Categorical resolutions of a class of derived categories  

作　　者：Pu Zhang 

机构地区：[1]School of Mathematical Sciences, Shanghai Jiao Tong University

出　　处：《Science China Mathematics》2018年第2期391-402,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11271251 and 11431010)

摘　　要：We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.We clarify the relation between the subcategory D_（hf）~b（A） of homological finite objects in D^b（A）and the subcategory K^b（P） of perfect complexes in D^b（A）, by giving two classes of abelian categories A with enough projective objects such that D_（hf）~b（A） = K^b（P）, and finding an example such that D_（hf）~b（A）≠K^b（P）. We realize the bounded derived category D^b（A） as a Verdier quotient of the relative derived category D_C^b（A）, where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT 〈∞ such that ~⊥T is finite, then D^b（modA） admits a categorical resolution; and that for a CM（Cohen-Macaulay）-finite Gorenstein algebra, such a categorical resolution is weakly crepant.

关 键 词：homologically finite object perfect complex smooth triangulated category （weakly crepant）categorical resolution （relative） derived category CM-finite Gorenstein algebra 

分 类 号：O154.1[理学—数学]