Lemma 82.10.4. In Situation 82.2.1 let $X, Y, Z/B$ be good. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of relative dimensions $r$ and $s$ over $B$. Then $g \circ f$ is flat of relative dimension $r + s$ and
as maps $Z_ k(Z) \to Z_{k + r + s}(X)$.
Proof. The composition is flat of relative dimension $r + s$ by Morphisms of Spaces, Lemmas 67.34.2 and 67.30.3. Suppose that
$A \subset Z$ is a closed integral subspace of $\delta $-dimension $k$,
$A' \subset Y$ is a closed integral subspace of $\delta $-dimension $k + s$ with $A' \subset g^{-1}(A)$, and
$A'' \subset Y$ is a closed integral subspace of $\delta $-dimension $k + s + r$ with $A'' \subset f^{-1}(W')$.
We have to show that the coefficient $n$ of $[A'']$ in $(g \circ f)^*[A]$ agrees with the coefficient $m$ of $[A'']$ in $f^*(g^*[A])$. We may choose a commutative diagram
where $U, V, W$ are schemes, the vertical arrows are étale, and there exist points $u \in U$, $v \in V$, $w \in W$ such that $u \mapsto v \mapsto w$ and such that $u, v, w$ map to the generic points of $A'', A', A$. (Details omitted.) Then we have flat local ring homorphisms $\mathcal{O}_{W, w} \to \mathcal{O}_{V, v}$, $\mathcal{O}_{V, v} \to \mathcal{O}_{U, u}$, and repeatedly using Lemma 82.4.1 we find
and
Hence the equality follows from Algebra, Lemma 10.52.14. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)