The Stacks project

Lemma 35.11.7. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, flat, and locally of finite presentation,

  2. $p$ is syntomic.

Then both $q$ and $f$ are syntomic.

Proof. By Lemma 35.11.3 we see that $q$ is of finite presentation. By Morphisms, Lemma 29.25.13 we see that $q$ is flat. By Morphisms, Lemma 29.30.10 it now suffices to show that the local rings of the fibres of $Y \to S$ and the fibres of $X \to Y$ are local complete intersection rings. To do this we may take the fibre of $X \to Y \to S$ at a point $s \in S$, i.e., we may assume $S$ is the spectrum of a field. Pick a point $x \in X$ with image $y \in Y$ and consider the ring map

\[ \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, x} \]

This is a flat local homomorphism of local Noetherian rings. The local ring $\mathcal{O}_{X, x}$ is a complete intersection. Thus may use Avramov's result, see Divided Power Algebra, Lemma 23.8.9, to conclude that both $\mathcal{O}_{Y, y}$ and $\mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ are complete intersection rings. $\square$


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