Calculating with topological André-Quillen theory, I: homotopical properties of universal derivations and free commutative S-algebras

Baker, A. (2012) Calculating with topological André-Quillen theory, I: homotopical properties of universal derivations and free commutative S-algebras. arXiv, (Unpublished)

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Abstract

We adopt a viewpoint that topological And\'e-Quillen theory for commutative $S$-algebras should provide usable (co)homology theories for doing calculations in the sense traditional within Algebraic Topology. Our main emphasis is on homotopical properties of universal derivations, especially their behaviour in multiplicative homology theories. There are algebraic derivation properties, but also deeper properties arising from the homotopical structure of the free algebra construction $\mathbb{P}_R$ and its relationship with extended powers of spectra. In the connective case in ordinary $\bmod{p}$ homology, this leads to useful formulae involving Dyer-Lashof operations in the homology of commutative $S$-algebras. Although some of our results should be obtainable using stabilisation, our approach seems more direct. We also discuss a reduced free algebra construction $\tilde{\mathbb{P}}_R$.

Item Type:Articles
Keywords:S-module, S-algebra, cell algebra, topological Andre-Quillen (co)homology, power operations
Status:Unpublished
Refereed:No
Glasgow Author(s) Enlighten ID:Baker, Dr Andrew
Authors: Baker, A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:arXiv
Published Online:09 August 2012
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