The Stacks project

110.57 Sheaves and specializations

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. Recall that if $x, x'$ are points of a scheme $X$ we say $x$ is a specialization of $x'$ or we write $x' \leadsto x$ if $x \in \overline{\{ x'\} }$. This is true in particular if $x = x'$.

Consider the functor $F : \mathit{Sch}_{\acute{e}tale}\to \textit{Ab}$ defined by the following rules:

\[ F(X) = \prod \nolimits _{x \in X} \prod \nolimits _{x' \in X, x' \leadsto x, x' \not= x} \mathbf{Z}/2\mathbf{Z} \]

Given a scheme $X$ we denote $|X|$ the underlying set of points. An element $a \in F(X)$ will be viewed as a map of sets $|X| \times |X| \to \mathbf{Z}/2\mathbf{Z}$, $(x, x') \mapsto a(x, x')$ which is zero if $x = x'$ or if $x$ is not a specialization of $x'$. Given a morphism of schemes $f : X \to Y$ we define

\[ F(f) : F(Y) \longrightarrow F(X) \]

by the rule that for $b \in F(Y)$ we set

\[ F(f)(b)(x, x') = \left\{ \begin{matrix} 0 & \text{if }x\text{ is not a specialization of }x' \\ b(f(x), f(x')) & \text{else.} \end{matrix} \right. \]

Note that this really does define an element of $F(X)$. We claim that if $f : X \to Y$ and $g : Y \to Z$ are composable morphisms then $F(f) \circ F(g) = F(g \circ f)$. Namely, let $c \in F(Z)$ and let $x' \leadsto x$ be a specialization of points in $X$, then

\[ F(g \circ f)(x, x') = c(g(f(x)), g(f(x'))) = F(g)(F(f)(c))(x, x') \]

because $f(x') \leadsto f(x)$. (This also works if $f(x) = f(x')$.)

Let $G$ be the sheafification of $F$ in the étale topology.

I claim that if $X$ is a scheme and $x' \leadsto x$ is a specialization and $x' \not= x$, then $G(X) \not= 0$. Namely, let $a \in F(X)$ be an element such that when we think of $a$ as a function $|X| \times |X| \to \mathbf{Z}/2\mathbf{Z}$ it is nonzero at $(x, x')$. Let $\{ f_ i : U_ i \to X\} $ be an étale covering of $X$. Then we can pick an $i$ and a point $u_ i \in U_ i$ with $f_ i(u_ i) = x$. Since generalizations lift along flat morphisms (see Morphisms, Lemma 29.25.9) we can find a specialization $u'_ i \leadsto u_ i$ with $f_ i(u'_ i) = x'$. By our construction above we see that $F(f_ i)(a) \not= 0$. Hence $a$ determines a nonzero element of $G(X)$.

Note that if $X = \mathop{\mathrm{Spec}}(k)$ where $k$ is a field (or more generally a ring all of whose prime ideals are maximal), then $F(X) = 0$ and for every étale morphism $U \to X$ we have $F(U) = 0$ because there are no specializations between distinct points in fibres of an étale morphism. Hence $G(X) = 0$.

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 it is always the case that $F(U) = F(U')$ in this situation we see that also $G(X) = G(X')$.

As a variant we can consider the presheaf $F_ n$ which associates to a scheme $X$ the collection of maps $a : |X|^{n + 1} \to \mathbf{Z}/2\mathbf{Z}$ where $a(x_0, \ldots , x_ n)$ is nonzero only if $x_ n \leadsto \ldots \leadsto x_0$ is a sequence of specializations and $x_ n \not= x_{n - 1} \not= \ldots \not= x_0$. Let $G_ n$ be the sheaf associated to $F_ n$. In exactly the same way as above one shows that $G_ n$ is nonzero if $\dim (X) \geq n$ and is zero if $\dim (X) < n$.

Lemma 110.57.1. There exists a sheaf of abelian groups $G$ on $\mathit{Sch}_{\acute{e}tale}$ with the following properties

  1. $G(X) = 0$ whenever $\dim (X) < n$,

  2. $G(X)$ is not zero if $\dim (X) \geq n$, and

  3. if $X \subset X'$ is a thickening, then $G(X) = G(X')$.

Proof. See the discussion above. $\square$

Remark 110.57.2. Here are some remarks:

  1. The presheaves $F$ and $F_ n$ are separated presheaves.

  2. It turns out that $F$, $F_ n$ are not sheaves.

  3. One can show that $G$, $G_ n$ is actually a sheaf for the fppf topology.

We will prove these results if we need them.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05LD. Beware of the difference between the letter 'O' and the digit '0'.