广义Tate空间的Sato Grassmannian
Sato Grassmannians for generalized Tate spaces
摘要：
将局部线性紧拓扑向量空间（Tate空间）的Sato Grassmannian概念推广到正合范畴A的“局部紧对象”范畴limA，并研究了它们的一些性质.这使我们能够推广Kapranov维torsor Dim（X）和行列式gerbe Det（X）的对象limA和统一的行列式torsor D（X）的治疗。然后，我们引入了一类正合范畴，称之为部分阿贝尔正合范畴，并证明了：如果A是部分阿贝尔正合的，那么Dim（X）和Det（X）在容许短正合序列中是乘法的.当A是域k上的有限维向量空间的范畴时，我们恢复了Kapranov定义的Tate空间的维数torsor和行列式gerbe的情形，并根据恰合范畴A的Waldhausen空间S（A）重新表述了它的性质.这种方法的优点是，它允许以同样的方式形式定义迭代范畴lim^nA的格拉斯曼。然后我们证明了Tate空间的范畴是部分阿贝尔精确的，这使得我们能够将关于Tate空间的Dim和Det的结果推广到2-Tate空间，例如Arkhipov-Kremnizer和Frenkel-Zhu所考虑的Dim和Det对2-Tate空间的乘法性。
英文摘要：
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.
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