Ring spectra and modules

Cohomology theories are almost never encountered as bare graded abelian groups: they usually carry multiplicative structure. Ordinary cohomology \(H^*(X;\Z )\) comes equipped with a cup product, turning it into a graded ring, and the K-theory of a space (introduced in the next chapter) carries a product coming from the tensor product of vector bundles. Combining Theorem 3.2.6 with the Yoneda lemma, the cup product on \(H^*(X;\Z )\) is induced by a multiplication \(H\Z \otimes H\Z \to H\Z \) on the representing spectrum itself, which is unital, associative and commutative up to homotopy. The resulting structure on \(H\Z \) is that of a homotopy ring spectrum, and the spectrum representing K-theory carries a similar multiplicative structure.

Once spectra rather than cohomology theories are taken as the primary objects of study, it becomes natural to view this multiplication on \(H\Z \) as the shadow of a much richer structure, that of a ring spectrum: a spectrum \(R\) equipped with a multiplication map \(R \otimes R \to R\) which is coherently unital and associative, much like the coherent multiplication of a monoid object in Chapter 5. Alongside it comes the notion of an \(R\)-module: a spectrum equipped with a coherent action of \(R\).

Where ordinary algebra takes place over the base ring \(\Z \), the algebra of ring spectra takes place over the sphere spectrum \(\S \), the monoidal unit for the tensor product on \(\Sp \). As the homotopy groups of \(\S \) are the stable stems, algebra over the sphere is a substantially richer subject than algebra over \(\Z \). Ordinary algebra can be recovered as a fragment of this theory: the Eilenberg–MacLane construction \(A \mapsto HA\) promotes an ordinary ring to a ring spectrum. This also recovers the homological algebra from Chapter 6: for a commutative ring \(R\) we will prove an equivalence \[ \D (R) \simeq \Mod _{HR}(\Sp ), \] identifying the derived \(\infty \)-category of \(R\)-modules with the \(\infty \)-category of \(HR\)-modules. The tools from homological algebra all admit counterparts over an arbitrary ring spectrum.

We begin in Section 8.1 by recording the package of coherent algebra and module theory that we will use from Part II. Section 8.2 studies the interaction of modules with connectivity, and Section 8.3 carries out the comparison with ordinary rings and with differential graded algebras sketched above. We then construct localizations of commutative ring spectra in Section 8.4. The online version concludes with supplementary treatments of Ore localization, projective and flat modules, and Tor-amplitude.

The homotopy coherent algebra underlying all of this is the theory of \(\infty \)-operads, which is developed at length in Part II. In the present chapter we use that theory as a black box. More precisely, we record in Proposition 8.1.5 the constructions and properties that will be used below. A reader following Part I may take this package as given; none of the later arguments requires familiarity with operads.

This chapter also leans on [Lurie (2017)] more heavily than the rest of Part I. Results quoted without proof are flagged as black boxes where they occur. Several further results that are not used elsewhere in the book are retained in the online-only supplements.

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