Derived categories for functional analysis
type: Item
instance of: scholarly article
title: Derived categories for functional analysis
zbMATH Open document ID: 0973.46069
author: Fabienne Prosmans
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-11-29T00:00:00Z
review text: 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.
reviewed by: Liliana Răileanu
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46M40
Mathematics Subject Classification ID: 18G50
zbMATH DE Number: 1501166
zbMATH Keywords: category of locally convex topological vector spaces
zbMATH Keywords: homological algebra
zbMATH Keywords: derived category
zbMATH Keywords: projective and inductive limit functors
zbMATH Keywords: strictness
zbMATH Keywords: acyclicity
zbMATH Keywords: quasi-abelian category
zbMATH Keywords: cohomological completeness
zbMATH Keywords: derived completion functor
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143226
OpenAlex ID: W2093393265
cites work: Etale homotopy
cites work: Projective objects in the category of locally convex spaces
cites work: Q5569793
cites work: Homological methods in the theory of locally convex spaces
cites work: Q4519756
cites work: Derived projective limits of topological abelian groups
DOI: 10.2977/PRIMS/1195143226
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description: scientific article; zbMATH DE number 1501166