Mathematical Research Letters

Volume 23 (2016)

Number 6

Operator ideals in Tate objects

Pages: 1565 – 1631



O. Braunling (Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, Germany)

M. Groechenig (Department of Mathematics, Imperial College, London, United Kingdom)

J. Wolfson (Department of Mathematics, University of Chicago, Il., U.S.A.)


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.

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