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Čech complex for a two-open cover

Statement

Let $X$ be a topological space, let $F$ be a sheaf of abelian groups on $X$ and let $U, V \subseteq X$ be open subsets with $X = U \cup V$ (A sheaf on a topological space). Index the two-member family by $I : = {0, 1}$ with $0 < 1$ and put $U_{0} : = U$ , $U_{1} : = V$ , so that $U = (U_{0}, U_{1})$ is an open cover of $X$ indexed by a linearly ordered set with ordered Čech cochains $C^{∙} (U, F)$ and differential $δ^{∙}$ (Ordered Čech cochain complex of a cover). Then $C^{0} (U, F) = F (U) \oplus F (V), C^{1} (U, F) = F (U \cap V), C^{p} (U, F) = 0 (p \geq 2),$ the only possibly nonzero component of the differential is $δ^{0} : F (U) \oplus F (V) ⟶ F (U \cap V), δ^{0} (s_{U}, s_{V}) = s_{V} ∣_{U \cap V} - s_{U} ∣_{U \cap V},$ and $δ^{1} = 0$ . Consequently, with ${\overset{ˇ}{H}}^{p} : = {\overset{ˇ}{H}}^{p} (U, F)$ the Čech cohomology of the cover (Fixed-cover Čech cohomology), ${\overset{ˇ}{H}}^{0} = \ker δ^{0}, {\overset{ˇ}{H}}^{1} = F (U \cap V) / im δ^{0}, {\overset{ˇ}{H}}^{p} = 0 (p \geq 2) .$ The differential is the difference of the two restrictions, in the order of the chosen linear order $0 < 1$ ; exchanging the roles of the two members changes its sign and leaves $\ker δ^{0}$ , $im δ^{0}$ and all three cohomology groups unchanged.

Facts & Assumptions

[F1]

The ordered Čech $p$ -cochains are the product $C^{p} (U, F) = \prod_{i_{0} < \dots < i_{p}} F (U_{i_{0}} \cap \dots \cap U_{i_{p}})$ over increasing tuples, with the convention that an empty product is the zero group (Ordered Čech cochain complex of a cover).

[F2]

The Čech differential is $(δ^{p} s)_{i_{0} \dots i_{p + 1}} = \sum_{j = 0}^{p + 1} (- 1)^{j} s_{i_{0} \dots \hat{i_{j}} \dots i_{p + 1}} ∣_{U_{i_{0}} \cap \dots \cap U_{i_{p + 1}}}$ , a sum of restrictions inside one section group (Ordered Čech cochain complex of a cover).

[F3]

The cohomology of the fixed cover is ${\overset{ˇ}{H}}^{p} (U, F) = \ker δ^{p} / im δ^{p - 1}$ , the cohomology of the cochain complex in degree $p$ (Fixed-cover Čech cohomology).

Proof

Given: A topological space $X$ , a sheaf of abelian groups $F$ on $X$ and open subsets $U, V \subseteq X$ with $X = U \cup V$ , indexed as $U_{0} = U < U_{1} = V$ .

1.1

The increasing tuples of the two-element linearly ordered set ${0, 1}$ are: the two singletons $(0)$ and $(1)$ in degree $0$ , the single pair $(0, 1)$ in degree $1$ , and none at all in degrees $p \geq 2$ . Evaluating the product formula of [F1] at these tuples gives $C^{0} (U, F) = F (U_{0}) \times F (U_{1}) = F (U) \oplus F (V)$ , then $C^{1} (U, F) = F (U_{0} \cap U_{1}) = F (U \cap V)$ , and then $C^{p} (U, F) = 0$ for $p \geq 2$ because the product over the empty set of tuples is the zero group by [F1]; the direct product of two groups is their direct sum.

1.2

For $(s_{U}, s_{V}) \in F (U) \oplus F (V)$ the differential formula of [F2] at the unique increasing pair $(0, 1)$ reads $(δ^{0} s)_{01} = \sum_{j = 0}^{1} (- 1)^{j} s_{0 \dots \hat{j} \dots 1} = (+ 1) s_{1} ∣_{U_{0} \cap U_{1}} + (- 1) s_{0} ∣_{U_{0} \cap U_{1}} = s_{V} ∣_{U \cap V} - s_{U} ∣_{U \cap V}$ , the two restrictions being taken along $U_{1} \cap U_{0} \subseteq U_{1}$ and $U_{0} \cap U_{1} \subseteq U_{0}$ ; there are no other components in degree $0$ because $δ^{0}$ has target $C^{1} (U, F)$ , which has the single component $(0, 1)$ . Hence $δ^{0} (s_{U}, s_{V}) = s_{V} ∣_{U \cap V} - s_{U} ∣_{U \cap V}$ .

2.1

Since $C^{2} (U, F) = 0$ by [step 1.1], the map $δ^{1} : C^{1} \to C^{2}$ is the zero map, and the complex is $F (U) \oplus F (V) \overset{δ^{0}}{\to} F (U \cap V) \to 0 \to \dots$ . By the description of the cohomology of a fixed cover in [F3] this gives ${\overset{ˇ}{H}}^{0} = \ker δ^{0}$ , ${\overset{ˇ}{H}}^{1} = F (U \cap V) / im δ^{0}$ and ${\overset{ˇ}{H}}^{p} = 0$ for $p \geq 2$ . Reversing the linear order interchanges the two summands of $C^{0}$ and multiplies $δ^{0}$ by $- 1$ in the sense that the new differential is the negative of the old one composed with the swap of the summands, which leaves kernel, image and cohomology unchanged. ∎

F3step 1.1step 1.2

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Sources

The Stacks Project, Cohomology of Sheaves (standard reference, not scraped)