The Stacks project

Exercise 111.32.6. Let $X = \mathbf{R}$ with the usual topology. Let $\mathcal{O}_ X = \underline{\mathbf{Z}/2\mathbf{Z}}_ X$. Let $i : Z = \{ 0\} \to X$ be the inclusion and let $\mathcal{O}_ Z = \underline{\mathbf{Z}/2\mathbf{Z}}_ Z$. Prove the following (the first three follow from the definitions but if you are not clear on the definitions you should elucidate them):

  1. $i_*\mathcal{O}_ Z$ is a skyscraper sheaf.

  2. There is a canonical surjective map from $\underline{\mathbf{Z}/2\mathbf{Z}}_ X \to i_*\underline{\mathbf{Z}/2\mathbf{Z}}_ Z$. Denote the kernel $\mathcal{I} \subset \mathcal{O}_ X$.

  3. $\mathcal{I}$ is an ideal sheaf of $\mathcal{O}_ X$.

  4. The sheaf $\mathcal{I}$ on $X$ cannot be locally generated by sections (as in Modules, Definition 17.8.1.)

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