The Stacks project

110.58 Sheaves and constructible functions

In the following we fix a big étale site $\mathit{Sch}_{\acute{e}tale}$ as constructed in Topologies, Definition 34.4.6. Moreover, a scheme will be an object of this site. In this section we say that a constructible partition of a scheme $X$ is a locally finite disjoint union decomposition $X = \coprod _{i \in I} X_ i$ such that each $X_ i \subset X$ is a locally constructible subset of $X$. Locally finite means that for any quasi-compact open $U \subset X$ there are only finitely many $i \in I$ such that $X_ i \cap U$ is not empty. Note that if $f : X \to Y$ is a morphism of schemes and $Y = \coprod Y_ j$ is a constructible partition, then $X = \coprod f^{-1}(Y_ j)$ is a constructible partition of $X$. Given a set $S$ and a scheme $X$ a constructible function $f : |X| \to S$ is a map such that $X = \coprod _{s \in S} f^{-1}(s)$ is a constructible partition of $X$. If $G$ is an (abstract group) and $a, b : |X| \to G$ are constructible functions, then $ab : |X| \to G, x \mapsto a(x)b(x)$ is a constructible function too. The reason is that given any two constructible partitions there is a third one refining both.

Let $A$ be any abelian group. For any scheme $X$ we define

We think of an element $a$ of $F(X)$ simply as a function well defined up to adding a locally constant one. Given a morphism of schemes $f : X \to Y$ and an element $b \in F(Y)$, then we define $F(f)(b) = b \circ f$. Thus $F$ is a presheaf on $\mathit{Sch}_{\acute{e}tale}$.

Note that if $\{ f_ i : U_ i \to X\} $ is an fppf covering, and $a \in F(X)$ is such that $F(f_ i)(a) = 0$ in $F(U_ i)$, then $a \circ f_ i$ is a locally constant function for each $i$. This means in turn that $a$ is a locally constant function as the morphisms $f_ i$ are open. Hence $a = 0$ in $F(X)$. Thus we see that $F$ is a separated presheaf (in the fppf topology hence a fortiori in the étale topology).

Let $G$ be the sheafification of $F$ in the étale topology. Since $F$ is separated, and since $F(X) \not= 0$ for example when $X$ is the spectrum of a discrete valuation ring, we see that $G$ is not zero.

Let $X = \mathop{\mathrm{Spec}}(k)$ where $k$ is a field. Then any étale covering of $X$ can be dominated by a covering $\{ \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)\} $ with $k'/k$ a finite separable extension of fields. Since $F(\mathop{\mathrm{Spec}}(k')) = 0$ we see that $G(X) = 0$.

Suppose that $X \subset X'$ is a thickening, see More on Morphisms, Definition 37.2.1. Then the category of schemes étale over $X'$ is equivalent to the category of schemes étale over $X$ by the base change functor $U' \mapsto U = U' \times _{X'} X$, see Étale Cohomology, Theorem 59.45.2. Since $F(U) = F(U')$ in this situation we see that also $G(X) = G(X')$.

The sheaf $G$ is limit preserving, see Limits of Spaces, Definition 70.3.1. Namely, let $R$ be a ring which is written as a directed colimit $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$ of rings. Set $X = \mathop{\mathrm{Spec}}(R)$ and $X_ i = \mathop{\mathrm{Spec}}(R_ i)$, so that $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$. Then $G(X) = \mathop{\mathrm{colim}}\nolimits _ i G(X_ i)$. To prove this one first proves that a constructible partition of $\mathop{\mathrm{Spec}}(R)$ comes from a constructible partitions of some $\mathop{\mathrm{Spec}}(R_ i)$. Hence the result for $F$. To get the result for the sheafification, use that any étale ring map $R \to R'$ comes from an étale ring map $R_ i \to R_ i'$ for some $i$. Details omitted.