Mathematical Research Letters
Volume 23 (2016)
Operator ideals in Tate objects
Pages: 1565 – 1631
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.