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Tannakization in derived algebraic geometry

Published online by Cambridge University Press:  22 May 2015

Isamu Iwanari
Isamu Iwanari*
Affiliation:
Mathematical Institute, Tohoku University, Sendai, Miyagi, 980-8578, Japan, [email protected]
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Abstract

In this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

Keywords

tannakian constructionsymmetric monoidal ∞-categorymotives18D1018G5555U30
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 3 , December 2014 , pp. 642 - 700
Copyright
Copyright © ISOPP 2014 

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