Moduli spaces of commutative ring spectra.

*(English)*Zbl 1086.55006
Baker, Andrew (ed.) et al., Structured ring spectra. Cambridge: Cambridge University Press (ISBN 0-521-60305-6/pbk). London Mathematical Society Lecture Note Series 315, 151-200 (2004).

The notions of \(A_\infty\) and \(E_\infty\) ring spectra arose in the 1970’s and were carefully developed in work of J. P. May and his collaborators and codify the strict multiplicative structure on many naturally occurring ring spectra such as those associated with \(K\)-theory and cobordism theories [J. P. May, \(E_\infty\) ring spaces and \(E_\infty\) ring spectra I, Lect. Notes Math. 577. Springer Verlag (1977; Zbl 0345.55007)]. Amongst the motivation for the work in the paper under review lies the ever increasing importance of the cohomology theories derived by applications of formal group techniques and Landweber’s Exact Functor Theorem, in particular various versions of elliptic cohomology were produced this way. As these theories lack geometric descriptions, there were no known associated rigid multiplications.

In the late 1980’s, A. Robinson [Lond. Math. Soc. Lect. Note Ser. 139, 143–152 (1989; Zbl 0688.55008)] produced an obstruction theory for the existence of \(A_\infty\) structures on ring spectra and showed that at odd primes \(p\) each Morava \(K\)-theory \(K(n)\) admits uncountably many such structures. Shortly thereafter, in [Q. J. Math., Oxf. II. Ser. 42, No. 168, 403–419 (1991; Zbl 0772.55003)] the reviewer extended this result by showing that the \(I_n\)-adic completion \(\widehat{E(n)}\) of the Johnson-Wilson spectrum \(E(n)\) admitted an essentially unique \(A_\infty\) structure. This result was followed by the Hopkins-Miller theorem [C. Rezk, Contemp. Math. 220, 313–366 (1998; Zbl 0910.55004)] on the related Lubin-Tate spectrum \(E_n\) and similar results on elliptic spectra. Robinson and S. A. Whitehouse later developed an obstruction theory for \(E_\infty\) structures on ring spectra [A. Robinson and S. A. Whitehouse, Math. Proc. Camb. Philos. Soc. 132, No. 2, 197–234 (2002; Zbl 0997.18004), A. Robinson, London Mathematical Society Lecture Note Series 315, 133–149 (2004; Zbl 1082.55005)] based on \(\Gamma\)-cohomology, which has many relations with André-Quillen cohomology of commutative algebras.

It is worth remarking that while these developments were occurring, new insights into the construction of categories of spectra before passage to derived homotopy categories led to the introduction of such categories with strictly associative and commutative smash products. In the category of \(S\)-modules of [A. D. Elmendorf, I. Kriz, M. Mandell and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs. 47. AMS (1997; Zbl 0894.55001)], the monoid objects correspond to \(A_\infty\) ring spectra, while the commutative monoid objects correspond to \(E_\infty\) ring spectra. Thus it is now much more meaningful to study objects with these properties in a manner similar to rings and their modules in algebra. It should noted that subsequently other approaches to these foundational questions have emerged, but it appears to be the case that their model categories are essentially equivalent.

In the present important paper (which existed in preprint form for several years) the authors build an obstruction theory for realizing \(A_\infty\) or \(E_\infty\) structures on ring spectra. The input to this theory includes work of Dwyer and Kahn on moduli problems, \(E_2\) model category structures of Dwyer, Kahn and Stover and André-Quillen type cohomology theory developed earlier by the authors [Contemp. Math. 265, 41–85 (2000; Zbl 0999.18009)]. The authors are careful to spell out the properties of a category of spectra required for their work, so readers should be able to apply it in any reasonable version. The machinery developed is described carefully and leads to algebraic cohomology groups which are now known to be equivalent to the \(\Gamma\)-cohomology of Robinson-Whitehouse, see the companion article [M. Basterra and B. Richter, London Mathematical Society Lecture Note Series 315, 115–131 (2004; Zbl 1079.13008)]. The authors apply their obstruction theory to the Lubin-Tate spectra which are shown to have unique \(E_\infty\) structures. Further applications have been made: see for example [A. Baker and B. Richter, Comment. Math. Helv. 80, No.4, 691-723 (2005; Zbl 1094.55010)] and recent work of J. Rognes and others on the Galois theory of commutative \(S\)-algebras.

For the entire collection see [Zbl 1051.55001].

In the late 1980’s, A. Robinson [Lond. Math. Soc. Lect. Note Ser. 139, 143–152 (1989; Zbl 0688.55008)] produced an obstruction theory for the existence of \(A_\infty\) structures on ring spectra and showed that at odd primes \(p\) each Morava \(K\)-theory \(K(n)\) admits uncountably many such structures. Shortly thereafter, in [Q. J. Math., Oxf. II. Ser. 42, No. 168, 403–419 (1991; Zbl 0772.55003)] the reviewer extended this result by showing that the \(I_n\)-adic completion \(\widehat{E(n)}\) of the Johnson-Wilson spectrum \(E(n)\) admitted an essentially unique \(A_\infty\) structure. This result was followed by the Hopkins-Miller theorem [C. Rezk, Contemp. Math. 220, 313–366 (1998; Zbl 0910.55004)] on the related Lubin-Tate spectrum \(E_n\) and similar results on elliptic spectra. Robinson and S. A. Whitehouse later developed an obstruction theory for \(E_\infty\) structures on ring spectra [A. Robinson and S. A. Whitehouse, Math. Proc. Camb. Philos. Soc. 132, No. 2, 197–234 (2002; Zbl 0997.18004), A. Robinson, London Mathematical Society Lecture Note Series 315, 133–149 (2004; Zbl 1082.55005)] based on \(\Gamma\)-cohomology, which has many relations with André-Quillen cohomology of commutative algebras.

It is worth remarking that while these developments were occurring, new insights into the construction of categories of spectra before passage to derived homotopy categories led to the introduction of such categories with strictly associative and commutative smash products. In the category of \(S\)-modules of [A. D. Elmendorf, I. Kriz, M. Mandell and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs. 47. AMS (1997; Zbl 0894.55001)], the monoid objects correspond to \(A_\infty\) ring spectra, while the commutative monoid objects correspond to \(E_\infty\) ring spectra. Thus it is now much more meaningful to study objects with these properties in a manner similar to rings and their modules in algebra. It should noted that subsequently other approaches to these foundational questions have emerged, but it appears to be the case that their model categories are essentially equivalent.

In the present important paper (which existed in preprint form for several years) the authors build an obstruction theory for realizing \(A_\infty\) or \(E_\infty\) structures on ring spectra. The input to this theory includes work of Dwyer and Kahn on moduli problems, \(E_2\) model category structures of Dwyer, Kahn and Stover and André-Quillen type cohomology theory developed earlier by the authors [Contemp. Math. 265, 41–85 (2000; Zbl 0999.18009)]. The authors are careful to spell out the properties of a category of spectra required for their work, so readers should be able to apply it in any reasonable version. The machinery developed is described carefully and leads to algebraic cohomology groups which are now known to be equivalent to the \(\Gamma\)-cohomology of Robinson-Whitehouse, see the companion article [M. Basterra and B. Richter, London Mathematical Society Lecture Note Series 315, 115–131 (2004; Zbl 1079.13008)]. The authors apply their obstruction theory to the Lubin-Tate spectra which are shown to have unique \(E_\infty\) structures. Further applications have been made: see for example [A. Baker and B. Richter, Comment. Math. Helv. 80, No.4, 691-723 (2005; Zbl 1094.55010)] and recent work of J. Rognes and others on the Galois theory of commutative \(S\)-algebras.

For the entire collection see [Zbl 1051.55001].

Reviewer: Andrew Baker (Glasgow)