The Stacks project

Lemma 42.14.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X, Y, Z$ be locally of finite type over $S$. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of relative dimensions $r$ and $s$. Then $g \circ f$ is flat of relative dimension $r + s$ and

\[ f^* \circ g^* = (g \circ f)^* \]

as maps $Z_ k(Z) \to Z_{k + r + s}(X)$.

Proof. The composition is flat of relative dimension $r + s$ by Morphisms, Lemma 29.29.3. Suppose that

  1. $W \subset Z$ is a closed integral subscheme of $\delta $-dimension $k$,

  2. $W' \subset Y$ is a closed integral subscheme of $\delta $-dimension $k + s$ with $W' \subset g^{-1}(W)$, and

  3. $W'' \subset Y$ is a closed integral subscheme of $\delta $-dimension $k + s + r$ with $W'' \subset f^{-1}(W')$.

We have to show that the coefficient $n$ of $[W'']$ in $(g \circ f)^*[W]$ agrees with the coefficient $m$ of $[W'']$ in $f^*(g^*[W])$. That it suffices to check the lemma in these cases follows from Lemma 42.13.1. Let $\xi '' \in W''$, $\xi ' \in W'$ and $\xi \in W$ be the generic points. Consider the local rings $A = \mathcal{O}_{Z, \xi }$, $B = \mathcal{O}_{Y, \xi '}$ and $C = \mathcal{O}_{X, \xi ''}$. Then we have local flat ring maps $A \to B$, $B \to C$ and moreover

\[ n = \text{length}_ C(C/\mathfrak m_ AC), \quad \text{and} \quad m = \text{length}_ C(C/\mathfrak m_ BC) \text{length}_ B(B/\mathfrak m_ AB) \]

Hence the equality follows from Algebra, Lemma 10.52.14. $\square$

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