Section 62.4 (0H4H): Specialization of cycles

62.4 Specialization of cycles

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 $r \geq 0$ . There is a specialization map

$sp_{X/R} : Z_ r(X_ K) \longrightarrow Z_ r(X_\kappa )$

defined as follows. For an integral closed subscheme $Z \subset X_ K$ of dimension $r$ we denote $\overline{Z}$ the scheme theoretic image of $Z \to X$ . Then we let $sp_{X/R}$ be the unique $\mathbf{Z}$ -linear map such that

$sp_{X/R}([Z]) = [\overline{Z}_\kappa ]_ r$

We briefly discuss why this is well defined. First, observe that the morphism $X_ K \to X$ is quasi-compact and hence the morphism $Z \to X$ is quasi-compact. Thus taking the scheme theoretic image of $Z \to X$ commutes with flat base change by Morphisms, Lemma 29.25.16. In particular, base changing back to $X_ K$ we see that $Z = \overline{Z}_ K$ . Since $Z$ is integral, of course $\overline{Z}$ is integral too and in fact is equal to the unique integral closed subscheme whose generic point is the (image of the) generic point of $Z$ . It follows from Varieties, Lemma 33.19.2 that $Z_\kappa$ is equidimensional of dimension $r$ .

Lemma 62.4.1. 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 $r \geq 0$ . Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$ -module flat over $R$ . Assume $\dim (\text{Supp}(\mathcal{F}_ K)) \leq r$ . Then $\dim (\text{Supp}(\mathcal{F}_\kappa )) \leq r$ and

$sp_{X/R}([\mathcal{F}_ K]_ r) = [\mathcal{F}_\kappa ]_ r$

Proof. The statement on dimension follows from More on Morphisms, Lemma 37.18.4. Let $x$ be a generic point of an integral closed subscheme $Z \subset X_\kappa$ of dimension $r$ . To finish the proof we will show that the coefficient of $[Z]$ in the left (L) and right hand side (R) of equality are the same.

Let $A = \mathcal{O}_{X, x}$ and $M = \mathcal{F}_ x$ . Observe that $M$ is a finite $A$ -module flat over $R$ . Let $\pi \in R$ be a uniformizer so that $A/\pi A = \mathcal{O}_{X_\kappa , x}$ . By Chow Homology, Lemma 42.3.2 we have

$\sum \nolimits _ i \text{length}_ A(A/(\pi , \mathfrak q_ i)) \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) = \text{length}_ A(M/\pi M)$

where the sum is over the minimal primes $\mathfrak q_ i$ in the support of $M$ . Since $\pi$ is a nonzerodivisor on $M$ we see that $\pi \not\in \mathfrak q_ i$ and hence these primes correspond to those generic points $y_ i \in X_ K$ of the support of $\mathcal{F}_ K$ which specialize to our chosen $x \in X_\kappa$ . Thus the left hand side is the coefficient of $[Z]$ in (L). Of course $\text{length}_ A(M/\pi M)$ is the coefficient of $[Z]$ in (R). This finishes the proof. $\square$

Lemma 62.4.2. 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 $r \geq 0$ . Let $W \subset X$ be a closed subscheme flat over $R$ . Assume $\dim (W_ K) \leq r$ . Then $\dim (W_\kappa ) \leq r$ and

$sp_{X/R}([W_ K]_ r) = [W_\kappa ]_ r$

Proof. Taking $\mathcal{F} = \mathcal{O}_ W$ this is a special case of Lemma 62.4.1. See Chow Homology, Lemma 42.10.3. $\square$

Lemma 62.4.3. Let $R'/R$ be an extension of discrete valuation rings inducing fraction field extension $K'/K$ and residue field extension $\kappa '/\kappa$ (More on Algebra, Definition 15.111.1). Let $X$ be locally of finite type over $R$ . Denote $X' = X_{R'}$ . Then the diagram

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

commutes where $r \geq 0$ and the vertical arrows are base change maps.

Proof. Observe that $X'_{K'} = X_{K'} = X_ K \times _{\mathop{\mathrm{Spec}}(K)} \mathop{\mathrm{Spec}}(K')$ and similarly for closed fibres, so that the vertical arrows indeed make sense (see Section 62.3). Now if $Z \subset X_ K$ is an integral closed subscheme with scheme theoretic image $\overline{Z} \subset X$ , then we see that $Z_{K'} \subset X_{K'}$ is a closed subscheme with scheme theoretic image $\overline{Z}_{R'} \subset X_{R'}$ . The base change of $[Z]$ is $[Z_{K'}]_ r = [\overline{Z}_{K'}]_ r$ by definition. We have

$sp_{X/R}([Z]) = [\overline{Z}_\kappa ]_ r \quad \text{and}\quad sp_{X'/R'}([\overline{Z}_{K'}]_ r) = [(\overline{Z}_{R'})_{\kappa '}]_ r$

by Lemma 62.4.1. Since $(\overline{Z}_{R'})_{\kappa '} = (\overline{Z}_\kappa )_{\kappa '}$ we conclude. $\square$

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. $\square$

Lemma 62.4.5. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa$ . Let $f : X \to Y$ be a proper morphism of schemes locally of finite type over $R$ . Then the diagram

$\xymatrix{ Z_ r(X_ K) \ar[rr]_{sp_{X/R}} \ar[d] & & Z_ r(X_\kappa ) \ar[d] \\ Z_ r(Y_ K) \ar[rr]^{sp_{Y/R}} & & Z_ r(Y_\kappa ) }$

commutes where $r \geq 0$ and the vertical arrows are given by proper pushforward.

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 $Z' = f(Z) \subset Y$ . Then $Z'$ is an integral closed subscheme of $Y$ dominating $R$ . Thus $sp_{Y/R}([Z'_ K]) = [Z'_\kappa ]_ r$ .

We can think of $[Z]$ as an element of $Z_{r + 1}(X)$ . By definition we have $f_*[Z] = 0$ if $\dim (Z') < r + 1$ and $f_*[Z] = d[Z']$ if $Z \to Z'$ is generically finite of degree $d$ . Since proper pushforward commutes with flat pullback by $Y_ K \to Y$ (Chow Homology, Lemma 42.15.1) we see that correspondingly $f_{K, *}[Z_ K] = 0$ or $f_{K, *}[Z_ K] = d[Z'_ K]$ . Let us apply Chow Homology, Lemma 42.29.8 to the commutative diagram

$\xymatrix{ X_\kappa \ar[d] \ar[r]_ i & X \ar[d] \\ Y_\kappa \ar[r]^ j & Y }$

We obtain that $f_{\kappa , *}[Z_\kappa ]_ r = 0$ or $f_{\kappa , *}[Z_\kappa ] = d[Z'_\kappa ]_ r$ because clearly $i^*[Z] = [Z_ k]_ r$ and $j^*[Z'] = [Z'_\kappa ]_ r$ . Putting everything together we conclude. $\square$

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The Stacks project

62.4 Specialization of cycles

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