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# Mathematical Research Letters

## Volume 23 (2016)

### Number 6

### Operator ideals in Tate objects

Pages: 1565 – 1631

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n6.a2

#### Authors

#### Abstract

Tate’s central extension originates from 1968 and has since found many applications to curves. In the 80s Beilinson found an $n$-dimensional generalization: cubically decomposed algebras, based on ideals of bounded and discrete operators in ind-pro limits of vector spaces. Kato and Beilinson independently defined ‘($n$-)Tate categories’ whose objects are formal iterated ind-pro limits in general exact categories. We show that the endomorphism algebras of such objects often carry a cubically decomposed structure, and thus a (higher) Tate central extension. Even better, under very strong assumptions on the base category, the $n$-Tate category turns out to be just a category of projective modules over this type of algebra.