Lemma 62.4.4 (0H4L)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 62: Relative Cycles
Section 62.4: Specialization of cycles
Lemma 62.4.4 (cite)

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Lemma 62.4.4. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa$ . Let $X$ be a scheme locally of finite type over $R$ . Let $f : X' \to X$ be a morphism which is locally of finite type, flat, and of relative dimension $e$ . Then the diagram

$\xymatrix{ Z_{r + e}(X'_ K) \ar[rr]_{sp_{X'/R}} & & Z_{r + e}(X'_\kappa ) \\ Z_ r(X_ K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_ r(X_\kappa ) \ar[u] }$

commutes where $r \geq 0$ and the vertical arrows are given by flat pullback.

Proof. Let $Z \subset X$ be an integral closed subscheme dominating $R$ . By the construction of $sp_{X/R}$ we have $sp_{X/R}([Z_ K]) = [Z_\kappa ]_ r$ and this characterizes the specialization map. Set $Z' = f^{-1}(Z) = X' \times _ X Z$ . Since $R$ is a valuation ring, $Z$ is flat over $R$ . Hence $Z'$ is flat over $R$ and $sp_{X'/R}([Z'_ K]_{r + e}) = [Z'_\kappa ]_{r + e}$ by Lemma 62.4.2. Since by Chow Homology, Lemma 42.14.4 we have $f_ K^*[Z_ K] = [Z'_ K]_{r + e}$ and $f_\kappa ^*[Z_\kappa ]_ r = [Z'_\kappa ]_{r + e}$ we win. $\square$

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