# S-Modules in the Category of Schemes

by Po Hu

ISBN-10: 0821829564

ISBN-13: 9780821829561

Pub. Date: 01/02/2003

Publisher: American Mathematical Society

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.

## Overview

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.

## Product Details

ISBN-13:
9780821829561
Publisher:
American Mathematical Society
Publication date:
01/02/2003
Series:
Memoirs of the American Mathematical Society Series, #161
Pages:
125
Product dimensions:
6.90(w) x 9.80(h) x 0.40(d)

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