G-dimension over local homomorphisms. Applications to the Frobenius endomorphism

By S. Iyengar, Sean Sather-Wagstaff. 2003

74 citations recorded by the source corpus.

We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it pos- sesses a nonzero finite module of finite projective dimension that has finite G-dimension when considered as an R-module via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3ecdccf1a9238106f646a1fb29eabbc0d4124bc9

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

On Gorenstein projective, injective and flat dimensions—A functorial description with applications (2004, methodology): “The ﬁrst result in this direction is inspired by [32, Theorem 8.6].”
Beyond totally reflexive modules and back (2008, background): “The next definition is due to Iyengar and Sather-Wagstaff [ 76, Section 3 ]; it is proved ibid.”
The set of semidualizing complexes is a nontrivial metric space (2004, background): “ity properties with respect to the ring. The modules and complexes of ﬁnite G-dimension have been studied extensively in recent years; see, e.g., [6, 12, 38, 42] for treatments of ﬁnite complexes and [9, 14, 15, 21, 29, 32, 39] for the nonﬁnite case. These examples are antipodal in the sense that we”