Sato Grassmannians for generalized Tate spaces

TL;DR

This paper extends the concept of Sato Grassmannians to generalized Tate spaces within exact categories, establishing properties of dimensional torsors and determinantal gerbes, and applies these to 2-Tate spaces.

Contribution

It generalizes Sato Grassmannians to the category of locally compact objects in an exact category, unifying and extending properties of dimensional torsors and determinantal gerbes.

Findings

Generalization of Sato Grassmannians to limA categories

Introduction of partially abelian exact categories

Extension of properties to 2-Tate spaces

Abstract

We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the category limA of the "locally compact objects" of an exact category A, and study some of their properties. This allows us to generalize the Kapranov dimensional torsor Dim(X) and determinantal gerbe Det(X) for the objects of limA and unify their treatment in the determinantal torsor D(X). We then introduce a class of exact categories, that we call partially abelian exact, and prove that if A is partially abelian exact, Dim(X) and Det(X) are multiplicative in admissible short exact sequences. When A is the category of finite dimensional vector spaces on a field k, we recover the case of the dimensional torsor and of the determinantal gerbe of a Tate space, as defined by Kapranov and reformulate its properties in terms of the Waldhausen space S(A) of the exact category A. The advantage of this approach is that it allows to define formally in the same way the Grassmannians of the iterated categories lim^nA. We then prove that the category of Tate spaces is partially abelian exact, which allows us to extend the results on Dim and Det already known for Tate spaces to 2-Tate spaces, such as the multiplicativity of Dim and Det for 2-Tate spaces, as considered by Arkhipov-Kremnizer and Frenkel-Zhu.