Gorenstein ring
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| Gorenstein ring | |
|---|---|
| Name | Gorenstein ring |
| Field | Commutative algebra |
| Introduced | 1960s |
| Introduced by | Daniel Gorenstein |
Gorenstein ring
A Gorenstein ring is a Noetherian commutative ring with finite injective dimension as a module over itself, notable for symmetric homological behavior and duality properties. It arises in the study of singularities, algebraic geometry, and invariant theory, connecting with Cohen–Macaulay rings, canonical modules, and local duality. Key contributors include Daniel Gorenstein, Jean-Pierre Serre, Alexander Grothendieck, and Maurice Auslander.
Definition and basic properties
A Noetherian local ring (R, m) is Gorenstein if it has finite injective dimension as an R-module; equivalently, its local cohomology and Ext functors satisfy symmetry conditions discovered in work of Jean-Pierre Serre, Alexander Grothendieck, Maurice Auslander, Idun Reiten, and David Eisenbud. For a Cohen–Macaulay local ring, being Gorenstein can be characterized by the canonical module being isomorphic to R, a viewpoint related to duality theorems of Grothendieck and local duality developed by Robin Hartshorne and Aise Johan de Jong. In the graded setting, standard graded algebras over a field studied by Pierre Deligne and Jean-Louis Koszul also admit Gorenstein criteria via graded injective dimensions and symmetry of Hilbert series, linking to work of Richard Stanley and Mark Haiman. Important properties include self-injectivity in the artinian case examined by Maurice Auslander and connections with complete intersections studied by Michel Lazard and Sergey Gelfand.
Examples and classes
Classical examples include regular local rings (studied by Oscar Zariski and André Weil), which are Gorenstein, and hypersurface rings investigated by John Milnor and René Thom in singularity theory. Quotients of polynomial rings by a single equation yield graded Gorenstein algebras examined by David Buchsbaum and David Eisenbud. Artinian Gorenstein rings appear in the work of Gerhard Hochschild and Emmy Noether via duality for zero-dimensional schemes; canonical graded examples relate to the theory of level algebras explored by Ethan I. Craig and Anthony Iarrobino. Determinantal rings and rings of invariants treated by David Mumford and Bertram Kostant supply families of Gorenstein rings. Cluster algebras and Calabi–Yau algebras studied by Bernhard Keller and Maxim Kontsevich provide noncommutative analogues and motivate Gorenstein conditions in representation theory analyzed by Idun Reiten and Osamu Iyama.
Homological characterizations
Homological criteria involve vanishing of Ext and Tor groups, with foundational contributions from Maurice Auslander and Idun Reiten in Auslander–Reiten theory, and from David Buchsbaum and David Eisenbud in free resolution theory. A commutative Noetherian ring R is Gorenstein if and only if every finitely generated R-module has finite Gorenstein dimension, a concept introduced by Maurice Auslander and further developed by Henning Christensen and Luchezar Avramov. The Bass numbers of local rings, studied by Hyman Bass, characterize Gorensteinness via symmetry of injective resolutions, and Tate cohomology techniques of John Tate are applied in stable module categories around Gorenstein conditions. Connections with syzygies and Betti numbers appear in work by Mark Green and Tony Pantev.
Canonical module and duality
The canonical module concept, central to characterization, originates in duality theories of Alexander Grothendieck and was applied by Jean-Pierre Serre in local algebra. For a Cohen–Macaulay ring R with canonical module ω_R, R is Gorenstein iff ω_R ≅ R. Grothendieck duality, Serre duality and local duality theorems relating Ext and local cohomology illuminate this equivalence; contributors include Robin Hartshorne, Joseph Lipman, and Paul Roberts. Canonical modules play key roles in birational geometry studied by Shigefumi Mori and in classification problems in algebraic geometry pursued by Shigeru Mukai and Francesco Catanese, where Gorenstein conditions affect dualizing sheaves and adjunction formulas appearing in the work of Federigo Enriques and Kunihiko Kodaira.
Applications and connections
Gorenstein rings appear in singularity theory investigated by John Milnor, Vladimir Arnold, and Kyoji Saito; in moduli problems and deformation theory studied by Pierre Deligne and Barry Mazur; in invariant theory of actions considered by David Mumford and Hilbert; in algebraic topology via Poincaré duality algebras explored by Edwin Spanier and René Thom; and in representation theory of algebras developed by Idun Reiten, Maurice Auslander, and Bernard Keller. They are important in computational algebra systems used by David Cox and Michael Stillman for Hilbert series computations, and in mirror symmetry and Calabi–Yau studies led by Maxim Kontsevich and Cumrun Vafa where Gorenstein singularities influence string-theoretic models. Number-theoretic appearances involve local rings at primes in arithmetic geometry as in work by Jean-Pierre Serre and Alexander Grothendieck.
Historical remarks and nomenclature
The term originates from foundational contributions by Daniel Gorenstein in the mid-20th century; subsequent formalization involved Hyman Bass and Jean-Pierre Serre who linked homological notions to duality. Developments through the 1960s–1980s by Alexander Grothendieck, Maurice Auslander, Idun Reiten, and David Eisenbud solidified the modern theory, while later work by Luchezar Avramov, Henning Christensen, and Mark Haiman expanded homological and combinatorial perspectives. The nomenclature reflects the link between representation-theoretic self-duality and geometric dualizing sheaves studied across algebraic geometry and commutative algebra traditions in the schools of Oscar Zariski, André Weil, and David Mumford.