Operator ideals in Tate objects

Abstract

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.

Reciprocity laws and K-theory

“…Let us record here that relevant and independent work has been done in [3] and [4]. There are several further directions which one may consider.…”

On the homology of Lie algebras like $\mathfrak{gl}(\infty, R)$

“…Proof. This is [BGW16a], Theorem 1, applied for C = P f (R) and n = 1. Note that the countable collection {S i } of the theorem loc.…”

Geometric and analytic structures on the higher adèles

“…Our paper [5] shows that Arrow (2) induces an isomorphism E Beil ∼ = E Tate . Yekutieli's Conjecture asks whether Arrow (4) induces an isomorphism E Beil ∼ = E Yek .…”

“…Without split exactness, one then has to be careful with the property I + 1 + I − 1 = A however, which may fail in general. The introduction of [5] provides a reasonably short survey to what extent the above theorem can be stretched, and which seemingly plausible generalizations turn out to be problematic.…”

“…1.5 automatically carry a cubical endomorphism algebra. See [5] for the construction of the algebra structure and for further background. The above result is not given in the broadest possible formulation, e.g.…”