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(∞,1)-category of (∞,1)-modules

Contents

Contents

Idea

For RCRing R \in CRing_\infty an E-∞ ring, the (∞,1)-modules over RR with homomorphisms between them form an (∞,1)-category, the (,1)(\infty,1)-category of (,1)(\infty,1)-modules over RR.

Properties

Compact generation, dualizable and perfect 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

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 chain complexes this also appears as (BFN 08, lemma 3.5).

Stable Dold-Kan correspondence

For RR an ordinary ring, write HRH R for the corresponding Eilenberg-MacLane spectrum.

Theorem

For RR any ring (or ringoid, even) there is a Quillen equivalence

HRModCh (RMod) H R Mod \simeq Ch_\bullet(R Mod)

between model structure on HRH R-module spectra and the model structure on chain complexes (unbounded) of ordinary RR-modules.

This presents a corresponding equivalence of (∞,1)-categories. If RR is a commutative ring, then this is an equivalence of symmetric monoidal (∞,1)-categories.

This equivalence on the level of homotopy categories is due to (Robinson). The refinement to a Quillen equivalence is (SchwedeShipley, theorem 5.1.6). See also the discussion at stable model categories. A direct description as an equivalence of (,1)(\infty,1)-categories appears as (Lurie, theorem 7.1.2.13).

Remark

This is a stable version of the Dold-Kan correspondence.

See at algebra spectrum_ for the corresponding statement for HRH R-algebra spectra and dg-algebras.

Periodicity

For EE a periodic ring spectrum, then EModE Mod ought to inherit a /2\mathbb{Z}/2\mathbb{Z}-∞-action. See at periodic ring spectrum – Periodicity of modules

References

Modules over algebras over an arbitrary (∞,1)-operad are discussed in section 3.3 of

Modules specifically over A-∞ algebras are discussed in section 4.2 there.

The equivalence between the homotopy categories of HRH R-module spectra and Ch (RMod)Ch_\bullet(R Mod) is due to

  • Alan Robinson, The extraordinary derived category , Math. Z. 196 (2) (1987) 231–238.

The refinement of this statement to a Quillen equivalence is due to

Discussion in the context of derived algebraic geometry includes

Last revised on November 1, 2014 at 23:31:09. See the history of this page for a list of all contributions to it.