Ladders of recollements of triangulated and of abelian categories

这个提议是关于三角分类和阿贝尔分类的集合。Beilinson， Bernstein和Deligne引入了三角化回忆，将三角化分类分解为开放部分和封闭部分，例如与Grothendieck的六个函子相关；集合可以看作是三个三角化（如派生）范畴的短精确序列，因此也可以看作是由派生等价提供的两个代数之间联系的深远推广。三角分类的阶梯是由进一步的伴随延伸的集合；这些是由Beilinson， Ginzburg和Schechtman引入的，作为Koszul对偶的设置。我们也引入了阿贝尔范畴的集合，但是阿贝尔范畴的阶梯需要有相当不同的定义；这是在这个项目中要做的。然后，该项目旨在连接和结合三角测量和阿贝尔技术，进一步发展梯子方法和各种内在相关的应用。预期的应用包括-表征，描述和构造代数李理论中经常出现的代数（如准遗传代数）以及重要的表示（如特征倾斜模块）；-将梯子与Serre函子关联起来，并使用它们来构造这样的函子；-充分描述自注入（如对称或Frobenius）代数的集合，并使用这种描述来分类自注入（或Frobenius或对称）细胞分层图代数（如Brauer， BMW或分区代数）；-联系集合中代数的Gorenstein和（Fg）条件的有效性，描述集合中函子下的模或复合体的支持变体。

This proposal is about recollements of triangulated and of abelian categories. Triangulated recollements have been introduced by Beilinson, Bernstein and Deligne to deconstruct triangulated categories into open and closed parts, related for instance by Grothendieck's six functors; recollements can be seen as short exact sequences of three triangulated (such as derived) categories, hence also as a far reaching generalisation of the connections between two algebras provided by derived equivalences . Ladders of triangulated categories are recollements extended by further adjoints; these have been introduced by Beilinson, Ginzburg and Schechtman as a setup for Koszul duality. Recollements of abelian categories have been introduced, too, but ladders of abelian categories need to be defined rather differently; this is to be done in this project. The project then aims at connecting and combining triangulated and abelian techniques, at further developing the method of ladders and at a variety of intrinsically related applications.The intended applications include- to characterise, describe and construct algebras frequently arising in algebraic Lie theory (such as quasi-hereditary algebras) as well as important representations (such as characteristic tilting modules);- to relate ladders with Serre functors and to use them to construct such functors;- to fully describe recollements of self-injective (eg symmetric or Frobenius) algebras and to use this description to classify self-injective (or Frobenius or symmetric) cellularly stratified diagram algebras (such as Brauer, BMW or partition algebras);- to relate being Gorenstein and validity of the (Fg) condition for algebras in a recollement and to describe support varieties of modules or complexes under the functors in a recollement.