perfect chain complex
objects such that commutes with certain colimits
A perfect complex over a commutative ring is a perfect module over the Eilenberg-Mac Lane spectrum . 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.
Relation to compact objects
For instance (Stacks Project, 07LT).
Perfect complexes on a ringed space
Let be a ringed space. A chain complex of -modules is called perfect if it is locally quasi-isomorphic to a bounded complex of free? -modules of finite type.
Let be the derived category of -modules. Let denote the full subcategory of perfect complexes. This is a triangulated subcategory, see triangulated categories of sheaves.
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.
Revised on November 11, 2015 14:25:52
by Urs Schreiber