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.
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.
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
\begin{equation} \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 \end{equation}
(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.slogan 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
X has affine diagonal \Delta : X \to X \times X,
for U, V \subset X affine open, the intersection U \cap V is affine, and
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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