Derived Categories for Functional Analysis

Abstract

In this paper, we study the homological algebra of the category Tc of locally convex topological vector spaces from the point of view of derived categories. We start by showing that Tc is a quasi-abelian category in which products and direct sums are exact. This allows us to derive projective and inductive limit functors and to clarify their homological properties. In particular, we obtain strictness and acyclicity criteria. Next, we establish that the category formed by the separated objects of Jc is quasi-abelian and has the same derived category as Tc. Since complete objects of Tc do not form a quasi-abelian category, we are lead to introduce the notion of cohomological completeness and to study the derived completion functor. Our main result in this context is an equivalence between the subcategory of D(Tc) formed by cohomologically complete complexes and the derived category of the category of pro-Banach spaces. We show also that, under suitable assumptions, we can reduce the computation of Ext's in Tc to their computation in 'Ban by means of derived projective limits. We conclude the paper by studying derived duality functors.