Abstract
The main result asserts that a local commutative Noetherian ring is Gorenstein, if it possesses a non-zero cyclic module of finite Gorenstein injective dimension. From this follows a classical result by Peskine and Szpiro stating that the ring is Gorenstein, if it admits a non-zero cyclic module of finite (classical) injective dimension. The main result applies to local homomorphisms of local rings and yields the next: if the source is a homomorphic image of a Gorenstein local ring and the target has finite Gorenstein injective dimension over the source, then the source is a Gorenstein ring. This, in turn, applies to the Frobenius endomorphism when the local ring is of prime equicharacteristic and is a homomorphic image of a Gorenstein local ring.
Citation
Hans-Bjørn Foxby. Anders J. Frankild. "Cyclic modules of finite Gorenstein injective dimension and Gorenstein rings." Illinois J. Math. 51 (1) 67 - 82, Spring 2007. https://doi.org/10.1215/ijm/1258735325
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