On Gorenstein projective, injective and flat dimensions—A functorial description with applications

By L. Christensen, Anders J. Frankild, Henrik Holm. 2004

391 citations recorded by the source corpus.

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions:

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fb4d37d40e6fb93455a94ba61c58de07acf5bfeb

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

G-dimension over local homomorphisms. Applications to the Frobenius endomorphism (2003, methodology): “This remark and an elementary argument based on the definition of G-flat dimension entail: Gfd R̂ (R̂⊗RW ) ≤ GfdR(W ); see also Holm [22, (3.10)].”
Acyclicity versus total acyclicity for complexes over noetherian rings (2005, background): “The theorem below contains a recent result of Christensen, Frankild, and Holm; more precisely, the equivalence of (i) and (ii) in [6, (4.1)], albeit in the case when R is commutative; however, see (8.3).”
Gorenstein Projective, Injective, and Flat Complexes (2011, background): “such that: (1) Pi is a projective complex for all i ∈ ; (2) Ker 0 = G; (3) Hom P is exact for any projective complex P.”
Semi-dualizing modules and related Gorenstein homological dimensions (2004, background): “These are usually denoted GidA(−), GpdA(−) and GfdA(−), respectively. Some references are [2, 4 ,10,14].”
Algebras that satisfy Auslander’s condition on vanishing of cohomology (2008, background): “Theorem C Let A be a two-sided noetherian ring that has a dualizing complex (as defined in[13]) or is commutative.”
Strongly Gorenstein projective, injective, and flat modules (2006, background): “Several decades later, Enochs, Jenda, and Torrecillas [10, 11, 12] extended the ideas of Auslander and Bridger, and introduced three homological dimensions, called Gorenstein projective, injective, and flat dimensions, which all have been studied extensively by their founders and by Avramov, Christe”