Operator ideals in Tate objects

TL;DR

This paper explores the structure of endomorphism algebras in n-Tate categories, revealing they often have cubically decomposed structures leading to higher Tate central extensions, with implications for algebraic geometry.

Contribution

It demonstrates that endomorphism algebras of n-Tate objects possess cubically decomposed structures and under certain conditions, form categories of projective modules over these algebras.

Findings

Endomorphism algebras often carry cubically decomposed structures.

n-Tate categories can be equivalent to categories of projective modules.

Higher Tate central extensions naturally arise from these structures.

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.