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Which abelian tensor categories are geometric?
- Source :
- Journal für die Reine und Angewandte Mathematik; Jan2018, Vol. 2017 Issue 734, p145-186, 42p
- Publication Year :
- 2018
-
Abstract
- For a large class of geometric objects, the passage to categories of quasicoherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties-expressible entirely within the language of tensor categories-which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne's result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products. [ABSTRACT FROM AUTHOR]
- Subjects :
- TENSOR algebra
LINEAR algebra
EQUATIONS
NUMERICAL analysis
MATHEMATICAL analysis
Subjects
Details
- Language :
- English
- ISSN :
- 00754102
- Volume :
- 2017
- Issue :
- 734
- Database :
- Complementary Index
- Journal :
- Journal für die Reine und Angewandte Mathematik
- Publication Type :
- Academic Journal
- Accession number :
- 127788775
- Full Text :
- https://doi.org/10.1515/crelle-2014-0053