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
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
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
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 commutes where $r \geq 0$ and the vertical arrows are base change maps.
\[ \xymatrix{ Z_ r(X'_{K'}) \ar[rr]_{sp_{X'/R'}} & & Z_ r(X'_{\kappa '}) \\ Z_ r(X_ K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_ r(X_\kappa ) \ar[u] } \]
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
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 commutes where $r \geq 0$ and the vertical arrows are given by flat pullback.
\[ \xymatrix{ Z_{r + e}(X'_ K) \ar[rr]_{sp_{X'/R}} & & Z_{r + e}(X'_\kappa ) \\ Z_ r(X_ K) \ar[rr]^{sp_{X/R}} \ar[u] & & Z_ r(X_\kappa ) \ar[u] } \]
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 commutes where $r \geq 0$ and the vertical arrows are given by proper pushforward.
\[ \xymatrix{ Z_ r(X_ K) \ar[rr]_{sp_{X/R}} \ar[d] & & Z_ r(X_\kappa ) \ar[d] \\ Z_ r(Y_ K) \ar[rr]^{sp_{Y/R}} & & Z_ r(Y_\kappa ) } \]
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
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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