30.2 Čech cohomology of quasi-coherent sheaves

Let $X$ be a scheme. Let $U \subset X$ be an affine open. Recall that a standard open covering of $U$ is a covering of the form $\mathcal{U} : U = \bigcup _{i = 1}^ n D(f_ i)$ where $f_1, \ldots , f_ n \in \Gamma (U, \mathcal{O}_ X)$ generate the unit ideal, see Schemes, Definition 26.5.2.

Lemma 30.2.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{U} : U = \bigcup _{i = 1}^ n D(f_ i)$ be a standard open covering of an affine open of $X$. Then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$.

Proof. Write $U = \mathop{\mathrm{Spec}}(A)$ for some ring $A$. In other words, $f_1, \ldots , f_ n$ are elements of $A$ which generate the unit ideal of $A$. Write $\mathcal{F}|_ U = \widetilde{M}$ for some $A$-module $M$. Clearly the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is identified with the complex

$\prod \nolimits _{i_0} M_{f_{i_0}} \to \prod \nolimits _{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod \nolimits _{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots$

We are asked to show that the extended complex

30.2.1.1
$$\label{coherent-equation-extended} 0 \to M \to \prod \nolimits _{i_0} M_{f_{i_0}} \to \prod \nolimits _{i_0i_1} M_{f_{i_0}f_{i_1}} \to \prod \nolimits _{i_0i_1i_2} M_{f_{i_0}f_{i_1}f_{i_2}} \to \ldots$$

(whose truncation we have studied in Algebra, Lemma 10.24.1) is exact. It suffices to show that (30.2.1.1) is exact after localizing at a prime $\mathfrak p$, see Algebra, Lemma 10.23.1. In fact we will show that the extended complex localized at $\mathfrak p$ is homotopic to zero.

There exists an index $i$ such that $f_ i \not\in \mathfrak p$. Choose and fix such an element $i_{\text{fix}}$. Note that $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$. Similarly for a localization at a product $f_{i_0} \ldots f_{i_ p}$ and $\mathfrak p$ we can drop any $f_{i_ j}$ for which $i_ j = i_{\text{fix}}$. Let us define a homotopy

$h : \prod \nolimits _{i_0 \ldots i_{p + 1}} M_{f_{i_0} \ldots f_{i_{p + 1}}, \mathfrak p} \longrightarrow \prod \nolimits _{i_0 \ldots i_ p} M_{f_{i_0} \ldots f_{i_ p}, \mathfrak p}$

by the rule

$h(s)_{i_0 \ldots i_ p} = s_{i_{\text{fix}} i_0 \ldots i_ p}$

(This is “dual” to the homotopy in the proof of Cohomology, Lemma 20.10.4.) In other words, $h : \prod _{i_0} M_{f_{i_0}, \mathfrak p} \to M_\mathfrak p$ is projection onto the factor $M_{f_{i_{\text{fix}}}, \mathfrak p} = M_{\mathfrak p}$ and in general the map $h$ equal projection onto the factors $M_{f_{i_{\text{fix}}} f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p} = M_{f_{i_1} \ldots f_{i_{p + 1}}, \mathfrak p}$. We compute

\begin{align*} (dh + hd)(s)_{i_0 \ldots i_ p} & = \sum \nolimits _{j = 0}^ p (-1)^ j h(s)_{i_0 \ldots \hat i_ j \ldots i_ p} + d(s)_{i_{\text{fix}} i_0 \ldots i_ p}\\ & = \sum \nolimits _{j = 0}^ p (-1)^ j s_{i_{\text{fix}} i_0 \ldots \hat i_ j \ldots i_ p} + s_{i_0 \ldots i_ p} + \sum \nolimits _{j = 0}^ p (-1)^{j + 1} s_{i_{\text{fix}} i_0 \ldots \hat i_ j \ldots i_ p} \\ & = s_{i_0 \ldots i_ p} \end{align*}

This proves the identity map is homotopic to zero as desired. $\square$

The following lemma says in particular that for any affine scheme $X$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$ we have

$H^ p(X, \mathcal{F}) = 0$

for all $p > 0$.

Lemma 30.2.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. For any affine open $U \subset X$ we have $H^ p(U, \mathcal{F}) = 0$ for all $p > 0$.

Proof. We are going to apply Cohomology, Lemma 20.11.9. As our basis $\mathcal{B}$ for the topology of $X$ we are going to use the affine opens of $X$. As our set $\text{Cov}$ of open coverings we are going to use the standard open coverings of affine opens of $X$. Next we check that conditions (1), (2) and (3) of Cohomology, Lemma 20.11.9 hold. Note that the intersection of standard opens in an affine is another standard open. Hence property (1) holds. The coverings form a cofinal system of open coverings of any element of $\mathcal{B}$, see Schemes, Lemma 26.5.1. Hence (2) holds. Finally, condition (3) of the lemma follows from Lemma 30.2.1. $\square$

Here is a relative version of the vanishing of cohomology of quasi-coherent sheaves on affines.

Lemma 30.2.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $f$ is affine then $R^ if_*\mathcal{F} = 0$ for all $i > 0$.

Proof. According to Cohomology, Lemma 20.7.3 the sheaf $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto H^ i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)})$. By assumption, whenever $V$ is affine we have that $f^{-1}(V)$ is affine, see Morphisms, Definition 29.11.1. By Lemma 30.2.2 we conclude that $H^ i(f^{-1}(V), \mathcal{F}|_{f^{-1}(V)}) = 0$ whenever $V$ is affine. Since $S$ has a basis consisting of affine opens we win. $\square$

Lemma 30.2.4. Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $H^ i(X, \mathcal{F}) = H^ i(S, f_*\mathcal{F})$ for all $i \geq 0$.

Proof. Follows from Lemma 30.2.3 and the Leray spectral sequence. See Cohomology, Lemma 20.13.6. $\square$

The following two lemmas explain when Čech cohomology can be used to compute cohomology of quasi-coherent modules.

Lemma 30.2.5. Let $X$ be a scheme. The following are equivalent

1. $X$ has affine diagonal $\Delta : X \to X \times X$,

2. for $U, V \subset X$ affine open, the intersection $U \cap V$ is affine, and

3. there exists an open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ such that $U_{i_0 \ldots i_ p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots , i_ p \in I$.

In particular this holds if $X$ is separated.

Proof. Assume $X$ has affine diagonal. Let $U, V \subset X$ be affine opens. Then $U \cap V = \Delta ^{-1}(U \times V)$ is affine. Thus (2) holds. It is immediate that (2) implies (3). Conversely, if there is a covering of $X$ as in (3), then $X \times X = \bigcup U_ i \times U_{i'}$ is an affine open covering, and we see that $\Delta ^{-1}(U_ i \times U_{i'}) = U_ i \cap U_{i'}$ is affine. Then $\Delta$ is an affine morphism by Morphisms, Lemma 29.11.3. The final assertion follows from Schemes, Lemma 26.21.7. $\square$

Lemma 30.2.6. Let $X$ be a scheme. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering such that $U_{i_0 \ldots i_ p}$ is affine open for all $p \ge 0$ and all $i_0, \ldots , i_ p \in I$. In this case for any quasi-coherent sheaf $\mathcal{F}$ we have

$\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(X, \mathcal{F})$

as $\Gamma (X, \mathcal{O}_ X)$-modules for all $p$.

Proof. In view of Lemma 30.2.2 this is a special case of Cohomology, Lemma 20.11.6. $\square$

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