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module spectrum

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

A module spectrum is a module over an algebra over an (∞,1)-operad for the commutative operad:

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

Properties

General

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

Proposition

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

Stab(E /R)RMod(Spec) Stab(E_\infty/R) \simeq R Mod(Spec)

is equivalent to the category of RR-module spectra.

This is (Lurie, cor. 1.5.15).

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 stable Dold-Kan correspondence for more.

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

Example

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

Under the stable Dold-Kan correspondence, prop. 1, this ought to be identified with the smash product (Σ + X)HR(\Sigma^\infty_+ X) \wedge H R of the suspension spectrum of XX with the Eilenberg-MacLane spectrum. Notice that by the general theory of generalized homology the homotopy groups of the latter are again singular homology

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

While the correspondence (Σ + X)HRC (X,R)(\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 (,1)(\infty,1)-vector bundles

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

References

A comprehensive general discussion is in

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

Applications to string topology are discussed in

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

Revised on September 27, 2014 16:24:47 by Anonymous Coward (93.131.0.105)