nLab perfect chain complex

Idea

A perfect complex over a commutative ring $A$ is a perfect module over the Eilenberg-Mac Lane spectrum $H(A)$. Under the stable Dold-Kan correspondence, perfect complexes correspond to bounded chain complexes of finitely generated projective modules.

Viewing commutative rings as affine schemes, this definition generalizes to arbitrary stacks. In this generality, perfect modules still coincide with the dualizable objects, but not always with the compact objects. The latter does hold for quasi-compact quasi-separated schemes by work of Thomason, Neeman, Bondal-Van den Bergh.

Properties

Relation to compact objects

Proposition

Let $A$ be a commutative ring and let $D(A)$ denote the derived category of $A$-modules. A chain complex $M_\bullet$ of $A$-modules is perfect if and only if it is a compact object of $D(A)$.

For instance (Stacks Project, 07LT).

Perfect complexes on a ringed space

Let $(X, \mathcal{O}_X)$ be a ringed space. A chain complex of $\mathcal{O}_X$-modules is called perfect if it is locally quasi-isomorphic to a bounded complex of free? $\mathcal{O}_X$-modules of finite type.

Let $D(Mod(\mathcal{O}_X))$ be the derived category of $\mathcal{O}_X$-modules. Let $Pf(X) \subset D(Mod(\mathcal{O}_X))$ denote the full subcategory of perfect complexes. This is a triangulated subcategory, see triangulated categories of sheaves.

Related concepts

• perfect module

• perfect dg-module

• triangulated categories of sheaves

• determinant of a perfect complex

finite objects:

geometry	monoidal category theory	category theory
perfect module	(fully-)dualizable object	compact object

References

• R. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in The Grothendieck Festschrift, Vol. III (1990), pp. 247-436.

• Alexei Bondal, Michel Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Math. Vol. 3, no. 1 (2003), pp. 1-36, arXiv:math/0204218, pdf.

• Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., vol. 9, no. 1, 1996, pp. 205-236.

• Raphaël Rouquier, Dimensions of triangulated categories, Journal of K-theory, 1 (2008), pp. 1-36, arXiv:math/0310134, pdf.

• The Stacks Project, Characterizing perfect objects

• Jack Hall, David Rydh, Perfect complexes on algebraic stacks, arXiv:1405.1887.