Lemma 62.6.16. Let $g : S' \to S$ be a universal homeomorphism of locally Noetherian schemes which induces isomorphisms of residue fields. Let $f : X \to S$ be locally of finite type. Set $X' = S' \times _ S X$. Let $r \geq 0$. Then base change by $g$ determines a bijection $z(X/S, r) \to z(X'/S', r)$.
Proof. By Lemma 62.5.10 we have a bijection between the group of families of $r$-cycles on fibres of $X/S$ and the group of families of $r$-cycles on fibres of $X'/S'$. Say $\alpha $ is a families of $r$-cycles on fibres of $X/S$ and $\alpha ' = g^*\alpha $ is the base change. If $R$ is a discrete valuation ring, then any morphism $h : \mathop{\mathrm{Spec}}(R) \to S$ factors as $g \circ h'$ for some unique morphism $h' : \mathop{\mathrm{Spec}}(R) \to S'$. Namely, the morphism $S' \times _ S \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(R)$ is a universal homomorphism inducing bijections on residue fields, and hence has a section (for example because $R$ is a seminormal ring, see Morphisms, Section 29.47). Thus the condition that $\alpha $ is compatible with specializations (i.e., is a relative $r$-cycle) is equivalent to the condition that $\alpha '$ is compatible with specializations. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)