Memoirs of the American Mathematical Society 2003; 125 pp; softcover Volume: 161 ISBN10: 0821829564 ISBN13: 9780821829561 List Price: US$62 Individual Members: US$37.20 Institutional Members: US$49.60 Order Code: MEMO/161/767
 This paper gives a theory \(S\)modules for Morel and Voevodsky's category of algebraic spectra over an arbitrary field \(k\). This is a "pointset" category of spectra which are commutative, associative and unital with respect to the smash product. In particular, \(E{\infty}\)ring spectra are commutative monoids in this category. Our approach is similar to that of 7. We start by constructing a category of coordinatefree algebraic spectra, which are indexed on an universe, which is an infinitedimensional affine space. One issue which arises here, different from the topological case, is that the universe does not come with an inner product. We overcome this difficulty by defining algebraic spectra to be indexed on the subspaces of the universe with finite codimensions instead of finite dimensions, and show that this is equivalent to spectra indexed on the integers. Using the linear injections operad, we also define universe change functors, as well as other important constructions analogous to those in topology, such as the twisted halfsmash product. Based on this category of coordinatefree algebraic spectra, we define the category of \(S\)modules. In the homotopical part of the paper, we give closed model structures to these categories of algebraic spectra, and show that the resulting homotopy categories are equivalent to Morel and Voevodsky's algebraic stable homotopy category. Readership Graduate student and research mathematicians. Table of Contents  Introduction
 Preliminaries
 Coordinatefree spectra
 Coordinatized prespectra
 Comparison with coordinatized spectra
 The stable simplicial model structure
 The \(\mathbb{A}^1\)local model structure
 Characterization of \(\mathbb{A}^1\)weak equivalences
 Change of universe
 The space of linear injections preserving finite subspaces
 Twisted halfsmash products and twisted function spectra
 The category of \(\mathbb{L}\)spectra
 Unital properties of \(\mathbb{L}\)spectra
 The category of \(S\)modules
 \(S\)algebras and their modules
 Proofs of the model structure theorems
 Technical results on the extended injections operad
 Appendix: Small objects in the category of simplicial sheaves
 Bibliography
