Derived categories for functional analysis
Derived categories for functional analysis
复制标题
泛函分析的派生类别
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Fabienne Prosmans
中科院分区：
文献类型：
--
作者：
Fabienne Prosmans
本文从派生范畴的角度研究了局部凸拓扑向量空间范畴TC的同调代数。我们首先证明了TC是一个拟阿贝尔范畴，其中乘积和直和是精确的。这使我们能够推导出射影极限函子和归纳极限函子，并阐明它们的同调性质。特别地，我们得到了严格的和无循环的标准。其次，我们证明了由TC的分离对象构成的范畴是拟阿贝尔范畴，并且具有与TC相同的派生范畴.由于TC的完备对象不构成拟阿贝尔范畴，我们引入了上同调完备性的概念，并研究了导出的完备性函子.我们的主要结果是由上同调完备复形构成的D(TC)的子范畴与ProBanach空间范畴的派生范畴等价。我们还证明了，在适当的假设下，我们可以通过导出的射影极限将T_c中的Ext的计算简化为它们在BAN中的计算。我们通过研究派生的对偶函子来总结本文。
In this paper, we study the homological algebra of the category T c of locally convex topological vector spaces from the point of view of derived categories. We start by showing that T c 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 T c is quasi-abelian and has the same derived category as T c. Since complete objects of T c 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(T c) 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 T c to their computation in Ban by means of derived projective limits. We conclude the paper by studying derived duality functors.