# \(S\)-Modules in the Category of Schemes

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*Po Hu*

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.

#### Table of Contents

# Table of Contents

## $S$-Modules in the Category of Schemes

- Contents vii8 free
- Introduction 110 free
- Chapter 1. Preliminaries 514 free
- Chapter 2. Coordinate-free Spectra 918
- Chapter 3. Coordinatized Prespectra 1726
- Chapter 4. Comparison with Coordinatized Spectra 2534
- Chapter 5. The Stable Simplicial Model Structure 3544
- Chapter 6. The A[sub(1)]-local Model Structure 3948
- Chapter 7. Characterization of A[sub(1)]-Weak Equivalences 4554
- Chapter 8. Change of Universe 4958
- Chapter 9. The Space of Linear Injections Preserving Finite Subspaces 6776
- Chapter 10. Twisted Half-Smash Products and Twisted Function Spectra 7180
- Chapter 11. The Category of L-spectra 8998
- Chapter 12. Unital Properties of L-spectra 97106
- Chapter 13. The Category of S-modules 99108
- Chapter 14. S-algebras and their Modules 105114
- Chapter 15. Proofs of the Model Structure Theorems 109118
- Chapter 16. Technical Results on the Extended Injections Operad 117126
- Chapter 17. Appendix: Small Objects in the Category of Simplicial Sheaves 121130
- Bibliography 125134