Calculating with topological Andr\'e-Quillen theory, I: Homotopical properties of universal derivations and free commutative $S$-algebras

Abstract

We adopt the 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 functor $\mathbb{P}_R$ 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 many of our results could no doubt be obtained using stabilisation, our approach seems more direct. We also discuss a reduced free algebra functor $\tilde{\mathbb{P}}_R$.