When the presheaf inverse image of a sheaf is already a sheaf

Question

In Milne's book Etale cohomology, p93, he defines $0$-th cohomology with compact support of a separated variety $X$ (in this book a variety is a geometrically irreducible, geometrically reduce scheme of finite type over a field) to be \begin{equation} \Gamma_c(X,F) = \bigcup \operatorname{ker}(\Gamma(X,F) \to \Gamma(X-Z,F)) \end{equation} where $Z$ run over the complete subvarieties of $X$. Take a compactification $j\colon X \to \bar X$, then he claims (and we want) \begin{equation} \Gamma_c(X,F) = \Gamma(\bar X,j_!F). \end{equation} The proof goes as follows: From the exact sequence on $\bar X$ \begin{equation} 0 \to j_!F \to j_*F \to i_*i^*j_*F \to 0, \end{equation} where $i\colon \bar X - X \hookrightarrow \bar X$, we see that \begin{equation} \Gamma(X,j_!F) = \operatorname{ker}(\Gamma(X,F) \to \Gamma(\bar X - X,i^*j_*F)). \end{equation} The following formula in the book confuses me: \begin{equation} \Gamma(\bar X - X,i^*j_*F)) = \operatorname{colim}\Gamma(V\times_{\bar X} X,F) \end{equation} where $V\to \bar X$ is etale and contains $\bar X - X$ in its image. But chasing of definition shows \begin{equation} \operatorname{colim}\Gamma(V\times_{\bar X} X,F) = i^pj_*F(\bar X - X) \end{equation} where $i^p$ denotes the presheaf inverse image. Then it comes to my question in title: is it true that in the case $i\colon X_1 \to X$ being a closed immersion, for a sheaf $F$ on $X_{et}$, $i^pF$ is already a sheaf? If not, then how to fix the proof?

Answer 1

Let $s$ be a section of $\mathrm{H}^0(X,\mathcal{F})$ such that its image under $$ \varphi\colon \mathrm{H}^0(X,\mathcal{F})=\mathrm{H}^0(\overline{X},j_{\ast}\mathcal{F}) \to \mathrm{H}^{0}(\overline{X}\setminus X, i^{-1}j_{\ast}\mathcal{F}) $$ is zero. You want to show that $s$ has "compact support" in the sense that $\operatorname{Supp}s$ is complete.

The proof goes as follows.

1. The hypothesis implies that the germ of $\varphi(s)$ at any point of $\overline{X}\setminus X$ is zero. [A section is zero if and only if its germ at any stalk is zero.]

2. Thus, for each geometric point $z$ of $\overline{X}\setminus X$, there exists an étale morphism $f_{z}\colon V_z \to \overline{X}$ such that (a) $z \in \operatorname{Im}f_z$, and (b) $s|_{V_z} = 0$. [The stalk at $z$ is a colimit of $\mathcal{F}(U)$, where $U$ runs in the family of neighborhoods of $z$.]

3. Since $\overline{X}\setminus X$ is quasi-compact, we can find finitely many $z_i$ such that the image $\coprod_{i}V_{z_i} \to \overline{X}$ contains $\overline{X}\setminus X$. Let $V = \coprod_{i}V_{z_i}$. Then $V \to \overline{X}$ is étale, and $s|_{V} = 0$ by construction. It follows that the support of $s$ is contained in the closed set $\overline{X} \setminus \operatorname{Im}(V\to \overline{X})$, which is complete since $\overline{X}$ is.