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

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.

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.)

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