nLab perfect module

Context

Higher algebra

higher algebra

universal algebra

Contents

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 $R Mod$ is called a perfect $R$-module .

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. 1

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).

finite objects:

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

Last revised on February 5, 2015 at 10:35:04. See the history of this page for a list of all contributions to it.