nLab
module spectrum

Idea

A module 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 $E_{\mathrm{\infty }}$-modules over $R$ as objects in the stabilization of the over-(∞,1)-category over $R$:

Proposition

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

$$\mathrm{Stab}(E_{\mathrm{\infty }}/R)\simeq A\mathrm{Mod}(\mathrm{Spec})$$Stab(E_\infty/R) \simeq A Mod(Spec)

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

This is (Lurie, cor. 1.5.15).

Stable Dold-Kan correspondence

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

Theorem

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

$$HR\mathrm{Mod}\simeq {\mathrm{Ch}}_{•}(R\mathrm{Mod})$$H R Mod \simeq Ch_\bullet(R Mod)

between model structure on $HR$-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 $(\mathrm{\infty },1)$-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 $HR$-algebra spectra and dg-algebras.

Example

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

Under the stable Dold-Kan correspondence, prop. 1, this ought to be identified with the smash product $({\Sigma }_{+}^{\mathrm{\infty }}X)\wedge HR$ 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

$${\pi }_{•}(({\Sigma }_{+}^{\mathrm{\infty }}X)\wedge HR)\simeq H_{•}(X,R).$$\pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R) \,.

While the correspondence $({\Sigma }_{+}^{\mathrm{\infty }}X)\wedge HR\sim C_{•}(X,R)$ 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 $(\mathrm{\infty },1)$-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

• ring spectrum, algebra spectrum

References

A comprehensive general discussion is in

• Jacob Lurie, Deformation Theory

The equivalence between the homotopy categories of $HR$-mopdule spectra and ${\mathrm{Ch}}_{•}(R\mathrm{Mod})$ 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.