Derived categories for functional analysis
Let \({\mathcal T}c\) be the category of locally convex topological vector spaces. It is well-known that the category \({\mathcal T}c\) is not abelian so that the classical techniques of homological algebra may not be used for the study of the homological algebra of \({\mathcal T}c\). The author starts by showing that \({\mathcal T}c\) is quasi-abelian category in which products and direct sums are exact. So, the derived category of \({\mathcal T}c\) is constructed as explained in the author's paper ``Algèbre homologique quasi-abélienne'', Mémoire de DEA, Université Paris 13 (1995) and Schneider's paper ``Quasi-abelian categories and sheaves'', Mém. Soc. Math. France, (N.S.) 76 (1999). The projective and inductive limit functors are derived and their homological properties are clarified. In particular, strictness and acyclicity criteria are obtained. Next, the category of separated objects of \({\mathcal T}c\) is considered and the fact that this category is quasi-abelian and has the same derived category as \({\mathcal T}c\) is established. Since complete objects of \({\mathcal T}c\) do not form a quasi-abelian category, the author is leaded to introduce the notion of cohomological completeness and to study the derived completion functor. In this context, the main result of the paper is an equivalence between the subcategory of \({\mathcal D}({\mathcal T}c)\) formed by cohomologically complete complexes and the derived category of the category of pro-Banach spaces. Moreover, under suitable assumptions, the computation of Ext's in \({\mathcal T}c\) is reduced to their computation in Ban by means of derived projective limits. In the last part of the paper, the derived duality functors are studied. So, the inductive dual functors \({\mathcal D}_i:{\mathcal T}c \to{\mathcal T}c^{\text{op}}\) is introduced and for this the left exactness is proved and its right derived functors is studied.
category of locally convex topological vector spaces
scientific article; zbMATH DE number 1501166