nLab
perfect module

Context

Higher algebra

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

In the generality of higher algebra:

Definition

Let RR be an A-∞ ring. Write RMod perfRModR Mod^{perf} \hookrightarrow R Mod for the smallest stable (∞,1)-category inside that of all ∞-modules which contains RR and is closed under retracts. An object in RModR Mod is called a perfect RR-module .

(HA, def. 8.2.5.1)

Properties

Relation to compact and dualizable objects

Propositon

Let RR be an A-∞ ring. The (∞,1)-category of ∞-modules RModR Mod is a compactly generated (∞,1)-category and the compact objects coincide with the perfect modules, def. 1

If RR is commutative (E-∞) then the perfect modules (and hence the compact objects) also coincide with the dualizable objects.

The first statement is (HA, prop. 8.2.5.2), the second (HA, prop. 8.2.5.4). For perfect chain complexes this also appears as (BFN 08, lemma 3.5).

Examples

finite objects:

geometrymonoidal category theorycategory theory
perfect module(fully-)dualizable objectcompact object

References

For perfect chain complexes see the references there.

In the general context of higher algebra perfect modules are discussed in

For the properties of perfect modules in derived algebraic geometry, see

Revised on February 5, 2015 10:35:04 by Adeel Khan (132.252.63.131)