The Stacks project

111.41 Čech Cohomology

Exercise 111.41.1. 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 111.41.2. (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 $\delta (\tau )$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02AO. Beware of the difference between the letter 'O' and the digit '0'.