The Stacks project

Lemma 37.56.8. Let $i : X \to Y$ morphism of schemes locally of finite type over a base scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed subset of $Y$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.

Proof. By Morphisms, Lemma 29.45.4 the morphism $i$ is affine. Thus we may assume $S$, $Y$, and $X$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$, $Y = \mathop{\mathrm{Spec}}(A)$, and $X = \mathop{\mathrm{Spec}}(B)$. The condition means that $A/\text{rad}(A) \to B/\text{rad}(B)$ is surjective; here $\text{rad}(A)$ and $\text{rad}(B)$ denote the Jacobson radical of $A$ and $B$. As $B$ is of finite type over $A$, we can find $b_1, \ldots , b_ m \in \text{rad}(B)$ which generate $B$ as an $A$-algebra. Say $b_ j^ N = 0$ for all $j$. Consider the diagram of rings

\[ \xymatrix{ B & R[x_ i, y_ j]/(y_ j^ N) \ar[l] & R[x_ i, y_ j] \ar[l] \\ A \ar[u] & R[x_ i] \ar[l] \ar[u] \ar[ru] } \]

which translates into a diagram

\[ \xymatrix{ X \ar[d] \ar[r] & T \ar[d] \ar[r] & \mathbf{A}^{n + m}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

of affine schemes. By Lemma 37.56.6 we see that $E$ is $m$-pseudo-coherent relative to $S$ if and only if its pushforward to $\mathbf{A}^{n + m}_ S$ is $m$-pseudo-coherent. By Derived Categories of Schemes, Lemma 36.12.5 we see that this is true if and only if its pushforward to $T$ is $m$-pseudo-coherent. The same lemma shows that this holds if and only if the pushforward to $\mathbf{A}^ n_ S$ is $m$-pseudo-coherent. Again by Lemma 37.56.6 this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$. $\square$

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