The Stacks project

Proof. Let $g : S' \to S$ be the inclusion of an irreducible component viewed as an integral closed subscheme. By Lemmas 62.8.2 and 62.5.7 it suffices to show that the support of the base change $g^*\alpha$ is closed in $S' \times _ S S$. Thus we may assume $S$ is an integral scheme with generic point $\eta$. We will show that $\text{Supp}(\alpha )$ is the closure of $\text{Supp}(\alpha _\eta )$. To do this, pick any $s \in S$. We can find a morphism $g : S' \to S$ where $S'$ is the spectrum of a discrete valuation ring mapping the generic point $\eta ' \in S'$ to $\eta$ and the closed point $0 \in S'$ to $s$, see Properties, Lemma 28.5.10. Then it suffices to prove that the support of $g^*\alpha$ is equal to the closure of $\text{Supp}((g^\alpha )_{\eta '})$. This reduces us to the case discussed in the next paragraph.

Here $S$ is the spectrum of a discrete valuation ring with generic point $\eta$ and closed point $0$. We have to show that $\text{Supp}(\alpha )$ is the closure of $\text{Supp}(\alpha _\eta )$. Since $\alpha$ is effective we may write $\alpha _\eta = \sum n_ i[Z_ i]$ with $n_ i > 0$ and $Z_ i \subset X_\eta$ integral closed of dimension $r$. Since $\alpha _0 = sp_{X/S}(\alpha _\eta )$ we know that $\alpha _0 = \sum n_ i [\overline{Z}_{i, 0}]_ r$ where $\overline{Z}_ i$ is the closure of $Z_ i$. By Varieties, Lemma 33.19.2 we see that $\overline{Z}_{i, 0}$ is equidimensional of dimension $r$. Since $n_ i > 0$ we conclude that $\text{Supp}(\alpha _0)$ is equal to the union of the $\overline{Z}_{i, 0}$ which is the fibre over $0$ of $\bigcup \overline{Z}_ i$ which in turn is the closure of $\bigcup Z_ i$ as desired. $\square$