Algebraic & Geometric Topology

Commutative $\mathbb{S}$–algebras of prime characteristics and applications to unoriented bordism

Markus Szymik

Full-text: Open access


The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the versal examples Sp for prime numbers p. These can be realized as Thom spectra, and therefore relate to other Thom spectra such as the unoriented bordism spectrum MO. We compute the Hochschild and André–Quillen invariants of the Sp. Among other applications, we show that Sp is not a commutative algebra over the Eilenberg–Mac Lane spectrum HFp, although the converse is clearly true, and that MO is not a polynomial algebra over S2.

Article information

Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3717-3743.

Received: 14 May 2014
Accepted: 3 June 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22] 55P20: Eilenberg-Mac Lane spaces 55P42: Stable homotopy theory, spectra

commutative $\mathbb{S}$–algebra characteristic p unoriented bordism


Szymik, Markus. Commutative $\mathbb{S}$–algebras of prime characteristics and applications to unoriented bordism. Algebr. Geom. Topol. 14 (2014), no. 6, 3717--3743. doi:10.2140/agt.2014.14.3717.

Export citation


  • M Ando, J Blumberg, Andrew, D Gepner, M J Hopkins, C Rezk, An $\infty$–categorical approach to $R$–line bundles, $R$–module Thom spectra, and twisted $R$–homology, J. Topol. 7 (2014) 869–893
  • V Angeltveit, Topological Hochschild homology and cohomology of $A\sb \infty$ ring spectra, Geom. Topol. 12 (2008) 987–1032
  • S Araki, H Toda, Multiplicative structures in ${\rm mod}\,q$ cohomology theories, I, Osaka J. Math. 2 (1965) 71–115
  • S Araki, H Toda, Multiplicative structures in ${\rm mod}\sb{q}$ cohomology theories, II, Osaka J. Math. 3 (1966) 81–120
  • L Astey, Commutative $2$–local ring spectra, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 1–10
  • A Baker, Calculating with topological André–Quillen theory, I: Homotopical properties of universal derivations and free commutative $S$–algebras
  • A Baker, $BP$: Close encounters of the $E\sb \infty$ kind, J. Homotopy Relat. Struct. 9 (2014) 553–578
  • A Baker, H Gilmour, P Reinhard, Topological André–Quillen homology for cellular commutative $S$–algebras, Abh. Math. Semin. Univ. Hambg. 78 (2008) 27–50
  • A Baker, B Richter, Uniqueness of $E\sb \infty$ structures for connective covers, Proc. Amer. Math. Soc. 136 (2008) 707–714
  • A Baker, B Richter, Some properties of the Thom spectrum over loop suspension of complex projective space, from: “An Alpine expedition through algebraic topology”, (C Ausoni, K Hess, B Johnson, W Lück, J Scherer, editors), Contemporary mathematics 617, Amer. Math. Soc. (2014)
  • M Basterra, André–Quillen cohomology of commutative $S$–algebras, J. Pure Appl. Algebra 144 (1999) 111–143
  • M Basterra, M A Mandell, Homology and cohomology of $E\sb \infty$ ring spectra, Math. Z. 249 (2005) 903–944
  • A J Blumberg, Topological Hochschild homology of Thom spectra which are $E\sb \infty$ ring spectra, J. Topol. 3 (2010) 535–560
  • J Boardman, Graded Eilenberg–Mac Lane ring spectra, Amer. J. Math. 102 (1980) 979–1010
  • R Bruner, J May, J McClure, M Steinberger, $H\sb \infty $ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer, Berlin (1986)
  • S Chadwick, M Mandell, $E\sb n$ genera
  • F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer, Berlin (1976)
  • A Elmendorf, I Kriz, M Mandell, J May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys Monographs 47, Amer. Math. Soc. (1997)
  • P Hu, I Kriz, J May, Cores of spaces, spectra, and $E\sb \infty$ ring spectra, Homology Homotopy Appl. 3 (2001) 341–354
  • S O Kochman, Homology of the classical groups over the Dyer–Lashof algebra, Trans. Amer. Math. Soc. 185 (1973) 83–136
  • N J Kuhn, Localization of André–Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces, Adv. Math. 201 (2006) 318–378
  • A Lazarev, Cohomology theories for highly structured ring spectra, from: “Structured ring spectra”, LMS Lecture Note Ser. 315, Cambridge Univ. Press (2004) 201–231
  • L Lewis Jr, J May, M Steinberger, J McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer, Berlin (1986)
  • M Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979) 549–559
  • M Mahowald, D C Ravenel, P Shick, The Thomified Eilenberg–Moore spectral sequence, from: “Cohomological methods in homotopy theory”, Progr. Math. 196, Birkhäuser, Basel (2001) 249–262
  • M Mandell, J May, S Schwede, B Shipley, Model categ.ories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • A Pazhitnov, Y Rudyak, On commutative ring spectra of characteristic $2$, Mat. Sb. 124(166) (1984) 486–494 In Russian; translated in Math. USSR–Sbornik, 52 (1985) 471–480
  • S Priddy, Dyer–Lashof operations for the classifying spaces of certain matrix groups, Quart. J. Math. Oxford Ser. 26 (1975) 179–193
  • B Richter, Symmetry properties of the Dold–Kan correspondence, Math. Proc. Cambridge Philos. Soc. 134 (2003) 95–102
  • B Richter, Homotopy algebras and the inverse of the normalization functor, J. Pure Appl. Algebra 206 (2006) 277–321
  • J Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 898, Amer. Math. Soc. (2008)
  • Y Rudyak, The spectra $k$ and $kO$ are not Thom spectra, from: “Group representations: cohomology, group actions and topology”, Proc. Sympos. Pure Math. 63, Amer. Math. Soc. (1998) 475–483
  • R Schwänzl, R Vogt, F Waldhausen, Adjoining roots of unity to $E\sb \infty$ ring spectra in good cases–-a remark, from: “Homotopy invariant algebraic structures”, Contemp. Math. 239, Amer. Math. Soc. (1999) 245–249
  • B Shipley, $H\Bbb Z$–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351–379
  • M Szymik, String bordism and chromtic characteristics
  • U Würgler, Commutative ring-spectra of characteristic $2$, Comment. Math. Helv. 61 (1986) 33–45
  • D Y Yan, On the Thom spectra over $\Omega({\rm SU}(n)/{\rm SO}(n))$ and Mahowald's $X\sb k$ spectra, Proc. Amer. Math. Soc. 116 (1992) 567–573
  • D Y Yan, The Brown–Peterson homology of Mahowald's $X\sb k$ spectra, Trans. Amer. Math. Soc. 344 (1994) 261–289