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
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 wil 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. $\square$
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