Sato Grassmannians for generalized 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 limnA. 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.