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$$S$$-Modules in the Category of Schemes
Po Hu, University of Chicago, IL
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Memoirs of the American Mathematical Society
2003; 125 pp; softcover
Volume: 161
ISBN-10: 0-8218-2956-4
ISBN-13: 978-0-8218-2956-1
List Price: US$59 Individual Members: US$35.40
Institutional Members: US\$47.20
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 "point-set" 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 coordinate-free algebraic spectra, which are indexed on an universe, which is an infinite-dimensional 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 half-smash product. Based on this category of coordinate-free 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.

• Introduction
• Preliminaries
• Coordinate-free 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 half-smash 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