Beyond Totally Reflexive Modules and Back
Lars Winther Christensen, Hans-Bj{\o}rn Foxby, and Henrik Holm
TL;DR
This paper surveys the theory of Gorenstein homological dimensions over commutative rings, highlighting its connections with relative homological algebra and characterizations of Gorenstein rings.
Contribution
It provides a comprehensive overview of Gorenstein homological dimensions, linking them with local ring homomorphisms and total acyclicity, extending the understanding of Gorenstein rings.
Findings
Characterization of Gorenstein rings via total acyclicity
Connections established between Gorenstein dimensions and relative homological algebra
Survey of the theory's development and key results
Abstract
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Rings, Modules, and Algebras