The Stacks project

Lemma 37.53.18. Let

\[ \xymatrix{ X \ar[rd] \ar[rr]_ i & & P \ar[ld] \\ & S } \]

be a commutative diagram of schemes. Assume $i$ is a closed immersion and $P \to S$ flat and locally of finite presentation. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then the following are equivalent

  1. $E$ is $m$-pseudo-coherent relative to $S$,

  2. $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and

  3. $Ri_*E$ is $m$-pseudo-coherent on $P$.

Proof. The equivalence of (1) and (2) is Lemma 37.53.9. The equivalence of (2) and (3) follows from Lemma 37.53.17 applied to $\text{id} : P \to P$ provided we can show that $\mathcal{O}_ P$ is pseudo-coherent relative to $S$. This follows from More on Algebra, Lemma 15.81.4 and the definitions. $\square$


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