^{n}.

Suppose X is a proper scheme over an algebraically closed field k with invertible sheaf L and generating sections s_{0}, ..., s_{n} ∈ Γ(X, L) as above. Let V ⊂ Γ(X, L) be the linear series (i.e. the subspace) generated by the s_{i}. Then the associated morphism f : X → P_{k}^{n} is a closed immersion if V separates points, meaning that for each pair p, q ∈ X of closed points there is an s ∈ V such that s ∈ m_{p}L but s ∉ m_{q}L, and V separates tangent directions, meaning that for each closed point p ∈ X, the vector space m_{p}L/m_{p}^{2}L is spanned by the sections s ∈ V that lie in m_{p}L. This is proved using Nakayama’s lemma again, and applies even to non-reduced schemes.

Serre’s Theorem A: If X is a projective scheme over a Noetherian ring A, let O_{X}(1) denote a very ample invertible sheaf on X. Then there is a d_{F} for each coherent sheaf F on X so that F(d) := F ⊗_{O_{X}} O_{X}(d) is generated by its global sections whenever d ≥ d_{F}.

Very ampleness of a line bundle is difficult to detect without knowing the space of global sections. Ampleness, however, is a much easier notion to work with, as evidenced by the following refinement of Serre’s Theorem A.

Criterion for Projectivity: Suppose A is Noetherian and X is a proper A-scheme with invertible sheaf L. Then L is ample (hence X is projective) if and only if there is a d_{F} for each coherent sheaf F so that F ⊗_{O_{X}} L^{⊗d} is generated by its global sections whenever d ≥ d_{F}.

Cohomology: Global properties of coherent sheaves are encoded in their cohomology spaces. Some important vanishing theorems are listed in this brief introduction to the most important tool in algebraic geometry.

The global section functor Γ is left exact, meaning that if:

0 → F′ →^{f′} F →^{f′′} F′′ → 0

is a short exact sequence of sheaves of abelian groups, then:

0 → Γ(X, F′) → Γ(X, F) → Γ(X, F′′)

is exact, but the last map may not be surjective. The cohomology groups H^{i}(X, F) are the “right derived functors” of the global section functor. In particular, H^{0}(X, F) = Γ(X, F) and given a short exact sequence as above, there are (functorial) connecting homomorphisms:

δ^{i} : H^{i}(X, F′′) → H^{i+1}(X, F′)

so that the following sequence is long-exact:

... →^{H^{i}(f′)} H^{i}(X, F) →^{H^{i}(f′′)} H^{i}(X, F′′) →^{δ^{i}} H^{i+1}(X, F′) →^{H^{i+1}(f′)} ...

A Noetherian scheme is a Noetherian space, meaning that it satisfies the descending chain condition on closed subsets. The maximal length of such a chain of irreducible closed subsets is the (Noetherian) dimension of the space. Hence the relevance of the following vanishing theorem of Grothendieck.

Vanishing Theorem: If X is a Noetherian space and i > dim(X), then H^{i}(X, F) = 0 for all sheaves F of abelian groups on X.

Another vanishing theorem of Serre gives the following:

Criterion for Affineness: If X is a Noetherian scheme, then the following are equivalent:

(i) X is affine.

(ii) H^{i}(X, F) = 0 for all quasi-coherent sheaves F and all i > 0.

(iii) H^{1}(X, I) = 0 for all coherent sheaves (of ideals) I ⊂ O_{X}.

On a Noetherian separated scheme X, an intersection of open affines is again affine. Using this fact, Serre’s vanishing theorem allows us to conclude that the cohomology groups H^{i}(X, F) of a quasi-coherent sheaf F on such an X can be computed as the Cech cohomology with respect to any open affine cover. For example, one can use the standard cover of P_{A}^{n} to check:

(a) A[x_{0}, ..., x_{n}] ≅ Γ_{∗}O_{P_{A}^{n}} = ⊕H^{0}(P_{A}^{n}, O_{P_{A}^{n}}(d)).

(b) H^{i}(P_{A}^{n}, O_{P_{A}^{n}}(d)) = 0 for all d and all 0 < i < n.

(c) H^{n}(P_{A}^{n}, O_{P_{A}^{n}}(−n − 1)) ≅ A.

(d) For each d, there is a perfect pairing of free A-modules:

H^{0}(P_{A}^{n}, O_{P_{A}^{n}}(d)) × H^{n}(P_{A}^{n}, O_{P_{A}^{n}}(−d − n − 1)) → H^{n}(P_{A}^{n}, O_{P_{A}^{n}}(−n − 1)) ≅ A

Serre’s Theorem B: If X is a projective scheme over a Noetherian ring A, let O_{X}(1) be a very ample invertible sheaf on X. Then for each coherent sheaf F, the cohomology groups H^{i}(X, F) are all finitely generated A-modules, and there is a d_{F} such that H^{i}(X, F(d)) = 0 for each d ≥ d_{F} and i > 0.

Once again, this is refined by replacing “very ample” with “ample.”

Criterion for Projectivity (2): Suppose A is Noetherian and X is a proper A-scheme with invertible sheaf L. Then L is ample if and only if there is a d_{F} for each coherent sheaf F so that H^{i}(X, F ⊗ L^{⊗d}) = 0 whenever i > 0 and d ≥ d_{F}.

Remark: Note that the criteria are independent of the Noetherian ring A. Thus ampleness, unlike very ampleness, only depends upon X, and not the map to Spec(A). I.e. ampleness is an “absolute” property of X.

Let X be a projective scheme over a field k. Then the Hilbert function:

d ↦ dim_{k}Γ(X, F(d))

is polynomial in d for sufficiently large d. This polynomial P_{F}(d) is the Hilbert polynomial of F. We can see the polynomial growth of the Hilbert function in one of two ways.

The projection formula states that if f : X → Y is a morphism of locally ringed spaces and E is a locally free sheaf of O_{Y} -modules, then:

f_{∗}(F ⊗_{O_{X}} f^{∗}E) ≅ f_{∗}F ⊗_{O_{Y}} E

If we apply this to the closed immersion f : X → P_{k}^{n} with E = O_{P_{k}^{n}}(d), we get:

dim_{k}Γ(X, F(d)) = dim_{k}Γ(P_{k}^{n}, f_{∗}F ⊗ O_{P_{k}^{n}}(d))

It follows from Serre’s theorem A that the graded module M := Γ̂_{∗}f_{∗}F over k[x_{0}, ..., x_{n}] is finitely generated, therefore the dimensions of the M_{d} grow as a polynomial in d for sufficiently large d. (See e.g. Atiyah-Macdonald)

Alternatively, we may define the Euler characteristic of F(d) by:

χ(X, F(d)) = ∑_{i=0}^{dim(X)}(−1)^{i} dim_{k}H^{i}(X, F(d))

Then χ(X, F(d)) = dim_{k}Γ(X, F(d)) for all sufficiently large d by Serre’s theorem B. Thus it suffices (and gives us much more information!) to prove that the Euler characteristic χ(X, F(d)) is polynomial in d for all d, hence equal to the Hilbert polynomial. This follows from the fact that the Euler characteristic is additive on exact sequences. If X_{1}, ..., X_{m} ⊂ X are the irreducible components of maximal dimension, then multiplication by any section s ∈ O_{X}(1) that is nonzero on each of these components gives rise to an exact sequence:

0 → i_{∗}N → F →^{·s} F(1) → i_{∗}Q → 0

where N and Q are coherent sheaves on the zero scheme D ⊂ X of s. Since dim(D) = dim(X) − 1, and since tensoring by O_{X}(d) is exact, the polynomiality follows by induction on dimension, the projection formula, and the observation that cohomology commutes with closed immersions. We also see in this way that the degree of the Hilbert polynomial is bounded above by the dimension of X.