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Homological Algebra of Gorenstein Rings

Creative Commons 'BY' version 4.0 license

We study three triangulated categories associated to a Gorenstein ring, that is, a right- and left-noetherian ring with finite right and left injective dimension. After a survey of exact categories and cotorsion theory, we discuss the homological algebra of the category of finitely generated modules over a Gorenstein ring, concluding that the subcategory of maximal Cohen-Macaulay (MCM) modules is both a Frobenius category and a cotorsion class. Immediate corollaries include a triangulated structure on the projective stabilization of the subcategory of MCM modules and structure theorems for finitely generated modules. Then we describe two triangulated categories equivalent to the stable category of MCM modules. The homotopy category of acyclic complexes of projective modules is the first, making precise a connection observed between MCM modules and existence of projective co-resolutions. The second, the singularity category, is a Verdier quotient of the bounded derived category that allows us to study modules up to MCM approximation.

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