Ladders of recollements of triangulated and of abelian categories

这个建议是关于三角范畴和交换范畴的重元素的。三角化的复元已经被贝林森、伯恩斯坦和德利涅引入，用来将三角化的范畴解构为开和闭的部分，例如由格罗滕迪克的六个函子所关联;复元可以被看作是三个三角化的（如导出的）范畴的短正合序列，因此也是由导出的等价所提供的两个代数之间的联系的一个深远的推广。 三角范畴的阶梯是由进一步的伴随所扩展的基本元素;这些已经由Beilinson、Ginzburg和Schechtman引入作为Koszul对偶的一个设置。阿贝尔范畴的重新校正也被引入，但是阿贝尔范畴的阶梯需要被相当不同的定义;这将在这个项目中完成。该项目的目标是连接和结合三角和阿贝尔技术，进一步发展梯子的方法，并在各种内在相关的应用。预期的应用包括-代数李理论中经常出现的代数，描述和构造（例如准遗传代数）以及重要的表示（如特征倾斜模）;-将梯子与Serre函子联系起来，并使用它们来构造这样的函子;- 充分描述自我内射的基本要素（例如对称或Frobenius）代数，并使用这种描述来分类自内射（或Frobenius或对称）胞腔分层图代数（如Brauer、BMW或划分代数）;- 将Gorenstein与（Fg）的有效性联系起来条件的代数在一个复元和描述支持品种的模或复函子下的复。

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.