导出模范畴上的粘贴结构及其应用

The technique of gluing triangulated categories was invented by Beilinson, Bernstein and Deligne in 1982. Using Grothendieck's six functors, the derived category of constructible sheaves on an algebraic variety can be glued from the derived categories of constructible sheaves on a closed subvariety and on its open complement. BBD called this 'recollement' in French. It decomposes a triangulated category into two smaller triangulated categories. Cline, Parshall and Scott introduced this concept to algebra representation theory. They defined a class of finite dimensional algebras, namely quasi-hereditary algebras or more generally stratified algebras, whose derived module categories admit recollements induced by idempotent ideals. Such idempotent ideals are called stratifying ideals by CPS. Recollements induced by stratifying ideals seem to be very special. There are plenty of recollements which are not of this form. In known examples however, we are able to replace the recollement by an equivalent one which is induced by a stratifying ideal. The following question was asked by Changchang Xi: Up to equivalence is every recollement of derived module categories induced by a stratifying ideal of the algebra sitting in the middle? We expect to have a positive answer to this question (with a suitable choice of 'equivalence', see later sections for more discussion). It would provide a normal form for recollements of derived module categories. This is the first part of the proposal. The other part is an application for constructing tilting objects. The motivation is taken up from a short note by Beilinson, Bezrukavnikov and Mirkovic called Tilting Exercises. They proved that a tilting perverse sheaf on an open subvariety of an algebraic variety can be extended to a tilting perverse sheaf on the whole space, using derived functors and canonical triangles in the recollement. We would like to complete the picture on the algebra side, namely to construct characteristic tilting modules for quasi-hereditary algebras or stratifying algebras by using their canonical stratification. The existing construction of characteristic tilting modules is for classical Schur algebras due to Donkin, via tensor product of exterior powers of the natural representation of general linear groups.

三角范畴上的粘贴结构是Beilinson，Bernstein和Deligne与1982年定义的，使用Grothendieck的六个函子可以把两个三角范畴粘贴起来得到一个更大的三角范畴。Cline，Parshall和Scott把这个概念引入到代数领域，他们发现拟遗传代数和分层代数的导出模范畴上自然地存在由幂等理想诱导的粘贴结构，这样的幂等理想被成为分层理想。该项目研究的第一个问题是：在相差等价意义下，是否每个粘贴结构都由分层理想诱导？任意给定的一个粘贴结构本身未必是由幂等理想诱导的，但是在已知的例子中，我们可以选取合适的等价条件从而把该粘贴结构转化为由幂等理想诱导的形式。如果一般情况下可行的话，我们将得到代数导出模范畴上粘贴结构的一个标准形式。作为应用，我们将考虑如何在导出模范畴上粘贴t-结构和倾斜对象，并对拟遗传代数和分层代数给出它们标准倾斜对象的一个构造。

我们的主要研究对象是导出模范畴上的粘贴结构、代数的导出单性、半倾斜对象及倾斜对象，所取得的主要研究结果包括以下几个方面：证明了两个顶点的有限维代数在导出等价意义下要么是导出单的，要么是拟遗传的；在粘贴结构上实现了半倾斜对象的粘贴，并刻划了由倾斜对象粘贴出来的半倾斜对象何时仍为倾斜对象；给出了不同导出范畴层次上的粘贴结构可以限制的充要条件，并利用“梯子”的概念刻画了代数的导出单性；给出了粘贴结构何时可由幂等元诱导的一个充要条件；构造了一类不满足导出Jordan-Hoelder性质有限维代数；研究了三角范畴上的半倾斜约化与Calabi-Yau约化之间的关系。