The Stacks project

Lemma 29.30.16. 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 and syntomic,

  2. $p$ is syntomic, and

  3. $q$ is locally of finite presentation1.

Then $q$ is syntomic.

Proof. By Lemma 29.25.13 we see that $q$ is flat. Hence it suffices to show that the fibres of $Y \to S$ are local complete intersections, see Lemma 29.30.11. Let $s \in S$. Consider the morphism $X_ s \to Y_ s$. This is a base change of the morphism $X \to Y$ and hence surjective, and syntomic (Lemma 29.30.4). For the same reason $X_ s$ is syntomic over $\kappa (s)$. Moreover, $Y_ s$ is locally of finite type over $\kappa (s)$ (Lemma 29.15.4). In this way we reduce to the case where $S$ is the spectrum of a field.

Assume $S = \mathop{\mathrm{Spec}}(k)$. Let $y \in Y$. Choose an affine open $\mathop{\mathrm{Spec}}(A) \subset Y$ neighbourhood of $y$. Let $\mathop{\mathrm{Spec}}(B) \subset X$ be an affine open such that $f(\mathop{\mathrm{Spec}}(B)) \subset \mathop{\mathrm{Spec}}(A)$, containing a point $x \in X$ such that $f(x) = y$. Choose a surjection $k[x_1, \ldots , x_ n] \to A$ with kernel $I$. Choose a surjection $A[y_1, \ldots , y_ m] \to B$, which gives rise in turn to a surjection $k[x_ i, y_ j] \to B$ with kernel $J$. Let $\mathfrak q \subset k[x_ i, y_ j]$ be the prime corresponding to $y \in \mathop{\mathrm{Spec}}(B)$ and let $\mathfrak p \subset k[x_ i]$ the prime corresponding to $x \in \mathop{\mathrm{Spec}}(A)$. Since $x$ maps to $y$ we have $\mathfrak p = \mathfrak q \cap k[x_ i]$. Consider the following commutative diagram of local rings:

\[ \xymatrix{ \mathcal{O}_{X, x} \ar@{=}[r] & B_{\mathfrak q} & k[x_1, \ldots , x_ n, y_1, \ldots , y_ m]_{\mathfrak q} \ar[l] \\ \mathcal{O}_{Y, y} \ar@{=}[r] \ar[u] & A_{\mathfrak p} \ar[u] & k[x_1, \ldots , x_ n]_{\mathfrak p} \ar[l] \ar[u] } \]

We claim that the hypotheses of Algebra, Lemma 10.135.12 are satisfied. Conditions (1) and (2) are trivial. Condition (4) follows as $X \to Y$ is flat. Condition (3) follows as the rings $\mathcal{O}_{Y, y}$ and $\mathcal{O}_{X_ y, x} = \mathcal{O}_{X, x}/\mathfrak m_ y\mathcal{O}_{X, x}$ are complete intersection rings by our assumptions that $f$ and $p$ are syntomic, see Lemma 29.30.10. The output of Algebra, Lemma 10.135.12 is exactly that $\mathcal{O}_{Y, y}$ is a complete intersection ring! Hence by Lemma 29.30.10 again we see that $Y$ is syntomic over $k$ at $y$ as desired. $\square$

[1] In fact, if $f$ is surjective, flat, and locally of finite presentation and $p$ is syntomic, then both $q$ and $f$ are syntomic, see Descent, Lemma 35.14.7.

Comments (2)

Comment #6024 by Student on

In the footnote, it is enough to have surjective, flat, and locally of finite presentation, as in the statement of Lemma 05B7.

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