An algebraic model for commutative \(H\mathbb{Z}\)-algebras.

*(English)*Zbl 1381.55007The main purpose of this paper is to show that the homotopy category of commutative algebra spectra over the Eilenberg-Mac Lane spectrum \(HR\) of an arbitrary commutative ring \(R\) is equivalent to the homotopy category of \(E_\infty\)-monoids in unbounded chain complexes over \(R\) by providing an explicit chain of Quillen equivalences. A similar equivalence was established by B. Shipley [Am. J. Math. 129, No. 2, 351–379 (2007; Zbl 1120.55007)] comparing algebra spectra over \(HR\) to differential graded \(R\)-algebras; in this paper the commutative case is established. The existence of an explicit chain of Quillen equivalences was suggested by M. A. Mandell [Adv. Math. 177, No. 2, 227–279 (2003; Zbl 1027.55009)]; this paper provides the desired chain.

The authors proceed by demonstrating a chain of Quillen equivalences between various categories as follows:

(1) There is a Quillen equivalence between commutative \(HR\)-algebra spectra and commutative symmetric ring spectra of simplicial \(R\)-modules induced by the Quillen equivalence between \(HR\)-module spectra and symmetric spectra of simplicial \(R\)-modules. (2) Using a variation of the Dold-Kan correspondence, the authors establish a Quillen equivalence between commutative symmetric ring spectra of simplicial \(R\)-modules and commutative symmetric ring spectra in nonnegatively graded chain complexes over \(R\). (3) The inclusion functor from nonnegatively graded chain complexes to unbounded chain complexes, together with its adjoint the truncation functor, is used to establish a Quillen equivalence between commutative symmetric ring spectra in nonnegatively graded chain complexes and commutative symmetric ring spectra of unbounded chain complexes. (4) The existence of certain right-induced model structures on categories of algebras over operads in symmetric spectra implies that commuative monoids of symmetric spectra of unbounded chain complexes and \(E_\infty\)-monoids in symmetric spectra of unbounded chain complexes are Quillen equivalent. (5) The Quillen equivalence between \(E_\infty\)-monoids in symmetric spectra of unbounded chain complexes over \(R\) and \(E_\infty\)-monoids in unbounded chain complexes over \(R\) is induced by the functor which evaluates a symmetric spectrum at its 0-th object, and its adjoint.

Finally, an interpretation of these results in terms of the diagram categories indexed by the category of finite sets and injections is provided. These diagram categories can be thought of as an alternative approach to symmetric spectra, see Proposition 9.1.

Many of the equivalences in this chain are modifications or extensions of Quillen equivalence used elsewhere in the literature. For example, a Dold-Kan correspondence similar to the one in step (2) for commutative monoids in symmetric sequences is established by B. Richter [Isr. J. Math. 209, Part 2, 651–682 (2015; Zbl 1378.13009)], but in the pointed case. Section 5 of this paper establishes this for the setting of positive model structures. The existence of right-induced model structures used in step (4) uses results of D. Pavlov and J. Scholbach [“Admissibility and rectification of coloured symmetric operads”, Preprint, arXiv:1410.5675; “Symmetric operads in abstract symmetric spectra”, Preprint, arXiv:1410.5699], but an alternative proof of the necessary results based on S. G. Chadwick and M. A. Mandell’s work in [Geom. Topol. 19, No. 6, 3193–3232 (2015; Zbl 1335.55008)] is presented.

Throughout, the paper gives a clear exposition of the construction of the Quillen equivalences, and presents the various categories and model structures needed for explicit constructions well.

The authors proceed by demonstrating a chain of Quillen equivalences between various categories as follows:

(1) There is a Quillen equivalence between commutative \(HR\)-algebra spectra and commutative symmetric ring spectra of simplicial \(R\)-modules induced by the Quillen equivalence between \(HR\)-module spectra and symmetric spectra of simplicial \(R\)-modules. (2) Using a variation of the Dold-Kan correspondence, the authors establish a Quillen equivalence between commutative symmetric ring spectra of simplicial \(R\)-modules and commutative symmetric ring spectra in nonnegatively graded chain complexes over \(R\). (3) The inclusion functor from nonnegatively graded chain complexes to unbounded chain complexes, together with its adjoint the truncation functor, is used to establish a Quillen equivalence between commutative symmetric ring spectra in nonnegatively graded chain complexes and commutative symmetric ring spectra of unbounded chain complexes. (4) The existence of certain right-induced model structures on categories of algebras over operads in symmetric spectra implies that commuative monoids of symmetric spectra of unbounded chain complexes and \(E_\infty\)-monoids in symmetric spectra of unbounded chain complexes are Quillen equivalent. (5) The Quillen equivalence between \(E_\infty\)-monoids in symmetric spectra of unbounded chain complexes over \(R\) and \(E_\infty\)-monoids in unbounded chain complexes over \(R\) is induced by the functor which evaluates a symmetric spectrum at its 0-th object, and its adjoint.

Finally, an interpretation of these results in terms of the diagram categories indexed by the category of finite sets and injections is provided. These diagram categories can be thought of as an alternative approach to symmetric spectra, see Proposition 9.1.

Many of the equivalences in this chain are modifications or extensions of Quillen equivalence used elsewhere in the literature. For example, a Dold-Kan correspondence similar to the one in step (2) for commutative monoids in symmetric sequences is established by B. Richter [Isr. J. Math. 209, Part 2, 651–682 (2015; Zbl 1378.13009)], but in the pointed case. Section 5 of this paper establishes this for the setting of positive model structures. The existence of right-induced model structures used in step (4) uses results of D. Pavlov and J. Scholbach [“Admissibility and rectification of coloured symmetric operads”, Preprint, arXiv:1410.5675; “Symmetric operads in abstract symmetric spectra”, Preprint, arXiv:1410.5699], but an alternative proof of the necessary results based on S. G. Chadwick and M. A. Mandell’s work in [Geom. Topol. 19, No. 6, 3193–3232 (2015; Zbl 1335.55008)] is presented.

Throughout, the paper gives a clear exposition of the construction of the Quillen equivalences, and presents the various categories and model structures needed for explicit constructions well.

Reviewer: Kristine Bauer (Calgary)

##### MSC:

55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |

##### Keywords:

Eilenberg-Mac Lane spectra; symmetric spectra; \(E_\infty\)-differential graded algebras; Dold-Kan correspondence; model categories
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\textit{B. Richter} and \textit{B. Shipley}, Algebr. Geom. Topol. 17, No. 4, 2013--2038 (2017; Zbl 1381.55007)

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