The Stacks project

Lemma 62.6.3. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a family of $r$-cycles on fibres of $X/S$. Let $\{ g_ i : S_ i \to S\} $ be a h covering (More on Flatness, Definition 38.34.2). Then $\alpha $ is a relative $r$-cycle if and only if each base change $g_ i^*\alpha $ is a relative $r$-cycle.

Proof. If $\alpha $ is a relative $r$-cycle, then each base change $g_ i^*\alpha $ is a relative $r$-cycle by Lemma 62.6.2. Assume each $g_ i^*\alpha $ is a relative $r$-cycle. Let $g : S' \to S$ be a morphism where $S'$ is the spectrum of a discrete valuation ring. After replacing $S$ by $S'$, $X$ by $X' = X \times _ S S'$, and $\alpha $ by $\alpha ' = g^*\alpha $ and using that the base change of a h covering is a h covering (More on Flatness, Lemma 38.34.9) we reduce to the problem studied in the next paragraph.

Assume $S$ is the spectrum of a discrete valuation ring with closed point $0$ and generic point $\eta $. We have to show that $sp_{X/S}(\alpha _\eta ) = \alpha _0$. Since a h covering is a V covering (by definition), there is an $i$ and a specialization $s' \leadsto s$ of points of $S_ i$ with $g_ i(s') = \eta $ and $g_ i(s) = 0$, see Topologies, Lemma 34.10.13. By Properties, Lemma 28.5.10 we can find a morphism $h : S' \to S_ i$ from the spectrum $S'$ of a discrete valuation ring which maps the generic point $\eta '$ to $s'$ and maps the closed point $0'$ to $s$. Denote $\alpha ' = h^*g_ i^*\alpha $. By assumption we have $sp_{X'/S'}(\alpha '_{\eta '}) = \alpha '_{0'}$. Since $g = g_ i \circ h : S' \to S$ is the morphism of schemes induced by an extension of discrete valuation rings we conclude that $sp_{X/S}$ and $sp_{X'/S'}$ are compatible with base change maps on the fibres, see Lemma 62.4.3. We conclude that $sp_{X/S}(\alpha _\eta ) = \alpha _0$ because the base change map $Z_ r(X_0) \to Z_ r(X'_{0'})$ is injective as discussed in Section 62.3. $\square$

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