The Stacks project

109.41 Čech Cohomology

Exercise 109.41.1. Čech cohomology. Here $k$ is a field.

  1. Let $X$ be a scheme with an open covering ${\mathcal U} : X = U_1 \cup U_2$, with $U_1 = \mathop{\mathrm{Spec}}(k[x])$, $U_2 = \mathop{\mathrm{Spec}}(k[y])$ with $U_1 \cap U_2 = \mathop{\mathrm{Spec}}(k[z, 1/z])$ and with open immersions $U_1 \cap U_2 \to U_1$ resp. $U_1 \cap U_2 \to U_2$ determined by $x \mapsto z$ resp. $y \mapsto z$ (and I really mean this). (We've seen in the lectures that such an $X$ exists; it is the affine line with zero doubled.) Compute ${\check H}^1({\mathcal U}, {\mathcal O})$; eg. give a basis for it as a $k$-vectorspace.

  2. For each element in ${\check H}^1({\mathcal U}, {\mathcal O})$ construct an exact sequence of sheaves of ${\mathcal O}_ X$-modules

    \[ 0 \to {\mathcal O}_ X \to E \to {\mathcal O}_ X \to 0 \]

    such that the boundary $\delta (1) \in {\check H}^1({\mathcal U}, {\mathcal O})$ equals the given element. (Part of the problem is to make sense of this. See also below. It is also OK to show abstractly such a thing has to exist.)

Definition 109.41.2. (Definition of delta.) Suppose that

\[ 0 \to {\mathcal F}_1 \to {\mathcal F}_2 \to {\mathcal F}_3 \to 0 \]

is a short exact sequence of abelian sheaves on any topological space $X$. The boundary map $\delta : H^0(X, {\mathcal F}_3) \to {\check H}^1(X, {\mathcal F}_1)$ is defined as follows. Take an element $\tau \in H^0(X, {\mathcal F}_3)$. Choose an open covering ${\mathcal U} : X = \bigcup _{i\in I} U_ i$ such that for each $i$ there exists a section $\tilde\tau _ i \in {\mathcal F}_2$ lifting the restriction of $\tau $ to $U_ i$. Then consider the assignment

\[ (i_0, i_1) \longmapsto \tilde\tau _{i_0}|_{U_{i_0i_1}} - \tilde\tau _{i_1}|_{U_{i_0i_1}}. \]

This is clearly a 1-coboundary in the Čech complex ${\check C}^\ast ({\mathcal U}, {\mathcal F}_2)$. But we observe that (thinking of ${\mathcal F}_1$ as a subsheaf of ${\mathcal F}_2$) the RHS always is a section of ${\mathcal F}_1$ over $U_{i_0i_1}$. Hence we see that the assignment defines a 1-cochain in the complex ${\check C}^\ast ({\mathcal U}, {\mathcal F}_2)$. The cohomology class of this 1-cochain is by definition $\delta (\tau )$.


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