nLab (∞,1)-category of (∞,1)-modules

Idea

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

Properties

Compact generation, dualizable and perfect 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

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. 7.2.4.2), the second (HA, prop. 7.2.4.4). For chain complexes this also appears as (BFN 08, lemma 3.5).

Stable Dold-Kan correspondence

For $R$ an ordinary ring, write $H R$ for the corresponding Eilenberg-MacLane spectrum.

Theorem

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

$$H R Mod \simeq Ch_\bullet(R Mod)$$

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

This presents a corresponding equivalence of (∞,1)-categories. If $R$ 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 $(\infty,1)$-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 $H R$-algebra spectra and dg-algebras.

Periodicity

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

Related concepts

• Mod

• 2Mod

• nMod

References

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

• Jacob Lurie, Higher Algebra

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

The equivalence between the homotopy categories of $H R$-module spectra and $Ch_\bullet(R Mod)$ 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

• Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules , Topology 42 (2003), 103-153 (pdf)

Discussion in the context of derived algebraic geometry includes

• 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)