报告人：Bernhard Keller， 巴黎西岱大学

时间：12月16日、23日、30日，10：00-11：30

地点：东区管理科研楼1418

题目: An introduction to exact dg categories, after Xiaofa Chen

摘要: In this series of lectures, we will report on results from Xiaofa Chen's ongoing Ph. D. thesis. His notion of an exact dg category is a simultaneous generalization of the notions of exact category in the sense of Quillen and of pretriangulated dg category in the sense of Bondal-Kapranov. It is also a dg enhancement of the notion of extriangulated category recently introduced by Nakaoka-Palu. Via Lurie's dg nerve, it is related to Barwick's notion of exact infinity-category.

We will start by recalling the definition and the main examples of extriangulated categories. We will then give the definition of an exact dg category in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We will give two descriptions of this category using respectively dg functor categories and A-infinity functor categories. We will then present examples related to Yilin Wu's Higgs categories and Haibo Jin's categories of dg Cohen-Macaulay modules.There will follow a number of fundamental results concerning the dg nerve, the dg derived category, tensor products and functor categories with exact dg target and the existence of the greatest exact structure on a dg category with additive H^0. This generalizes a Theorem of Rump for Quillen exact categories. Under certain hypotheses, it allows to classify all exact structures on a given dg category with additive zeroth homology. Finally, we will present work in progress with Yilin Wu and Xiaofa Chen on the subtle notion of projective respectively injective resolution for objects in exact dg categories.

报告人简介：Bernhard Keller是巴黎西岱大学教授、图书馆VIP客座教授、著名代数学家，在微分分次理论、丛理论以及Hochschild同调理论中均做出了奠基性的学术成果。Keller教授是法国科学院“索菲·热尔曼”2014年度大奖得主、法国大学研究院资深成员、挪威皇家科学通讯院士、比利时安特卫普大学荣誉博士、国际数学家大会ICM邀请报告人以及美国数学会会士，现任国际知名杂志Advances in Mathematics以及Forum of Mathematics Pi 编委。