Ladders of recollements of triangulated and of abelian categories

这项提议是关于三角分类和阿贝尔分类的重新归集。Beilinson，Bernstein和Deligne已经引入了三角重集，将三角范畴解构为开放和闭合部分，例如通过Grothendieck的六个函子相关；重集可以被视为三个三角(如派生)范畴的短精确序列，因此也是由派生等价提供的两个代数之间联系的深远推广。三角范畴的阶梯是由更多的伴随项扩展的重合；Beilinson，Ginzburg和Scheck htman已经引入这些阶梯作为Koszul对偶的设置。阿贝尔类别的回收也被引入，但阿贝尔类别的阶梯需要有相当不同的定义；这将在本项目中完成。该项目的目标是连接和结合三角化和阿贝尔技术，进一步发展阶梯方法和各种本质上相关的应用。预期的应用包括-刻画、描述和构造在代数李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)；-将梯子与Serre函子联系起来并使用它们来构造这样的函子；-完全描述自内射(如对称或Frobenius)代数的回忆，并使用这种描述对自内射(或Frobenius或对称)胞元分层图代数(如Brauer、BMW或划分代数)进行分类；-联系在一个集合中代数的(FG)条件的有效性和成为Gorenstein，并描述在一个集合中函子下的模或复形的支撑簇。

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.