Remark 36.35.14 (0DI9)—The Stacks project

Remark 36.35.14. Our Definition 36.35.1 of a relatively perfect complex is equivalent to the one given in [lieblich-complexes] whenever our definition applies ¹. Next, suppose that $f : X \to S$ is only assumed to be locally of finite type (not necessarily flat, nor locally of finite presentation). The definition in the paper cited above is that $E \in D(\mathcal{O}_ X)$ is relatively perfect if

locally on $X$ the object $E$ should be quasi-isomorphic to a finite complex of $S$-flat, finitely presented $\mathcal{O}_ X$-modules.

On the other hand, the natural generalization of our Definition 36.35.1 is

$E$ is pseudo-coherent relative to $S$ (More on Morphisms, Definition 37.59.2) and $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$ (Cohomology, Definition 20.48.1).

The advantage of condition (B) is that it clearly defines a triangulated subcategory of $D(\mathcal{O}_ X)$, whereas we suspect this is not the case for condition (A). The advantage of condition (A) is that it is easier to work with in particular in regards to limits.

[1] To see this, use Lemma 36.35.3 and More on Algebra, Lemma 15.83.4.

Comments (1)

Comment #9834 by Noah Olander on October 29, 2024 at 20:19

Is there maybe an unstated fact that if $X \to S$ is pseudocoherent then $E \in D(\mathcal{O}_X)$ is pseudocoherent iff it's pseudocoherent relative to $S$ ? Maybe this is in the Stacks Project somewhere and I missed it. As it stands I don't think you've explained why (B) is equivalent to $S$ -perfect when $X \to S$ is finitely presented and flat because in the definition of $S$ -perfect you say $E$ is pseudocoherent and in (B) you say $E$ is pseudocoherent relative to $S$ .

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