nLab perfect module

Definition

In the generality of higher algebra:

Definition

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

(HA, def. 8.2.5.1)

Properties

Relation to compact and dualizable objects

Propositon

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

If $R$ 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

• perfect chain complex

Related concepts

finite objects:

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

References

For perfect chain complexes see the references there.

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

• Jacob Lurie, sectin 8.2.5 of Higher Algebra

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

• David Ben-Zvi, John Francis, David Nadler, section 3.1 of Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 (arXiv:0805.0157)

• B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.

• Dennis Gaitsgory, Notes on geometric Langlands: Quasi-coherent sheaves.