nLab module spectrum

Idea

A module spectrum is a module over a ring spectrum. As for ring spectra, this has different interpretations according to how much coherence is considered:

A homotopy module spectrum over a homotopy commutative ring spectrum is a module object in the stable homotopy category equipped with its derived smash product of spectra.

On the other hand if “ring spectrum” means E-∞ ring (or A-∞ ring) then a module over a ring spectrum is a module over an algebra over an (∞,1)-operad for the commutative operad:

for $R$ an E-∞ ring (an ∞-algebra over Comm), an $R$-module spectrum is a spectrum equipped with an $R$-action.

Properties

General

By the discussion an tangent (∞,1)-category we may realize $E_\infty$-modules over $R$ as objects in the stabilization of the over-(∞,1)-category over $R$:

Proposition

Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the (∞,1)-category of E-∞ rings and let $R \in E_\infty$. Then the stabilization of the over-(∞,1)-category over $R$

$$Stab(E_\infty/R) \simeq R Mod(Spec)$$

is equivalent to the category of $R$-module spectra.

This is (Lurie, cor. 1.5.15).

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, remark 7.1.1.16). If $R$ is a commutative ring, then this equivalence underlies an equivalence of symmetric monoidal $(\infty,1)$-categories by (Lurie, theorem 7.1.2.13).

Remark

This is a stable version of the Dold-Kan correspondence. See at stable Dold-Kan correspondence for more.

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

Example

For $X$ a topological space and $R$ a ring, let $C_\bullet(X, R)$ be the standard chain complex for singular homology $H_\bullet(X, R)$ of $X$ with coefficients in $R$.

Under the stable Dold-Kan correspondence, prop. , this ought to be identified with the smash product $(\Sigma^\infty_+ X) \wedge H R$ of the suspension spectrum of $X$ with the Eilenberg-MacLane spectrum. Notice that by the general theory of generalized homology the homotopy groups of the latter are again singular homology

$$\pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R) \,.$$

While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on p. 15 of (BCT).

As fibers for $(\infty,1)$-vector bundles

An ordinary vector bundle is a bundle of $k$-modules for $k$ some ring (which should be a field, or otherwise we’d rather say “module bundle”). Generalizing $k$ here from a ring to a ring spectrum, we may hence regard $K$-module spectra as (∞,1)-vector spaces, and ∞-bundles of these as (∞,1)-vector bundles. See there for more details.

Related concepts

algebra	homological algebra	higher algebra
abelian group	chain complex	spectrum
ring	dg-ring	ring spectrum
module	dg-module	module spectrum

• comodule spectrum

• ring spectrum, algebra spectrum

• (∞,1)-category of (∞,1)-modules

References

The notion of homotopy module spectra (module objects in the stable homotopy category):

• Frank Adams, part III, section 10 of Stable homotopy and generalised homology, 1974

Review:

• Urs Schreiber, Introduction to Stable homotopy theory – Homotopy ring spectra, 2016

Discussion in terms of symmetric spectra:

• Stefan Schwede, chapter IV Symmetric spectra, 2012 (pdf)

and in terms of (infinity,1)-category theory in

• Jacob Lurie, chapter 7 of Higher Algebra.

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)

Applications to string topology are discussed in

• Andrew Blumberg, Ralph Cohen, Constantin Teleman, Open-closed field theories, string topology and Hochschild homology (arXiv:0906.5198)

See the section on string topology at sigma model for more on this.