Illinois Journal of Mathematics

Cyclic modules of finite Gorenstein injective dimension and Gorenstein rings

Hans-Bjørn Foxby and Anders J. Frankild

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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.

Article information

Source
Illinois J. Math., Volume 51, Number 1 (2007), 67-82.

Dates
First available in Project Euclid: 20 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258735325

Digital Object Identifier
doi:10.1215/ijm/1258735325

Mathematical Reviews number (MathSciNet)
MR2346187

Zentralblatt MATH identifier
1121.13015

Subjects
Primary: 13D05: Homological dimension
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Citation

Foxby, Hans-Bjørn; Frankild, Anders J. Cyclic modules of finite Gorenstein injective dimension and Gorenstein rings. Illinois J. Math. 51 (2007), no. 1, 67--82. doi:10.1215/ijm/1258735325. https://projecteuclid.org/euclid.ijm/1258735325


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