Categorical resolutions of a class of derived categories

本文刊于:《Science China Mathematics》 2018年第2期

homologically finite object;perfect comp

homologically finite object;perfect complex;smooth triangulated category;(weakly crepant)categorical resolution;(relative) derived category;CM-finite Gorenstein algebra
     We clarify the relation between the subcategory Dhfb(A) of homological finite objects in Db(A)and the subcategory Kb(P) of perfect complexes in Db(A), by giving two classes of abelian categories A with enough projective objects such that Dhfb(A) = Kb(P), and finding an example such that Dhfb(A)≠Kb(P). We realize the bounded derived category Db(A) as a Verdier quotient of the relative derived category DCb(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 Db(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.


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