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

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 37.52.9. The equivalence of (2) and (3) follows from Lemma 37.52.17 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. $\square$

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).