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

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.

Comment #6024 by Student on

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

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02K3. Beware of the difference between the letter 'O' and the digit '0'.