Lemma 42.14.4. Let (S, \delta ) be as in Situation 42.7.1. Let X, Y be locally of finite type over S. Let f : X \to Y be a flat morphism of relative dimension r.
Let Z \subset Y be a closed subscheme with \dim _\delta (Z) \leq k. Then we have \dim _\delta (f^{-1}(Z)) \leq k + r and [f^{-1}(Z)]_{k + r} = f^*[Z]_ k in Z_{k + r}(X).
Let \mathcal{F} be a coherent sheaf on Y with \dim _\delta (\text{Supp}(\mathcal{F})) \leq k. Then we have \dim _\delta (\text{Supp}(f^*\mathcal{F})) \leq k + r and
f^*[{\mathcal F}]_ k = [f^*{\mathcal F}]_{k+r}
in Z_{k + r}(X).
Proof.
The statements on dimensions follow immediately from Lemma 42.13.1. Part (1) follows from part (2) by Lemma 42.10.3 and the fact that f^*\mathcal{O}_ Z = \mathcal{O}_{f^{-1}(Z)}.
Proof of (2). As X, Y are locally Noetherian we may apply Cohomology of Schemes, Lemma 30.9.1 to see that \mathcal{F} is of finite type, hence f^*\mathcal{F} is of finite type (Modules, Lemma 17.9.2), hence f^*\mathcal{F} is coherent (Cohomology of Schemes, Lemma 30.9.1 again). Thus the lemma makes sense. Let W \subset Y be an integral closed subscheme of \delta -dimension k, and let W' \subset X be an integral closed subscheme of dimension k + r mapping into W under f. We have to show that the coefficient n of [W'] in f^*[{\mathcal F}]_ k agrees with the coefficient m of [W'] in [f^*{\mathcal F}]_{k+r}. Let \xi \in W and \xi ' \in W' be the generic points. Let A = \mathcal{O}_{Y, \xi }, B = \mathcal{O}_{X, \xi '} and set M = \mathcal{F}_\xi as an A-module. (Note that M has finite length by our dimension assumptions, but we actually do not need to verify this. See Lemma 42.10.1.) We have f^*\mathcal{F}_{\xi '} = B \otimes _ A M. Thus we see that
n = \text{length}_ B(B \otimes _ A M) \quad \text{and} \quad m = \text{length}_ A(M) \text{length}_ B(B/\mathfrak m_ AB)
Thus the equality follows from Algebra, Lemma 10.52.13.
\square
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