Topological André-quillen Homology for Cellular Commutative S-algebras

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 homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic S-algebras. Introduction In this paper we give an account of some results on topological André-Quillen homology and cohomology for CW commutative A-algebras, where A is a commutative S-algebra. The main goal is to develop arguments based on skeletal filtrations with a view to continuing the work begun in [11] by extending results of [2] to the case of CW commutative S-algebras. Some of this work originally appeared in the second author’s PhD thesis [9], but we go further and use it to investigate some examples. Our main sources on topological André-Quillen (co)homology include [3–6,15]. For definiteness, we work in the model category of commutative S-algebras described in [7]; following the referee’s suggestions we write cofibration or fibration in place of q-cofibration or q-fibration. We use the standard notions ։, ֌ and ∼ −→ to denote fibrations, cofibrations and weak equivalences, respectively. For general notions of model categories, see [10]. Given a model category M , we will denote its homotopy category by hM . Thus for a commutative Salgebra A, MA denotes the category of A-modules and hMA denotes its homotopy category; the latter is written DA in [7], but we follow [3]. Where necessary, we will assume that (co)fibrant replacements are made. We will often consider a map of commutative S-algebras A −→ B. The notation B/A or B|A is used in Basterra [3] and Quillen [16] to indicate such a pair of S-algebras. This notation is not ideal given the appearance of algebras over and under a given one, therefore we follow the suggestions of the referee in adopting alternatives which we hope are more suitable. In particular, we use the traditional ‘pair’ notation when discussing (co)homology, writing (B,A) for A −→ B, thus following [4]. Date: 12/05/2008 Version 4 arXiv:0708.2041 . 2000 Mathematics Subject Classification. Primary 55P43; Secondary 13D03, 55N35, 55P48.