The Stacks project

Lemma 62.6.12. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ and $\alpha , \beta \in z(X/S, r)$. The set $E = \{ s \in S : \alpha _ s = \beta _ s\} $ is closed in $S$.

Proof. The question is local on $S$, thus we may assume $S$ is affine. Let $X = \bigcup U_ i$ be an affine open covering. Let $E_ i = \{ s \in S : \alpha _ s|_{U_{i, s}} = \beta _ s|_{U_{i, s}}\} $. Then $E = \bigcap E_ i$. Hence it suffices to prove the lemma for $U_ i \to S$ and the restriction of $\alpha $ and $\beta $ to $U_ i$. This reduces us to the case discussed in the next paragraph.

Assume $X$ and $S$ are quasi-compact. Set $\gamma = \alpha - \beta $. Then $E = \{ s \in S : \gamma _ s = 0\} $. By Lemma 62.6.8 there exists a jointly surjective finite family of proper morphisms $\{ g_ i : S_ i \to S\} $ such that $g_ i^*\gamma $ is in the image of (62.6.8.1). Observe that $E_ i = g_ i^{-1}(E)$ is the set of point $t \in S_ i$ such that $(g_ i^*\gamma )_ t = 0$. If $E_ i$ is closed for all $i$, then $E = \bigcup g_ i(E_ i)$ is closed as well. This reduces us to the case discussed in the next paragraph.

Assume $X$ and $S$ are quasi-compact and $\gamma = \sum n_ i[Z_ i/X/S]_ r$ for a finite number of closed subschemes $Z_ i \subset X$ flat and of relative dimension $\leq r$ over $S$. Set $X' = \bigcup Z_ i$ (scheme theoretic union). Then $i : X' \to X$ is a closed immersion and $X'$ has relative dimension $\leq r$ over $S$. Also $\gamma = i_*\gamma '$ where $\gamma ' = \sum n_ i[Z_ i/X'/S]_ r$. Since clearly $E = E' = \{ s \in S : \gamma '_ s = 0\} $ we reduce to the case discussed in the next paragraph.

Assume $X$ has relative dimension $\leq r$ over $S$. Let $s \in S$, $s \not\in E$. We will show that there exists an open neighbourhood $V \subset S$ of $s$ such that $E \cap V$ is empty. The assumption $s \not\in E$ means there exists an integral closed subscheme $Z \subset X_ s$ of dimension $r$ such that the coefficient $n$ of $[Z]$ in $\gamma _ s$ is nonzero. Let $x \in Z$ be the generic point. Since $\dim (Z) = r$ we see that $x$ is a generic point of an irreducible component (namely $Z$) of $X_ s$. Thus after replacing $X$ by an open neighbourhood of $x$, we may assume that $Z$ is the only irreducible component of $X_ s$. In particular, we have $\gamma _ s = n[Z]$.

At this point we apply More on Morphisms, Lemma 37.47.1 and we obtain a diagram

\[ \xymatrix{ X \ar[dd] & X' \ar[l]^ g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^ h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] } \]

with all the properties listed there. Let $\gamma ' = g^*\gamma $ be the flat pullback. Note that $E \subset E' = \{ s \in S: \gamma '_ s = 0\} $ and that $s \not\in E'$ because the coefficient of $Z'$ in $\gamma '_ s$ is nonzero, where $Z' \subset X'_ s$ is the closure of $x'$. Similarly, set $\gamma '' = \pi _*\gamma '$. Then we have $E' \subset E'' = \{ s \in S: \gamma ''_ s = 0\} $ and $s \not\in E''$ because the coefficient of $Z''$ in $\gamma ''_ s$ is nonzero, where $Z'' \subset Y_ s$ is the closure of $y$. By Lemma 62.6.11 and openness of $Y \to S$ we see that an open neighbourhood of $s$ is disjoint from $E''$ and the proof is complete. $\square$


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 0H5C. Beware of the difference between the letter 'O' and the digit '0'.