eBook ISBN:  9781470403652 
Product Code:  MEMO/161/767.E 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $37.20 
eBook ISBN:  9781470403652 
Product Code:  MEMO/161/767.E 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $37.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 161; 2003; 125 ppMSC: Primary 55; Secondary 14
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.
ReadershipGraduate student and research mathematicians.

Table of Contents

Chapters

Introduction

1. Preliminaries

2. Coordinatefree spectra

3. Coordinatized prespectra

4. Comparison with coordinatized spectra

5. The stable simplicial model structure

6. The $A^1$local model structure

7. Characterization of $A^1$weak equivalences

8. Change of universe

9. The space of linear injections preserving finite subspaces

10. Twisted halfsmash products and twisted function spectra

11. The category of $\mathbb {L}$spectra

12. Unital properties of $\mathbb {L}$spectra

13. The category of $S$modules

14. $S$algebras and their modules

15. Proofs of the model structure theorems

16. Technical results on the extended injections operad

17. Appendix: Small objects in the category of simplicial sheaves


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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.
Graduate student and research mathematicians.

Chapters

Introduction

1. Preliminaries

2. Coordinatefree spectra

3. Coordinatized prespectra

4. Comparison with coordinatized spectra

5. The stable simplicial model structure

6. The $A^1$local model structure

7. Characterization of $A^1$weak equivalences

8. Change of universe

9. The space of linear injections preserving finite subspaces

10. Twisted halfsmash products and twisted function spectra

11. The category of $\mathbb {L}$spectra

12. Unital properties of $\mathbb {L}$spectra

13. The category of $S$modules

14. $S$algebras and their modules

15. Proofs of the model structure theorems

16. Technical results on the extended injections operad

17. Appendix: Small objects in the category of simplicial sheaves