题目: Gorenstein projective dg modules and applications
报告人: Bernhard Keller（孔博恩），Université Paris Cité（巴黎西岱大学）
时间：11月26日至12月25日（每周三）上午9:45-11:20
地点：东区第二教学楼2306教室
摘要：
In a long series of papers, Geiss-Leclerc-Schroer have used certain categories of modules over preprojective algebras to (additively) categorify cluster algebras with coefficients appearing in Lie theory. Recent work by Yilin Wu provides a unifying conceptual framework for their theory. His approach shows that Geiss-Leclerc-Schroer's categories admit a unified description as *Higgs categories* and that these in turn may be constructed as categories of *Gorenstein projective modules* over the boundary algebras associated to ice quivers with potential. In his thesis (to be defended next month), Miantao Liu shows that Wu's Higgs categories also allow to extend Geiss-Leclerc-Schroeer's theory to new classes of examples, notably double Bruhat cells and varieties of triples of flags, which are fundamental in Fock-Goncharov's approach to higher Teichmuller theory. The Higgs categories occurring in these examples are exact dg categories (in the sense of Xiaofa Chen) which are no longer concentrated in degree 0 but usually have homologies in infinitely degrees to the left. One of Liu's important results is that they may still be described as categories of *dg* Gorenstein projective modules over *dg* boundary algebras. This description is particularly important in proving a conjecture by Merlin Christ and in categorifying Goncharov-Shen's symmetries of varieties of triples of flags (cyclic symmetry and braid group symmetry). Dg Gorenstein projective modules over *proper* Gorenstein dg algebras were previously studied by Haibo Jin. However, the dg boundary algebras which occur in the applications are almost never proper and it is still an open question in which sense they are Gorenstein.
In this series of lectures, we will give an introduction to Gorenstein projective dg modules and their applications in the categorification of cluster algebras with coefficients. We will begin with reminders on Frobenius extriangulated categories, dg algebras and their derived categories, Calabi-Yau completions and Higgs categories concentrated in degree 0. We will then introduce Gorenstein projective dg modules over dg algebras and establish their link to Gorenstein projective modules in the classical sense. We will next study the problem of describing Higgs categories as categories of Gorenstein projective dg modules. We will need to strengthen the notion of "enough projectives" (which makes sense for any extriangulated category and trivially holds for Higgs categories) to that of "projective domination". We will give sufficient conditions for exact dg categories to be projectively dominated and show that they hold in the examples arising from higher Teichmuller theory. In the final part of the lecture series, we will sketch a proof of Christ's conjecture and the construction of the categorical symmetries predicted in the work of Goncharov-Shen.
