diff --git a/CONTRIBUTORS b/CONTRIBUTORS index 9d17d790d..e5834a975 100644 --- a/CONTRIBUTORS +++ b/CONTRIBUTORS @@ -117,6 +117,7 @@ Xu Gao Toby Gee Anton Geraschenko Daniel Gerigk +Harry Gindi Alberto Gioia Charles Godfrey Julia Ramos Gonzalez diff --git a/etale-cohomology.tex b/etale-cohomology.tex index aa73a8a02..12e3c90fa 100644 --- a/etale-cohomology.tex +++ b/etale-cohomology.tex @@ -12308,7 +12308,7 @@ \section{Constructible sheaves on Noetherian schemes} There exist an injective map of sheaves $$ \mathcal{F} \longrightarrow -\coprod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} +\prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} $$ where $f_i : Y_i \to X$ is a finite morphism and $E_i$ is a finite set. \item Let $\mathcal{F}$ be a constructible abelian sheaf on $X_\etale$. @@ -12351,13 +12351,13 @@ \section{Constructible sheaves on Noetherian schemes} (Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}) on $\overline{\{x\}}$. Since $\mathcal{F}$ is constructible, there is a finite separable -extension $\kappa(x) \subset \Spec(K)$ such that +extension $K/\kappa(x)$ such that $\mathcal{F}|_{\Spec(K)}$ is the constant sheaf with value $E$ for some finite set $E$. Let $Y \to Z$ be the normalization of $Z$ in $\Spec(K)$. By Morphisms, Lemma \ref{morphisms-lemma-normal-normalization} we see that $Y$ is a normal integral scheme. -As $\kappa(x) \subset K$ is finite, it is clear that $K$ is the function +As $K/\kappa(x)$ is a finite extension, it is clear that $K$ is the function field of $Y$. Denote $g : \Spec(K) \to Y$ the inclusion. The map $\mathcal{F}|_{\Spec(K)} \to \underline{E}$ is adjoint to a map $\mathcal{F}|_Y \to g_*\underline{E} = \underline{E}$ @@ -12409,13 +12409,16 @@ \section{Constructible sheaves on Noetherian schemes} \begin{lemma} \label{lemma-constructible-maps-into-constant-general} +\begin{reference} +\cite[Exposee IX, Proposition 2.14]{SGA4} +\end{reference} Let $X$ be a quasi-compact and quasi-separated scheme. \begin{enumerate} \item Let $\mathcal{F}$ be a constructible sheaf of sets on $X_\etale$. There exist an injective map of sheaves $$ \mathcal{F} \longrightarrow -\coprod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} +\prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} $$ where $f_i : Y_i \to X$ is a finite and finitely presented morphism and $E_i$ is a finite set. @@ -12455,7 +12458,7 @@ \section{Constructible sheaves on Noetherian schemes} to find an injection $$ \mathcal{F}_t \longrightarrow -\coprod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} +\prod\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{E_i} \quad\text{or}\quad \mathcal{F}_t \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} f_{i, *}\underline{M_i}