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S-Modules in the Category of Schemes (Memoirs of the American Mathematical Society) by Po Hu is a nonfiction available to read on EtoBox.

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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 “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

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Author

Po Hu

Publisher

American Mathematical Society

Published

2003

Language

EN

ISBN

9781470403652

Category

nonfiction

Subjects

Mathematics, Science, History