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 Beilinson category of the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category. Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.