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. $\square$

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