Perfect complex - Knowlpedia

Definition

Let $(X,\mathcal O_X)$ be a ringed space, scheme, or algebraic stack. A complex $E$ of $\mathcal O_X$ -modules is perfect if every point has a neighborhood $U$ on which $E|_U$ is quasi-isomorphic to a bounded complex of finite-rank locally free $\mathcal O_U$ -modules.

For a ring $A$ , this says that $E\in D(A)$ is quasi-isomorphic to a bounded complex of finitely generated projective $A$ -modules.

Characterizations

Under standard quasi-compactness and quasi-separatedness hypotheses, perfect complexes are precisely the compact objects of the derived category of quasi-coherent complexes. They are also the dualizable objects for the derived tensor product. The derived dual is

E^\vee=R\mathcal Hom(E,\mathcal O_X).

The perfect complexes form a stable symmetric monoidal category $\operatorname{Perf}(X)$ .

Spectral actions

For the derived stack of local $L$ -parameters, the category $\operatorname{Perf}(\operatorname{LocSys}_{\widehat G})$ acts on sheaves on $\operatorname{Bun}_G$ in the Fargues–Scholze spectral action. Using perfect rather than arbitrary quasi-coherent complexes provides dualizability and finite behavior needed by the action.

References

The Stacks Project Authors, “Cohomology of Sheaves,” §20.49, “Perfect complexes.” Stacks Project.
R. W. Thomason and Thomas Trobaugh, “Higher algebraic K-theory of schemes and of derived categories,” in The Grothendieck Festschrift III, Progress in Mathematics 88, 1990, 247–435.