The Stacks project

Lemma 37.57.12. The property $\mathcal{P}(f) =$“$f$ is pseudo-coherent” is syntomic local on the source.

Proof. We will use the criterion of Descent, Lemma 35.25.4 to prove this. It follows from Lemmas 37.57.6 and 37.57.4 that being pseudo-coherent is preserved under precomposing with flat morphisms locally of finite presentation, in particular under precomposing with syntomic morphisms (see Morphisms, Lemmas 29.30.7 and 29.30.6). It is clear from Definition 37.57.2 that being pseudo-coherent is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma 35.25.4 it suffices to prove the following: Suppose $X' \to X \to Y$ are morphisms of affine schemes with $X' \to X$ syntomic and $X' \to Y$ pseudo-coherent. Then $X \to Y$ is pseudo-coherent. To see this, note that in any case $X \to Y$ is of finite presentation by Descent, Lemma 35.13.1. Choose a closed immersion $X \to \mathbf{A}^ n_ Y$. By Algebra, Lemma 10.136.18 we can find an affine open covering $X' = \bigcup _{i = 1, \ldots , n} X'_ i$ and syntomic morphisms $W_ i \to \mathbf{A}^ n_ Y$ lifting the morphisms $X'_ i \to X$, i.e., such that there are fibre product diagrams

\[ \xymatrix{ X'_ i \ar[d] \ar[r] & W_ i \ar[d] \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

After replacing $X'$ by $\coprod X'_ i$ and setting $W = \coprod W_ i$ we obtain a fibre product diagram

\[ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^ h \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

with $W \to \mathbf{A}^ n_ Y$ flat and of finite presentation and $X' \to Y$ still pseudo-coherent. Since $W \to \mathbf{A}^ n_ Y$ is open (see Morphisms, Lemma 29.25.10) and $X' \to X$ is surjective we can find $f \in \Gamma (\mathbf{A}^ n_ Y, \mathcal{O})$ such that $X \subset D(f) \subset \mathop{\mathrm{Im}}(h)$. Write $Y = \mathop{\mathrm{Spec}}(R)$, $X = \mathop{\mathrm{Spec}}(A)$, $X' = \mathop{\mathrm{Spec}}(A')$ and $W = \mathop{\mathrm{Spec}}(B)$, $A = R[x_1, \ldots , x_ n]/I$ and $A' = B/IB$. Then $R \to A'$ is pseudo-coherent. Picture

\[ \xymatrix{ A' = B/IB & B \ar[l] \\ A = R[x_1, \ldots , x_ n]/I \ar[u] & R[x_1, \ldots , x_ n] \ar[l] \ar[u] } \]

By Lemma 37.57.11 we see that $IB$ is pseudo-coherent as a $B$-module. The ring map $R[x_1, \ldots , x_ n]_ f \to B_ f$ is faithfully flat by our choice of $f$ above. This implies that $I_ f \subset R[x_1, \ldots , x_ n]_ f$ is pseudo-coherent, see More on Algebra, Lemma 15.64.15. Applying Lemma 37.57.11 one more time we see that $R \to A$ is pseudo-coherent. $\square$

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