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Locally Compact Objects in Exact Categories

Profile image of Luigi PrevidiLuigi Previdi

2007

Abstract

We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories Ind_{aleph_0}(C), Pro_{aleph_0}(C) of countably indexed ind/pro-objects over any category C can be described as localizations of categories of diagrams over C.

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