Commutative algebra

Gorenstein ring

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in). The name comes from a duality property of singular plane curves studied by Gorenstein (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ ⊃ complete intersection rings ⊃ regular local rings (Wikipedia).

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Related pages

Prime ideal | Smooth scheme | Integral domain | Krull dimension | Regular local ring | Alexander Grothendieck | Injective hull | Commutative algebra | Complete intersection ring | Cohen–Macaulay ring | Gorenstein scheme | Invertible sheaf | Poincaré duality | Socle (mathematics) | Canonical bundle | Noetherian ring | Wiles's proof of Fermat's Last Theorem | Ring theory | Residue field | Local cohomology | Grothendieck local duality | Pfaffian | Local ring | Serre duality