# 4 Cohomology of ${\mathscr O}_X$-modules

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

June 19,

2019

## 4.1 The formalism of derived functors

**(4.1) Complexes in abelian categories**

Reference: [ We ] Ch. 1.

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 $X$).

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 4.1**

*$i$-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 4.2**

**Proposition 4.3**

*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. Reference: [ We ] 1.4.

**Definition 4.4**

*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 4.5**

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 4.6**

*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 $i$.

June 24,

2019

**(4.2) 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 4.7**

*left exact*, if for every short exact sequence $0\to A’\to A\to A”\to 0$ in ${\mathcal A}$, the sequence

**Definition 4.8**

*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.

**(4.3) $\delta $-functors**

Reference: [ We ] 2.1.

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

**Definition 4.9**

*$\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 4.10**

*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 $i > 0$, 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 4.11**

*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 4.12**

Let $(T^i)_i$ be a $\delta $-functor from ${\mathcal A}$ to ${\mathcal B}$ such that for every $i>0$, the functor $T^i$ is effaceable. Then $(T^i)_i$ is a universal $\delta $-functor.

Reference: [ We ] Thm. 2.4.7, Ex. 2.4.5.

**(4.4) Injective objects**

Let ${\mathcal A}$ be an abelian category.

**Definition 4.13**

*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 4.14**

*injective resolution*of $X$ is an exact sequence

**Definition 4.15**

*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.

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.

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.

**(4.5) Right derived functors**

June 26,

2019

**Theorem 4.16**

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 $i>0$.

The family $(R^i F)_i$ is a universal $\delta $-functor.

We call the $R^iF$ the *right derived functors* of $F$.

**Definition 4.17**

*$F$-acyclic*, if $R^iF(A) = 0$ for all $i>0$.

**Definition 4.18**

*$F$-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 4.19**

## 4.2 Cohomology of sheaves

General reference: [ H ] Ch. III, [ Stacks ] Ch. 20, 29.

**(4.6) 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 4.20**

July 1,

2019

**(4.7) Flasque sheaves**

**Definition 4.21**

*flasque*(or

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

**Lemma 4.22**

**Proposition 4.23**

**Corollary 4.24**

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.

**(4.8) Grothendieck vanishing**

Reference: [ H ] III.2.

**Lemma 4.25**

**Theorem 4.26**(Grothendieck)

July 3,

2019

## 4.3 Čech cohomology

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

**(4.9) 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 ${\mathscr F}$*.

**Definition 4.27**

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

Since ${\mathscr F}$ is a sheaf, we have $\check{H}^0({\mathscr U}, {\mathscr F}) = \Gamma (X, {\mathscr F}) = H^0(X, {\mathscr F})$.

**(4.10) 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 $I$.

**(4.11) Passing to refinements**

**Definition 4.28**

*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 ${\mathscr F}$*.

**Proposition 4.29**

**(4.12) A sheaf version of the Čech complex**

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 4.30**

**Proposition 4.31**

**Proposition 4.32**

**Proposition 4.33**

July 8,

2019

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

**Theorem 4.34**

From this theorem, if 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 more is true:

**Theorem 4.35**

This follows from the above using Cartan’s Theorem (see e.g., [ Go ] II Thm. 5.9.2, [ Stacks ] 01EO):

**Theorem 4.36**

we have $H^i(U, {\mathscr F}) = 0$ for all $i>0$,

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 $i\ge 0$.

For $X$ noetherian, there is another approach which relies on the following result (see [ H ] III.3):

**Proposition 4.37**

From either approach, we also obtain the following consequence (of course, the second approach again works only in the noetherian situation):

**Theorem 4.38**

**Corollary 4.39**

July 10,

2019

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

References: [ H ] III.5, [ Stacks ] 01XS.

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 4.40**

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 $r$, there is a perfect pairing

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

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

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

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

**Definition 4.41**

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

Let $A$ be a noetherian ring.

**Lemma 4.42**

**Theorem 4.43**

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).

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

Reference: [ H ] III.7, IV.1.

Recall the Theorem of Riemann–Roch that we stated above (Thm. 2.9). 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 4.44**

*Euler characteristic*of ${\mathscr F}$.

Note that the sum is finite (by the Grothendieck vanishing theorem, Theorem 4.26) 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 4.45**

Now we can define the genus of $X$ as $g:= 1-\chi ({\mathscr O}_X) = \dim _k H^1(X, {\mathscr O}_X)$, and choose for $K$ a divisor with ${\mathscr O}(K) \cong \Omega ^1_{X/k}$.

From the above, we immediately get

**Corollary 4.46**(Theorem of Riemann)

Furthermore, the Theorem of Riemann–Roch will follow from the Serre duality theorem (which will be discussed in the sequel to this course, Algebraic Geometry 3).

**Theorem 4.47**(Serre duality)

A similar statement holds for every locally free sheaf ${\mathscr L}$ (and the theorem can be vastly further generalized).