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

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 arXiv, (Unpublished)

Full text not currently available from Enlighten.

## 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Item Type: Articles S-module, S-algebra, cell algebra, topological Andre-Quillen (co)homology, power operations Unpublished No Baker, Dr Andrew Baker, A. College of Science and Engineering > School of Mathematics and Statistics > Mathematics arXiv 09 August 2012 arXiv

University Staff: Request a correction | Enlighten Editors: Update this record