# 6 Cohomology of \({\mathscr O}_X\)-modules

General references: [ We ] , [ HS ] , [ Gr ] , [ KS ] , [ GW2 ] .

June 7,

2023

## The formalism of derived functors

**(6.1) Complexes in abelian categories**

Reference: [ We ] Ch. 1, [ GW2 ] App. F.

Let \({\mathcal A}\) be an abelian category (e.g., the category of abelian groups, the category of \(R\)-modules for a ring \(R\), the category of abelian sheaves on a topological space \(X\), the category of \({\mathscr O}_X\)-modules on a ringed space \(X\), or the category of quasi-coherent \({\mathscr O}_X\)-modules on a scheme \(X\); see [ GW2 ] Section (F.7)).

A *complex in \({\mathcal A}\)* is a sequence of morphisms

in \({\mathcal A}\) (\(i\in \mathbb {Z}\)), such that \(d^{i+1}\circ d^i = 0\) for every \(i\in \mathbb {Z}\). The maps \(d^i\) are called the *differentials* of the complex.

Given complexes \(A^\bullet \), \(B^\bullet \), a morphism \(A^\bullet \to B^\bullet \) of complexes is a family of morphisms \(f^i\colon A^i\to B^i\) such that the \(f^i\) commute with the differentials of \(A^\bullet \) and \(B^\bullet \) in the obvious way. With this notion of morphisms, we obtain the category \(C({\mathcal A})\) of complexes in \({\mathcal A}\). This is an abelian category (kernels, images, …are formed degree-wise); see [ We ] Thm. 1.2.3.

**Definition 6.1**

*\(i\)-th cohomology object*of \(A^\bullet \). We obtain functors \(H^i\colon C({\mathcal A})\to {\mathcal A}\). We say that \(A^\bullet \) is

*exact at \(i\)*, if \(H^i(A^\bullet ) = 0\). We say that \(A^\bullet \) is

*exact*, if \(H^i(A^\bullet )=0\) for all \(i\).

**Remark 6.2**

**Proposition 6.3**

*boundary maps*) such that, together with the maps induced by functoriality of the \(H^i\), we obtain the

*long exact cohomology sequence*

We need a criterion which ensures that two morphisms between complexes induce the same maps on all cohomology objects. See [ We ] 1.4.

**Definition 6.4**

*homotopic*, if there exists a family of maps \(k^i\colon A^i\to B^{i-1}\), \(i\in \mathbb {Z}\), such that

*homotopy*.

**Proposition 6.5**

In particular, if \(A^\bullet \) is a complex such that \(\operatorname{id}_{A^\bullet }\sim 0\), then \(H^i(A^\bullet )=0\) for all \(i\), i.e., \(A^\bullet \) is exact.

**Definition 6.6**

*homotopy equivalent*, if there exist morphisms \(f\colon A^\bullet \to B^\bullet \) and \(g\colon B^\bullet \to A^\bullet \) of complexes such that \(g\circ f \sim \operatorname{id}_A\) and \(f\circ g\sim \operatorname{id}_B\). In this case, \(f\) and \(g\) induce isomorphisms \(H^i(A^\bullet )\cong H^i(B^\bullet )\) for all \(i\).

**(6.2) Left exact functors**

Let \({\mathcal A}\), \({\mathcal B}\) be abelian categories. All functors \(F\colon {\mathcal A}\to {\mathcal B}\) that we consider are assumed to be additive, i.e., they induce group homomorphisms \(\operatorname{Hom}_{{\mathcal A}}(A, A')\to \operatorname{Hom}_{{\mathcal B}}(F(A), F(A'))\) for all \(A\), \(A'\) in \({\mathcal A}\).

**Definition 6.7**

*left exact*, if for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), the sequence

**Definition 6.8**

*left exact*if for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), the sequence

Similarly, we have the notion of right exact functor. A functor which is left exact and right exact (and hence preserves exactness of arbitrary sequences) is called exact.

Let \(A_0\in {\mathcal A}\). Then the functors \(A\mapsto \operatorname{Hom}_{{\mathcal A}}(A, A_0)\) and \(A\mapsto \operatorname{Hom}_{{\mathcal A}}(A_0, A)\) are left exact.

**(6.3) \(\delta \)-functors**

Reference: [ We ] 2.1, [ GW2 ] Section (F.48).

Let \({\mathcal A}\), \({\mathcal B}\) be abelian categories.

**Definition 6.9**

*\(\delta \)-functor*from \({\mathcal A}\) to \({\mathcal B}\) is a family \((T^i)_{i\ge 0}\) of functors \({\mathcal A}\to {\mathcal B}\) together with morphisms \(\delta ^i\colon T^i(A'')\to T^{i+1}(A')\) (called

*boundary morphisms*) for every short exact sequence \(0\to A'\to A\to A''\to 0\) in \({\mathcal A}\), such that the sequence

**Definition 6.10**

*universal*, if for every \(\delta \)-functor \((S^i)_i\) and every morphism \(f^0\colon T^0\to S^0\) of functors, there exist unique morphisms \(f^i\colon T^i\to S^i\) of functors for al \(i {\gt} 0\), such that the \(f^i\), \(i\ge 0\) are compatible with the boundary maps \(\delta ^i\) of the two \(\delta \)-functors for each short exact sequence in \({\mathcal A}\).

The definition implies that given a (left exact) functor \(F\), any two universal \(\delta \)-functors \((T^i)_i\), \(({T’}^i)_i\) with \(T^0 = {T’}^{0} = F\) are isomorphic (in the obvious sense) via a unique isomorphism.

**Definition 6.11**

*effaceable*, if for every \(X\) in \({\mathcal A}\) there exists a monomorphism \(\iota \colon X\hookrightarrow A\) with \(F(\iota )=0\).

A particular case is the situation where each \(X\) admits a monomorphism to an object \(I\) with \(F(I)=0\).

**Proposition 6.12**

**(6.4) Injective objects**

June 12,

2023

Let \({\mathcal A}\) be an abelian category.

**Definition 6.13**

*injective*, if the functor \(X\mapsto \operatorname{Hom}_{{\mathcal A}}(X, I)\) is exact.

If \(I\) is injective, then every short exact sequence \(0\to I\to A\to A''\to 0\) in \({\mathcal A}\) splits. Conversely, if \(I\) is an object such that every short exact sequence \(0\to I\to A\to A''\to 0\) splits, then \(I\) is injective.

**Definition 6.14**

*injective resolution*of \(X\) is an exact sequence

**Definition 6.15**

*has enough injectives*if for every object \(X\) there exists a monomorphism \(X\hookrightarrow I\) from \(X\) into an injective object \(I\). Equivalently: Every object has an injective resolution.

**Remark 6.16**

The categories of abelian groups, of \(R\)-modules for a ring \(R\), of abelian sheaves on a topological space, and more generally of \({\mathscr O}_X\)-modules on a ringed space \(X\) all have enough injective objects. For these categories, it is not too hard to show this statement directly (for \({\mathscr O}_X\)-modules on a ringed space \(X\), e.g., see [ H ] Proposition III.2.2). Alternatively, one can show that they all are “Grothendieck abelian categories”, and that those have enough injective objects (see [ GW2 ] Sections (F.12), (21.2)).

Dually, we have the notion of

*projective*object (i.e., \(P\) such that \(\operatorname{Hom}_{{\mathcal A}}(P, -)\) is exact), of*projective resolution*\(\cdots \to P_1\to P_0 \to A \to 0\), and of abelian categories with*enough projective objects*. For a ring \(R\), the category of \(R\)-modules clearly has enough projectives, since every free module is projective, and every module admits an epimorphism from a free module. Categories of sheaves of abelian groups or \({\mathscr O}_X\)-modules typically do not have enough projectives.

**Lemma 6.17**

( [ We ] Theorem 2.3.7, see also Porism 2.2.7) Let \({\mathcal A}\) be an abelian category and let \(f\colon A\to B\) be a morphism in \({\mathcal A}\). Let \(I^i\in {\mathcal A}\) be injective, \(i\ge 0\), and let \(0\to A\to A^0\to A^1\to \cdots \) be exact with \(A^i\in {\mathcal A}\). Then there exists a morphism \(g\colon A^\bullet \to I^\bullet \) of complexes such that the diagram

is commutative, and \(g\) is uniquely determined up to homotopy.

**Lemma 6.18**

**(6.5) Right derived functors**

**Theorem 6.19**

Let \(F\colon {\mathcal A}\to {\mathcal B}\) be a left exact functor, and assume that \({\mathcal A}\) has enough injectives.

For each \(A\in {\mathcal A}\), fix an injective resolution \(0\to A\to I^\bullet \), and define

where \(F(I^\bullet )\) denotes the complex obtained by applying the functor to all \(I^i\) and to the differentials of the complex \(I^\bullet \). Then:

The \(R^i F\) are additive functors \({\mathcal A}\to {\mathcal B}\), and \(R^iFX\) is independent of the choice of injective resolution of \(X\) up to natural isomorphism.

We have an isomorphism \(F\cong R^0F\) of functors.

For \(I\) injective, we have \(R^i FI=0\) for all \(i{\gt}0\).

The family \((R^i F)_i\) admits natural boundary maps making it a universal \(\delta \)-functor.

We call the \(R^iF\) the *right derived functors* of \(F\).

For part (1), the main ingredient is Lemma 6.17. Parts (2) and (3) are easy. To construct the boundary maps giving rise to long exact cohomology sequences in Part (4), use Lemma 6.18; since the exact sequence \(0\to I_1^\bullet \to I_2^\bullet \to I_3^\bullet \to 0\) is term-wise split, the exactness is preserved when applying the functor \(F\). Finally, this \(\delta \)-functor is effaceable by (c) and hence universal.

June 14,

2023

**Definition 6.20**

*\(F\)-acyclic*, if \(R^iF(A) = 0\) for all \(i{\gt}0\).

**Definition 6.21**

*\(F\)-acyclic resolution*of \(A\) is an exact sequence \(0\to A\to J^0\to J^1\to \cdots \) where all \(J^i\) are \(F\)-acyclic.

**Proposition 6.22**

The assertion is easy to see for \(i=0\). For \(i=1\), note that \(\operatorname{Ker}(F(J^1)\to F(J^2)) = F(J^0/A)\) by the left exactness of \(F\). From this, one gets \(R^1F(A) = H^1(F(J^\bullet ))\).

From the long exact cohomology sequence attached to the short exact sequence \(0\to A\to J^0 \to J^0/A \to 0\), we get isomorphisms \(R^iF(J^0/A)\cong R^{i+1}F(A)\) for \(i \ge 1\). Since \(0\to J^0/A\to J^1\to \cdots \) is an acyclic resolution of \(J^0/A\), we get

where the second isomorphism holds by induction.

**(6.6) Derived categories**

Sometimes it is useful to employ the language of derived categories which gives rise to a notion of derived functor \(RF\) that contains slightly more information than the family \((R^iF)_i\) discussed above (for a left exact functor \(F\colon {\mathcal A}\to {\mathcal B}\)). In fact, \(RF\) attaches to each object of \({\mathcal A}\), and more generally to each complex \(A\) in \(C({\mathcal A})\), a *complex* \(RF(A)\) (up to some equivalence relation; more precisely, an object \(RF(A)\in D({\mathcal B})\) in the derived category of \({\mathcal B}\)) whose cohomology objects are the \(R^iF(A)\), i.e., \(H^i(RF(A)) = R^iF(A)\).

See [ GW2 ] (and the references given there) for a systematic treatment: Appendix F for the general theory, in particular Sections (F.37) for the definition of the derived category of an abelian category, (F.42), (F.43), (F.44) for the notion of derived functor, (F.48) for a comparison with the notion of \(\delta \)-functor defined above, Chapter 21 for general results on cohomology of \({\mathscr O}_X\)-modules on a ringed space \(X\) (specifically, the derived functors of the global section and direct image functors), Chapter 22 for cohomology of quasi-coherent \({\mathscr O}_X\)-modules on a scheme \(X\), Chapter 23 for cohomology of proper schemes and the derived direct image functor of a proper morphism. While in these chapters one sometimes obtains stronger or clearer statements using derived categories, for a large part the key arguments are exactly the same as with the “naive approach” using derived functors that we will take. In Chapter 24 on the theorem on formal functions and in particular in Chapter 25 on Grothendieck duality, the machinery of derived categories really shows its full power (but it is unlikely that we will get there in this class).

## Cohomology of sheaves

General reference: [ H ] Ch. III, [ Stacks ] , [ GW2 ] Chapter 21.

**(6.7) Cohomology groups**

Let \(X\) be a topological space. Denote by \(({\rm Ab}_{X})\) the category of abelian sheaves (i.e., sheaves of abelian groups) on \(X\). We have the global section functor

to the category of abelian groups. This is a left exact functor, and we denote its right derived functors by \(H^i(X, -)\). We call \(H^i(X, {\mathscr F})\) the *\(i\)-th cohomology group of \(X\) with coefficients in \({\mathscr F}\)*.

**Example 6.23**

**(6.8) Flasque sheaves**

Reference: [ GW2 ] Section (21.7).

June 19,

2023

**Definition 6.24**

*flasque*(or

*flabby*), if all restriction maps \({\mathscr F}(U)\to {\mathscr F}(V)\) for \(V\subseteq U\subseteq X\) open are surjective.

For the next lemma, we use the *extension by zero* functor: Let \(j\colon U\to X\) be the inclusion of an open subspace into a topological space \(X\) (similarly one can work with an open immersion of ringed spaces). For an abelian sheaf \({\mathscr F}\) on \(U\) we define \(j_!{\mathscr F}\) as the sheafification of the presheaf

We obtain a left adjoint to the restriction functor \(j^*\).

**Lemma 6.25**

Identify \({\mathscr F}(U) = \operatorname{Hom}(j_{U, !}{\mathscr O}_U, {\mathscr F})\), where \(j_U\colon U\to X\) is the inclusion of an open subset \(U\) into \(X\), and \(j_{U, !}\) is the extension by zero functor, using the adjunction between \(j_{U, !}\) and \(j_U^*\).

**Proposition 6.26**

Given a flasque sheaf \({\mathscr F}\); embed it into an injective abelian sheaf and use the results of Problem 40 and dimension shift.

**Corollary 6.27**

Let \(X\) be a ringed space. The right derived functors of the global section functor from the category of \({\mathscr O}_X\)-modules to the category of abelian groups can naturally be identified with \(H^i(X, -)\).

It follows that for an \({\mathscr O}_X\)-module \({\mathscr F}\) the cohomology groups \(H^i(X, {\mathscr F})\) carry a natural \(\Gamma (X, {\mathscr O}_X)\)-module structure.

**(6.9) Grothendieck vanishing**

Reference: [ H ] III.2, [ GW2 ] Section (21.12).

**Lemma 6.28**

Use that \(\iota _*\) preserves the property of being flasque, and that the cohomology groups can be computed using a flasque resolution.

**Proposition 6.29**

First show the statement for global sections, i.e., for \(n=0\). Since filtered colimits are exact, both sides give \(\delta \)-functors from the abelian category of inductive systems of abelian sheaves to the category of abelian groups. To conclude the proof, it is enough to show that they are both effaceable (because this implies they are universal, and universal \(\delta \)-functors are determined by their \(0\)-th term).

To show effaceability, take a system \(({\mathscr F}_i)_i\) and consider functorial flasque resolutions of the \({\mathscr F}_i\) (such as the Godement resolution). Since filtered colimits are exact and, on noetherian spaces, preserve flasqueness, this gives rise to a flasque resolution of \(\mathop{\rm colim}\limits {\mathscr F}_i\). With this at hand, it is not hard to finish the proof.

June 21,

2023

Let \(X\) be a topological space, \(U\subseteq X\) open and \(Z = X\setminus U\) its closed complement. Denote by \(j\colon U\to X\) and \(i\colon Z\to X\) the inclusions. Recall the *extension by zero* functor \(j_!\colon ({\rm Ab}_{U})\to ({\rm Ab}_{X})\) (which is left adjoint to the pull-back \(j^{-1}\)). From this adjunction and the one between \(i^*\) and \(i_*\) we obtain the maps in the short exact sequence

of abelian sheaves on \(X\). The exactness can be checked on stalks where it is clear.

**Theorem 6.30**(Grothendieck)

The proof is omitted in these notes. It consists of a series of reduction steps using the above ingredients, ultimately reducing to the fact that the cohomology of the constant sheaf \(\mathbb {Z}_X\) on an irreducible space vanishes in positive degrees (since there a constant sheaf is flasque). See the references given above for more.

## Čech cohomology

Reference: [ GW2 ] Sections (21.14) to (21.17), [ H ] III.4, [ Stacks ] 01ED (and following sections); a classical reference is [ Go ] .

**(6.10) Čech cohomology groups**

Let \(X\) be a topological space, and let \({\mathscr F}\) be an abelian sheaf on \(X\). (The definitions below can be made more generally for presheaves.)

Let \({\mathscr U}= (U_i)_{i\in I}\) be an open cover of \(X\). We fix a total order of the index set \(I\) (but see below for a sketch that the results are independent of this). For \(i_0, \dots , i_p\in I\), we write \(U_{i_0\dots i_p} := \bigcap _{\nu =0}^p U_{i_\nu }\).

We define

and

where \(\widehat{\cdot }\) indicates that the corresponding index is omitted. One checks that \(d\circ d =0\), so we obtain a complex, the so-called *Čech complex for the cover \({\mathscr U}\) with coefficients in \({\mathscr F}\)*.

June 26,

2023

**Definition 6.31**

*Čech cohomology groups*for \({\mathscr U}\) with coefficients in \({\mathscr F}\) are defined as

Since \({\mathscr F}\) is a sheaf, we have \(\check{H}^0({\mathscr U}, {\mathscr F}) = \Gamma (X, {\mathscr F}) = H^0(X, {\mathscr F})\). In fact, a presheaf \({\mathscr F}\) is a sheaf if and only if for all open subsets \(U\subseteq X\) and all covers \({\mathscr U}\) of \(U\) the natural map \(\Gamma (U, {\mathscr F})\to \check{H}^0({\mathscr U}, {\mathscr F})\) is an isomorphism.

**(6.11) The “full” Čech complex**

Instead of the “alternating” (or “ordered”) Čech complex as above, we can also consider the “full” Čech complex

with differentials defined by the same formula as above. Then the projection \(C^\bullet _f({\mathscr U}, {\mathscr F}) \to C^\bullet ({\mathscr U}, {\mathscr F})\) is a homotopy equivalence, with “homotopy inverse” given by

where \(t_{\underline{i}} = 0\) whenever two entries in the multi-index \(\underline{i}\) coincide, and otherwise \(t_{\underline{i}} = \operatorname{sgn}(\sigma ) s_{\sigma (\underline{i})}\), where \(\sigma \) is the permutation such that \(\sigma (\underline{i})\) is in increasing order.

In particular, we have natural isomorphisms between the cohomology groups of the two complexes. So we also see that the Čech cohomology groups as defined above are independent of the choice of order on \(I\).

**(6.12) Passing to refinements**

**Definition 6.32**

*refinement*of a cover \({\mathscr U}= (U_i)_i\) of \(X\) is a cover \({\mathscr V}= (V_j)_{j\in J}\) (with \(J\) totally ordered) together with a map \(\lambda \colon J\to I\) respecting the orders on \(I\) and \(J\) such that \(V_j\subseteq U_{\lambda (j)}\) for every \(j\in J\).

Given a refinement \({\mathscr V}\) of \({\mathscr U}\), one obtains a natural map (using restriction of sections to smaller open subsets)

We can pass to the colimit over all these maps given by refinements, and define

the *\(p\)-th Čech cohomology group of \(X\) with coefficients in \({\mathscr F}\)*. More precisely, we here take the colimit over the category of refinements (see
[
GW2
]
Section (F.3)). Working with the full Čech complex, one can view this as a colimit over a partially ordered set “as usual”, for the cohomology groups and even for the Čech complexes, if one considers them in the homotopy category (i.e., the map between complexes attached to a refinement is independent of \(\lambda \) up to homotopy), cf.
[
GW2
]
Section (21.16), also for the discussion of set-theoretic issues.

**Proposition 6.33**

This can be checked “directly”. For instance, to construct the connecting homomorphism \(\delta \), take an element \(s\in \Gamma (X, {\mathscr F}'')\). Locally on \(X\), we can lift it to sections of \({\mathscr F}\), so we obtain an element \((s_i)_i\in C^1({\mathscr U}, {\mathscr F})\). Its image \((s_{ij})_{i,j}\) in \(C^2({\mathscr U}, {\mathscr F})\) will usually be different from \(0\) (in fact it is \(=0\) if and only if \(s\) is in the image of \(\Gamma (X, {\mathscr F})\)), but has image \(0\) in \(C^2({\mathscr U}, {\mathscr F}'')\), and hence comes from an element \((t_{ij})_{i,j}\in C^2({\mathscr U}, {\mathscr F}')\). Then \((t_{ij})_{i,j}\) has image \(0\) in \(C^3({\mathscr U}, {\mathscr F}')\) (because that is obviously true in \(C^3({\mathscr U}, {\mathscr F})\) and the morphism \({\mathscr F}'\to {\mathscr F}\) is injective), and so gives rise to a class in \(\check{H}^1({\mathscr U}, {\mathscr F}')\). Its image in \(\check{H}^1(X, {\mathscr F}')\) is the image of \(s\) under \(\delta \). One checks that this procedure is independent of choices and gives rise to the exact sequence in the proposition.

**(6.13) Comparison of cohomology and Čech cohomology**

In degrees \(0\) and \(1\), cohomology and Čech cohomology coincide. For degree \(0\), we have already shown this, so we proceed to the case of degree \(1\). We start with some preparations.

We define a sheaf version of the Čech complex as follows:

with differentials defined by (basically) the same formula as above. Here \(j_{\underline{i}}\) denotes the inclusion \(U_{\underline{i}}\hookrightarrow X\).

We have a natural map \({\mathscr F}\to {\mathscr C}^0({\mathscr U}, {\mathscr F})\), which on an open \(V\) is given by \(s\mapsto (s_{|U_i\cap V})_i\).

**Proposition 6.34**

The exactness can be checked on stalks, and one can show that for each point of \(X\), the stalks of the above complex form a complex that is homotopy equivalent to \(0\). We omit the details.

**Proposition 6.35**

It is not hard to check that the sheaves \({\mathscr C}^p({\mathscr U}, {\mathscr F})\) are flasque, since all the constructions involved preserve flasqueness.

We then obtain

where the second equality holds since \({\mathscr C}^\bullet ({\mathscr U}, {\mathscr F})\) is a flasque resolution of \({\mathscr F}\) by the above, and the third one follows since \({\mathscr F}\) itself is flasque.

**Proposition 6.36**

Let \({\mathscr U}\) be an open cover of \(X\). For every \(i\), there is a natural map \(\check{H}^i({\mathscr U}, {\mathscr F})\to H^i(X, {\mathscr F})\).

These maps are compatible with refinements, so we obtain a natural map \(\check{H}^i(X, {\mathscr F})\to H^i(X, {\mathscr F})\). These maps are functorial in \({\mathscr F}\).

For \(i=0, 1\), the natural map \(\check{H}^i(X, {\mathscr F})\to H^i(X, {\mathscr F})\) is an isomorphism.

Part (1) follows from Proposition 6.17 applied to the resolution \({\mathscr C}^\bullet ({\mathscr U}, {\mathscr F})\) of \({\mathscr F}\) and any injective resolution. We omit the proof of Part (2).

For Part (3), it remains to consider the case \(i=1\). Embed \({\mathscr F}\) into a flasque sheaf \({\mathscr G}\). We obtain short exact sequences

by Proposition 6.33 Proposition 6.35 and

since flasque sheaves are \(\Gamma (X, -)\)-acyclic. The statement follows from this.

One can also show that the natural map \(\check{H}^2(X, {\mathscr F})\to H^2(X, {\mathscr F})\) is always injective.

The following result will allow us to compute cohomology groups of separated schemes with coefficients in quasi-coherent modules as Čech cohomology.

**Theorem 6.37**

we have \(H^i(U, {\mathscr F}) = 0\) for all \(U\in {\mathscr B}\) and \(i{\gt}0\),

The natural homomorphisms \(\check{H}^i({\mathscr U},{\mathscr F})\to H^i(X, {\mathscr F})\) are isomorphisms for all \(i\ge 0\) and all covers \({\mathscr U}\) of \(X\) consisting of elements of \({\mathscr B}\).

The natural homomorphisms \(\check{H}^i(X,{\mathscr F})\to H^i(X, {\mathscr F})\) are isomorphisms for all \(i\ge 0\).

See e.g., [ Go ] II Thm. 5.9.2, [ Stacks ] 01EO or [ GW2 ] Section (21.17).

## Cohomology of affine schemes

General references: [ H ] Ch. III, [ Stacks ] , [ GW2 ] Chapter 22.

June 28,

2023

**(6.14) Vanishing of cohomology of quasi-coherent sheaves on affine schemes**

**Theorem 6.38**

It is enough to show that \(\check{H}^i({\mathscr U}, {\mathscr F}) = 0\) for all covers \({\mathscr U}\) of \(X\) by principal open subsets. This can be proved using “direct computation” (see [ GW1 ] Lemma 12.33), or can be viewed as a consequence of the theory of “faithfully flat descent” (specifically Problem 37, see also [ GW1 ] Lemma 14.64).

From this theorem, it follows immediately (using the above results) that \(H^1(X, {\mathscr F})=0\) for \(X\) affine and \({\mathscr F}\) quasi-coherent. In particular, the global section functor on \(X\) preserves exactness of every short exact sequence where the left hand term is a quasi-coherent \({\mathscr O}_X\)-module. But using Cartan’s Theorem, Theorem 6.37, we get more:

**Theorem 6.39**

**Theorem 6.40**

**Remark 6.41**

For \(X\) noetherian, the use of Cartan’s Theorem can be avoided by using the result (see [ H ] III.3) that for a noetherian ring \(A\), \(X=\operatorname{Spec}A\), and \(I\) an injective \(A\)-module, the \({\mathscr O}_X\)-module \(\widetilde{I}\) is a flasque \({\mathscr O}_X\)-module.

Together with the fact that for an affine scheme \(X\) the global section functor is exact on the category of quasi-coherent \({\mathscr O}_X\)-modules, this gives the vanishing of \(H^i(X, {\mathscr F})\) for \(X\) affine, \({\mathscr F}\) quasi-coherent and \(i {\gt} 0\).

It also implies that any quasi-coherent \({\mathscr O}_X\)-module on a noetherian scheme can be embedded into a flasque *quasi-coherent* sheaf. From this one can prove Theorem 6.40.

See [ H ] III.3, Theorem III.4.5; cf. also [ GW2 ] Section (22.18).

**Corollary 6.42**

**Lemma 6.43**

Let \(X\) be a scheme. For \(f\in \Gamma (X, {\mathscr O}_X)\) write

an open subset of \(X\) which we consider as an open subscheme.

If there exist \(f_1, \dots , f_n\in \Gamma (X, {\mathscr O}_X)\) such that \(X_{f_i}\) is affine for \(i=1,\dots , n\) and such that \(f_1, \dots , f_n\) generate the unit ideal in the ring \(\Gamma (X, {\mathscr O}_X)\), then \(X\) is affine.

**Theorem 6.44**

The scheme \(X\) is affine.

For every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and every \(i {\gt} 0\), \(H^i(X, {\mathscr F})=0\).

For every quasi-coherent ideal sheaf \({\mathscr I}\subseteq {\mathscr O}_X\), \(H^1(X, {\mathscr I})=0\).

## Cohomology of projective schemes

**(6.15) The cohomology of line bundles on projective space**

References: [ H ] III.5, [ GW2 ] Section (22.6), [ Stacks ] 01XS.

July 3,

2023

Using Čech cohomology, we can compute the cohomology of line bundles on projective space. It is best to aggregate the results for all \({\mathscr O}(d)\), as we have already seen for their global sections, a result which we repeat as the first statement below.

**Theorem 6.45**

There is a natural isomorphism \(S \cong \bigoplus _{d\in \mathbb {Z}} H^0(X, {\mathscr O}(d))\).

For \(i\ne 0, n\) and all \(d\in \mathbb {Z}\) we have \(H^i(X, {\mathscr O}(d)) = 0\).

There is a natural isomorphism \(H^n(X, {\mathscr O}(-n-1)) \cong A\).

For every \(d\), there is a perfect pairing

\[ H^0(X, {\mathscr O}(d)) \times H^n(X, {\mathscr O}(-d-n-1)) \to H^n(X, {\mathscr O}(-n-1))\cong A, \]i.e., a bilinear map which induces isomorphisms

\[ H^0(X, {\mathscr O}(d)) \cong H^n(X, {\mathscr O}(-d-n-1))^\vee \]and

\[ H^0(X, {\mathscr O}(d))^\vee \cong H^n(X, {\mathscr O}(-d-n-1)) \](where \(-^\vee = \operatorname{Hom}_A(-, A)\) denotes the \(A\)-module dual).

We compute the cohomology groups as Čech cohomology groups for the standard cover \({\mathscr U}= (D_+(T_i))_i\) of \(\mathbb {P}^n_A\). It simplifies the reasoning to do the computation for all \({\mathscr O}(d)\) at once, i.e., to compute the cohomology groups of \({\mathscr F}:=\bigoplus _{d\in \mathbb {Z}} {\mathscr O}(d)\) (and to – implicitly – keep track of the grading by \(d\)). Since cohomology is compatible with direct sums (cf. Proposition 6.29; we proved that for noetherian schemes, but it holds more generally for quasi-compact separated schemes, cf. [ GW2 ] Corollary 21.56), this also gives the result for the individual \({\mathscr O}(d)\).

The Čech complex \(C^\bullet ({\mathscr U}, {\mathscr F})\) is

(with non-zero entries in degrees \(0, \dots , n\)), cf. the computation of the global sections of the sheaves \({\mathscr O}(d)\), Proposition 3.19. Also note that that proposition proves Part (1) of the theorem here. From this, we see that we can identify

This easily implies Parts (3) and (4).

It remains to prove the vanishing statement of Part (2) for \(0 {\lt} i {\lt} n\). We do this by induction on \(n\). Note that in view of Part (1), all the cohomology groups \(H^i(\mathbb {P}^n_A, {\mathscr F})\) carry a natural \(S\)-module structure.

First note that for the localization we have \(H^i(\mathbb {P}^n_A, {\mathscr F})_{T_n} = 0\). In fact, this localization is the \(i\)-th cohomology group of the localized Čech complex \(C^\bullet ({\mathscr U}, {\mathscr F})_{T_n}\) which computes the cohomology \(H^\bullet (D_+(T_n), {\mathscr F}_{|D_+(T_n)})\) which vanishes in positive degrees. Therefore it suffices to show that multiplication by \(T_n\) is a bijection \(H^i(\mathbb {P}^n_A, {\mathscr F})\to H^i(\mathbb {P}^n_A, {\mathscr F})\) for all \(0{\lt}i{\lt}n\).

The global section \(T_n\in H^0(\mathbb {P}^n_A, {\mathscr O}(1))\) gives rise to a short exact sequence

and tensoring with the locally free module \({\mathscr F}\) we obtain a short exact sequence

where \({\mathscr F}' = {\mathscr F}\otimes _{{\mathscr O}_{\mathbb {P}^n_A}}{\mathscr O}_{V_+(T_n)}\) is (the push-forward from \(V_+(T_n) \cong \mathbb {P}^{n-1}_A\) to \(\mathbb {P}^n_A\)) of the sheaf analogous to \({\mathscr F}\) on \(\mathbb {P}^{n-1}_A\).

Multiplication by \(T_n\in \Gamma (\mathbb {P}^n_A, {\mathscr O}(1))\) gives an isomorphism \({\mathscr F}\otimes {\mathscr O}(-1)\to {\mathscr F}\). Therefore the long exact cohomology sequence attached to the above short exact sequence can be written as

where the maps \(H^i(\mathbb {P}^n_A, {\mathscr F}) \to H^i(\mathbb {P}^n_A, {\mathscr F})\) are given by multiplication by \(T_n\) (i.e., what we want to show is that these maps are isomorphisms). The induction hypothesis together with Lemma 6.28 and the observations that the map \(H^0(\mathbb {P}^n_A, {\mathscr F})\to H^0(\mathbb {P}^{n-1}_A, {\mathscr F}')\) is surjective (cf. Part (1)) and that the map \(H^{n-1}(\mathbb {P}^{n-1}_A, {\mathscr F}')\to H^n(\mathbb {P}^n_A,{\mathscr F})\) is injective, then allow us to conclude. For the injectivity cf. the above proof for Part (3). Looking at individual graded pieces of \(H^{n-1}(\mathbb {P}^{n-1}_A, {\mathscr F}')\) and the kernel of \(H^n(\mathbb {P}^n_A,{\mathscr F})\to H^n(\mathbb {P}^n_A,{\mathscr F})\), we have a surjective homomorphism of free \(A\)-modules of the same rank which is necessarily an isomorphism. (The kernel of \(H^n(\mathbb {P}^n_A,{\mathscr F})\to H^n(\mathbb {P}^n_A,{\mathscr F})\) is the free \(A\)-module spanned by all monomials of the form \(T_0^{i_0}\cdots T_{n-1}^{i_{n-1}}T_n^{-1}\) with all \(i_\nu {\lt} 0\).)

**Remark 6.46**

**(6.16) Finiteness of cohomology of coherent \({\mathscr O}_X\)-modules on projective schemes**

References: [ H ] III.5; [ GW2 ] Sections (23.1), (23.2).

July 5,

2023

**Definition 6.47**

*coherent*, if it is quasi-coherent and of finite type.

Let \(A\) be a noetherian ring. For an \({\mathscr O}_{\mathbb {P}^n_A}\)-module \({\mathscr F}\), we write \({\mathscr F}(d) := {\mathscr F}\otimes _{{\mathscr O}_{\mathbb {P}^n_A}}{\mathscr O}(d)\). We need the following general lemma.

**Lemma 6.48**

Let \(t \in \Gamma (X,{\mathscr F})\) be a global section such that \(t{}_{\vert }{}_{X_s} = 0\). Then there exists an integer \(n {\gt} 0\) such that \(t \otimes s^{\otimes n} = 0 \in \Gamma (X,{\mathscr F}\otimes {\mathscr L}^{\otimes n})\).

For every section \(t' \in \Gamma (X_s,{\mathscr F})\) there exist \(n {\gt} 0\) and a section \(t \in \Gamma (X,{\mathscr F}\otimes {\mathscr L}^{\otimes n})\) such that \(t{}_{\vert }{}_{X_s} = t' \otimes s^{\otimes n}\).

If \(X\) is affine and \({\mathscr L}= {\mathscr O}_X\), then this follows immediately from our results on quasi-coherent \({\mathscr O}_X\)-modules; namely we know that then \(\Gamma (X_s,{\mathscr F}) = \Gamma (X, {\mathscr F})_s\). For the general case, let \(X=\bigcup _i U_i\) be a finite affine open cover such that \({\mathscr L}_{|U_i}\cong {\mathscr O}_{U_i}\) for all \(i\) (and fix such isomorphisms). Then (1) can be checked on each \(U_i\) individually and thus follows from what was said in the beginning. To prove Part (2), we construct \(t\) by considering sections \(t_i\) on the \(U_i\) obtained from the restrictions \(t_{|U_i\cap X_s}\), using the result in the affine case. The \(t_i\) may not glue, but applying Part (1) to the intersections \(U_i\cap U_j\) and the elements \(t_{i|U_i\cap U_j}-t_{j|U_i\cap U_j}\) we find that for \(n\) sufficiently large, the \(t_i\otimes s^{\otimes n}\) will glue to a section of \({\mathscr F}\otimes {\mathscr L}^{\otimes n}\). See [ GW1 ] Theorem 7.22 or [ H ] Lemma II.5.14 for more details.

**Proposition 6.49**

There exist integers \(d_1, \dots , d_s\) and a surjective \({\mathscr O}_X\)-module homomorphism

\[ \bigoplus _{i=1}^n {\mathscr O}(d_i)\twoheadrightarrow {\mathscr F}. \]For \(d\) sufficiently large, the \({\mathscr O}_{\mathbb {P}^n_A}\)-module \({\mathscr F}(d)\) is globally generated, i.e., there exist \(N\ge 0\) and a surjective homomorphism \({\mathscr O}^N_{\mathbb {P}^n_A}\to {\mathscr F}(d)\).

It is easy to see that (1) and (2) are equivalent. We prove (2). For \(i\in \{ 0,\dots , n\} \), \({\mathscr F}_{|D_+(T_i)}\) has the form \(\widetilde{M_i}\) for some finitely generated \(A[\frac{T_0}{T_i},\cdots , \frac{T_n}{T_i}]\)-module \(M_i\). For any \(s\in M_i = \Gamma (D_+(T_i), {\mathscr F})\), for \(d\) sufficiently large, \(X_i^ds\) extends to a global section of \({\mathscr F}(d)\), by Lemma 6.48. This implies the claim.

**Theorem 6.50**

Using Lemma 6.28, we reduce to the case that \(X=\mathbb {P}^n_A\). By Proposition 6.49, the vanishing result Corollary 6.42 and descending induction, we may further reduce to the case that \({\mathscr F}\) is a finite direct sum of sheaves of the form \({\mathscr O}(d)\), but for these we already know the result.

At this point it is not hard to prove that higher derived images \(R^if_*{\mathscr F}\) of a coherent \({\mathscr O}_X\)-module under a projective morphism \(f\colon X\to Y\) are coherent (see [ H ] III.8).

Another useful result is the following vanishing statement.

**Proposition 6.51**

Using the *projection formula* (see below), one reduces to the case \(X=\mathbb {P}^n_A\). It is then clear from the above, that the statement holds whenever \({\mathscr F}\) is a direct sum of line bundles \({\mathscr O}(d)\). The general statement follows from this by descending induction, similarly as above.

**Proposition 6.52**

\({\mathscr G}\) is a locally free \({\mathscr O}_Y\)-module,

\(f\) is a closed immersion,

\({\mathscr F}\) and \({\mathscr G}\) are quasi-coherent, and for every affine open \(V\subseteq Y\), \(f^{-1}(V)\) is affine,

\({\mathscr F}\) and \({\mathscr G}\) are quasi-coherent, \(f\) is quasi-compact and separated, and \({\mathscr G}\) is a flat \({\mathscr O}_Y\)-module.

The homomorphism is obtained formally using the adjunction between \(f^*\) and \(f_*\). The statement that it is an isomorphism is local on \(Y\). Thus for (1), we may assume that \({\mathscr G}\) is free, and then the claim is easy to check. In situation (2) one can check that the homomorphism induces an isomorphism on each stalk, and hence is an isomorphism. Under the assumptions in (3), one may assume that \(Y\) and \(X\) are affine and the claim then follows easily from the description of pushforward and pullback in this case (Proposition 2.15). For further details, and for an argument to prove the statement in case (4), see [ GW2 ] Proposition 22.80.

## Serre duality

July 10,

2023

**(6.17) The Theorem of Riemann–Roch revisited**

References: [ H ] III.7, IV.1; [ GW2 ] Chapters 25, 26.

Recall the Theorem of Riemann–Roch that we stated above (Theorem 3.13). In this section, we prove a preliminary version, which also gives a more conceptual view on the “error term” \(\dim \Gamma (X, {\mathscr O}(K-D))\) (with notation as above).

Let \(k\) be an algebraically closed field.

**Definition 6.53**

*Euler characteristic*of \({\mathscr F}\).

Note that the sum is finite (by the Grothendieck vanishing theorem, Theorem 6.30) and that each term is finite by the results of the previous section.

Now let \(X/k\) be a smooth, projective, connected curve. Then \(\chi ({\mathscr F}) = \dim _k H^0(X, {\mathscr F}) - \dim _k H^1(X, {\mathscr F})\).

The following theorem is the preliminary version of the Theorem of Riemann–Roch mentioned above.

**Theorem 6.54**

The statement is clear for \({\mathscr L}= {\mathscr O}_X\). Since every line bundle is isomorphic to the line bundle attached to a Weil divisor, it is therefore enough to prove that for every closed point \(x\in X\), and every line bundle \({\mathscr L}\) on \(X\), we have

We write \({\mathscr L}(D):={\mathscr L}\otimes _{{\mathscr O}_X}{\mathscr O}_X(D)\) for any divisor \(D\). The short exact sequence

remains exact after tensoring with \({\mathscr L}\), so we obtain a short exact sequence

Since the Euler characteristic is additive in short exact sequences (use the long exact cohomology sequence) and since \(\chi (\kappa (x)) = 1\), the claim follows.

Now we can define the genus of \(X\) as \(g:= 1-\chi ({\mathscr O}_X) = \dim _k H^1(X, {\mathscr O}_X)\). From the above, we immediately get

**Corollary 6.55**(Theorem of Riemann)

Furthermore, the Theorem of Riemann–Roch, Theorem 3.13, follows from the above result on Euler characteristics and the Serre duality theorem (which however we cannot prove in this class).

**Theorem 6.56**(Serre duality)

*smooth*to mean that all local rings \({\mathscr O}_{X,x}\) are regular; in fact it is enough to assume that \(X\) is Cohen-Macaulay, i.e., that all local rings of \(X\) are Cohen-Macaulay rings.) There is a unique (up to isomorphism) coherent \({\mathscr O}_X\)-module \(\omega \), the so-called

*dualizing sheaf*, such that for every locally free \({\mathscr O}_X\)-module \({\mathscr E}\) of finite rank on \(X\), there is a natural isomorphism

July 12,

2023

If \(X\) is smooth over \(k\), then the dualizing sheaf is a line bundle and coincides with the so-called *canonical bundle*, the top exterior power of the sheaf \(\Omega ^1_{X/k}\) of differentials of \(X\) over \(k\), see Chapter 4.

A fairly elementary approach in the case where \(X\) is projective is to prove a similar duality theorem for the cohomology of line bundles on projective space \(\mathbb {P}^n_k\) (this we have basically done above, Theorem 6.45 (4), with \(\omega = {\mathscr O}(-n-1)\)), and then to derive the statement for (certain) closed subschemes of projective space; see [ H ] III.7. A more general (but technically more sophisticated) approach is to derive this duality result from the existence of a right adjoint functor \(f^\times \) of the derived push-forward functor \(Rf_*\colon D_{\rm qcoh}(X)\to D_{\rm qcoh}(\operatorname{Spec}k)\), where \(f\colon X\to \operatorname{Spec}k\) is the structure morphism and \(D_{\rm qcoh}(X)\) denotes the full triangulated subcategory of the derived category \(D(X)\) of the category of \({\mathscr O}_X\)-modules (and this generalizes to the case of arbitrary proper morphisms \(f\) between noetherian schemes). In fact, one shows that for \(X\) Cohen-Macaulay and equidimensional of dimension \(n\) the complex \(f^\times k\) is concentrated in degree \(-n\), and we denote by \(\omega \) the unique non-vanishing cohomology object. For every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr F}\) and integer \(i\) one then has

and if \({\mathscr F}\) is locally free of finite rank, then we can identify the right hand side with

This approach also gives a result (in terms of the derived category) without requiring that \(X\) be Cohen-Macaulay, and also for (proper) morphisms \(f\colon X\to Y\) where \(Y\) is not necessarily the spectrum of a field. See [ GW2 ] Chapter 25 for more on this and for further references.

As another application of Serre duality, we mention the following result. The first part is named the Lemma of Enriques-Severi-Zariski. To name an application, we note that this is one of the ingredients of the proof that every regular proper surface over a field is projective (see [ GW2 ] Theorem 25.151).

The result still holds (but in Part (1) only for \(H^1\), as stated, while for \(X\) Cohen-Macaulay one has an analogous result for all \(H^i\), \(i{\lt}n\)) if the assumption that \(X\) be Cohen-Macaulay is replaced by the assumption that \(X\) is normal (i.e., for every non-empty affine open \(U\subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is integrally closed in its field of fractions \(K(X)\)). On the other hand it is clear that the assumption that \(X\) has dimension at least \(2\) cannot be dropped.

**Theorem 6.57**

Let \(K\) be an (algebraically closed) field, let \(X\) be an integral Cohen-Macaulay (e.g., smooth) projective \(k\)-scheme and let \(\iota \colon X\to \mathbb {P}^n_k\) be a closed immersion of \(k\)-schemes.

Assume that \(\dim X \ge 2\).

Fix \(d {\gt} 0\) and let \({\mathscr L}:= \iota ^*{\mathscr O}_{\mathbb {P}^n_k}(d)\). Let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. We write \({\mathscr F}(n):={\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr L}^{\otimes n}\), \(n\in \mathbb {Z}\). Then for \(n \ll 0\), \(H^1(X, {\mathscr F}(n)) = 0\).

Let \(f\in k[T_0, \dots , T_n]\) be a non-constant homogeneous polynomial, and let \(Z := V_+(f)\), a closed subscheme of \(\mathbb {P}^n_k\). Then \(X\cap Z\) is connected.

Part (1) follows from Serre duality (Theorem 6.56) and the vanishing result of Proposition 6.51. To prove Part (2), let \(d\) denote the degree of \(f\). We view \(Z\) as an effective Cartier divisor. Then \({\mathscr O}_{\mathbb {P}^n_k}(Z)\cong {\mathscr O}(d)\). We set \({\mathscr L}= \iota ^*{\mathscr O}(d)\) and \({\mathscr F}= {\mathscr O}_X\) and apply Part (1) to find \(n\) such that \(H^1(X, {\mathscr L}^{-n}) = 0\). Let \(Z_n\) be the Cartier divisor \(n\cdot Z\) (with associated line bundle \(\cong {\mathscr O}(dn)\)). We then have a short exact sequence

where \(X\cap Z_n\) denotes the scheme-theoretic intersection of \(X\) and \(Z\), i.e., \({\mathscr O}_{X\cap Z_n} = \iota ^*{\mathscr O}_{Z_n}\). The underlying topological space of \(X\cap Z_n\) is independent of \(n\ge 1\) and equals the set-theoretic intersection \(X\cap Z\). It is therefore sufficient to show that the scheme \(X\cap Z_n\) is connected. But the above short exact sequence induces, in view of the vanishing of \(H^1(X, {\mathscr L}^{-n})\) given by Part (1), a surjective homomorphism \(\Gamma (X, {\mathscr O}_X)\to \Gamma (X, {\mathscr O}_{X\cap Z_n})\) of \(k\)-vector spaces. Since \(\Gamma (X, {\mathscr O}_X) = k\), it follows that \(\Gamma (X, {\mathscr O}_{X\cap Z_n}) = k\), as well, and in particular \(X\cap Z_n\) is connected. See also [ GW2 ] Section (25.28), [ H ] Section III.7.