Commutative S-algebras of prime characteristics and applications to unoriented bordism

@article{Szymik2012CommutativeSO,
title={Commutative S-algebras of prime characteristics and applications to unoriented bordism},
author={Markus Szymik},
journal={arXiv: Algebraic Topology},
year={2012}
}
• Markus Szymik
• Published 14 November 2012
• Mathematics
• arXiv: Algebraic Topology
The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the versal examples S//p 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 Andr\'e-Quillen invariants of the S//p. Among other applications, we show that S//p is not a commutative algebra over the Eilenberg-Mac Lane spectrum HF_p, although the converse is clearly true…
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References

SHOWING 1-10 OF 68 REFERENCES
String bordism and chromatic characteristics
• Markus Szymik
• Mathematics
Homotopy Theory: Tools and Applications
• 2019
We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism
Topological André-Quillen homology for cellular commutative S-algebras
• Mathematics
• 2007
Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a
SOME PROPERTIES OF THE THOM SPECTRUM OVER LOOP SUSPENSION OF COMPLEX PROJECTIVE SPACE
• Mathematics
• 2014
This note provides a reference for some properties of the Thom spectrum M over ΩΣCP 1 . Some of this material is used in recent work of Kitchloo and Morava. We determine the M -cohomology of CP 1 and
GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA
We introduce the notion of a Galois extension of commutative S-algebras (E1 ring spectra), often localized with respect to a flxed homology theory. There are numerous examples, including some
Uniqueness of E-infinity structures for connective covers
• Mathematics
• 2005
We refine our earlier work on the existence and uniqueness of E∞ structures on K- theoretic spectra to show that at each prime p, the connective Adams summand l has a unique structure as a
Cores of spaces, spectra, and E∞ ring spectra
• Mathematics
• 2001
In a paper that has attracted little notice, Priddy showed that the Brown-Peterson spectrum at a prime p can be constructed from the p-local sphere spectrum S by successively killing its odd
UNIQUENESS OF E∞ STRUCTURES FOR CONNECTIVE COVERS
We refine our earlier work on the existence and uniqueness of E∞ structures on K-theoretic spectra to show that at each prime p, the connective Adams summand ` has an essentially unique structure as
Correction to: HZ-algebra spectra are differential graded algebras
This correction article is actually unnecessary. The proof of Theorem 1.2, concerning commutative HQ-algebra spectra and commutative differential graded algebras, in the author's paper [American
Topological Hochschild homology of Thom spectra which are E∞‐ring spectra
We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E∞ classifying map X → BG for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison