The Stacks project

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 $r$.

  1. 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 $Z_{k + r}(X)$.

  2. 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. $\square$

Comments (2)

Comment #2319 by Daniel on

Two typos in the proof: We have to check that the coefficients of (instead of ) coincide

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