Abstract
In this paper we study the finiteness of global Gorenstein AC-homological dimensions for rings, and answer the questions posed by Becerril, Mendoza, Pérez and Santiago. As an application, we show that any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category, which improves the known result by Beligiannis.
Highlights
Throughout this work, all rings are assumed to be associative
Let R be a ring; we adopt the convention that an R-module is a left R-module, and we refer to right R-modules as modules over the opposite ring R◦
The Verdier quotient triangulated category Db(R)/Kb(Prj) was first studied by Buchweitz [10] under the name of “stable derived category”; it is named by “singularity category” to emphasize certain homological singularity of the ring R reflected by this quotient category
Summary
Throughout this work, all rings are assumed to be associative. Let R be a ring; we adopt the convention that an R-module is a left R-module, and we refer to right R-modules as modules over the opposite ring R◦. Any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category. The Verdier quotient triangulated category Db(R)/Kb(Prj) was first studied by Buchweitz [10] under the name of “stable derived category”; it is named by “singularity category” to emphasize certain homological singularity of the ring R reflected by this quotient category (see Orlov [31] and Chen [11]) As another immediate consequence of Theorem 1, we see that the singularity categories Db(R)/Kb(Prj) Db(R)/Kb(Inj) are compactly generated over left (or right) coherent rings of finite global Gorenstein dimension (or equivalently, left Gorenstein); see Corollaries 34 and 38.
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