Tate Objects in Exact Categories（附录作者：Jan \vS \vtov\'Jan \vcek and Jan Trlifaj）
Tate Objects in Exact Categories (with appendix by Jan \vS\vtov\'ı\vcek and Jan Trlifaj)
摘要：
我们在一个精确范畴中研究基本Tate对象。我们刻画了基本Tate对象范畴为可容许Ind-Pro对象范畴的最小子范畴，它包含可容许Ind-Pro对象范畴和可容许Ind-Pro对象范畴，并且在扩张下是闭的.我们比较贝林森的方法泰特模块德林费尔德的。我们建立了幂等完备正合范畴中的基本Tate对象的Sato Grassmannian的几个性质（例如它是一个有向偏序集）。最后，我们简要介绍了n-Tate模和n维Adèles。J. \vS \vtov\'\vcek和J. Trlifaj的附录确定了平坦Mittag-Leffler模的范畴，其可容许的Ind-objects的范畴的幂等完备性在可容许的生成投射模的范畴中。
英文摘要：
We study elementary Tate objects in an exact category. We characterize the category of elementary Tate objects as the smallest sub-category of admissible Ind-Pro objects which contains the categories of admissible Ind-objects and admissible Pro-objects, and which is closed under extensions. We compare Beilinson's approach to Tate modules to Drinfeld's. We establish several properties of the Sato Grassmannian of an elementary Tate object in an idempotent complete exact category (e.g. it is a directed poset). We conclude with a brief treatment of n-Tate modules and n-dimensional adèles. An appendix due to J. \vS\vtov\'ı\vcek and J. Trlifaj identifies the category of flat Mittag-Leffler modules with the idempotent completion of the category of admissible Ind-objects in the category of finitely generated projective modules.
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