Artin algebra
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| Artin algebra | |
|---|---|
| Name | Artin algebra |
| Field | Abstract algebra |
| Notable people | Emil Artin, Maurice Auslander, Idun Reiten, Hyman Bass, Gunnar Bergman, Karen Smith |
Artin algebra
An Artin algebra is a class of associative rings that are finite-length modules over commutative Artinian rings and that serve as central objects in representation theory and homological algebra. Originating in work related to Emil Artin and developed through contributions by Maurice Auslander, Idun Reiten, and others, Artin algebras connect to the representation theory of finite groups, algebras arising from quivers, and the homological methods used in modern algebraic geometry. They provide a tractable framework where module categories are abelian, Krull–Schmidt, and often amenable to categorical equivalences such as Morita and derived equivalences.
Definition and basic properties
An Artin algebra is typically defined as an associative unital algebra Λ that is finitely generated as a module over a commutative Artinian ring R (often a commutative Artinian local ring or an Artinian principal ideal ring). Fundamental structural properties include that the category of finitely generated Λ-modules is an abelian Krull–Schmidt category with finite length objects, and that Λ has a semiperfect idempotent theory enabling decompositions analogous to those in Wedderburn–Artin theorem. Important structural invariants include the Jacobson radical, projective cover existence, and the Loewy length, which interact with classical theorems by Jacobson and results studied by Hyman Bass and Peter Gabriel.
Examples and classes of Artin algebras
Key examples arise from finite-dimensional algebras over fields such as group algebras kG for finite groups G and path algebras of quivers with admissible relations studied by Gabriel and Happel. Specific families include hereditary Artin algebras (linking to the work of Henri Cartan and Samuel Eilenberg), self-injective algebras studied by H. Bass and K. Morita, Nakayama algebras named after Tadasi Nakayama, and representation-finite algebras classified in results related to Gabriel's theorem and the Dynkin diagram classification of type A_n, D_n, E_6, E_7, E_8. Other notable classes include tilted algebras arising from Bernšteĭn–Gelfand–Ponomarev reflection functors, cluster-tilted algebras related to Fomin–Zelevinsky cluster algebras, and canonical algebras introduced by Dieter Happel and C.M. Ringel.
Module theory and representation theory
The category mod-Λ of finitely generated Λ-modules is the central object of study, with concepts such as indecomposable modules, projective modules, injective modules, and simple modules paralleling classical representation theory of Emil Artin-related structures. Representation-finite versus representation-infinite dichotomies are analyzed through Auslander–Reiten quivers and homological dimensions influenced by work of Auslander and Reiten. Techniques draw on connections to the representation theory of quivers as developed by Bernšteĭn–Gelfand–Ponomarev, and to methods from Jean-Pierre Serre-style homological algebra. Tame and wild dichotomies, stemming from results of Donovan–Freislich and Drozd, classify module categories and link to broader classification problems encountered by Igor Dolgachev and Anthony J. Scholl in adjacent contexts.
Homological properties and functors
Homological invariants such as projective dimension, injective dimension, global dimension, and dominant dimension are central; they connect to classical homological theories by Samuel Eilenberg and Henri Cartan, and to modern invariants like Gorenstein dimension studied by Maurice Auslander and Markus Auslander. Derived functors Ext^n and Tor_n organize extension and torsion information and are calculable in mod-Λ via projective resolutions influenced by work of Jean-Louis Verdier and Alexandre Grothendieck. Functors such as Hom, tensor, and the Nakayama functor play key roles; Serre duality analogues in finite-dimensional settings link to results by Pierre Deligne and Maxim Kontsevich in categorical contexts.
Auslander–Reiten theory and almost split sequences
Auslander–Reiten theory provides tools like almost split sequences and the Auslander–Reiten translate τ, introduced by Maurice Auslander and Idun Reiten, to analyze the morphism structure in mod-Λ. The Auslander–Reiten quiver encodes irreducible morphisms and connected components classified in representation-finite cases often corresponding to Dynkin diagram combinatorics. Almost split sequences facilitate classification of irreducible morphisms between indecomposables and connect to tilting theory developed by M. Auslander, Bernhard Keller, and Christine M. Ringel. Techniques from Auslander–Reiten theory have been extended in work by Happel and Igusa–Todorov to study homological conjectures and cluster categories influenced by Fomin–Zelevinsky.
Morita equivalence and derived equivalence
Morita theory, originating from Kiiti Morita and expanded by B. Mitchell and Hyman Bass, characterizes when two Artin algebras have equivalent module categories via progenerators and module tensor equivalences. Derived equivalence, developed through the language of derived categories by Jean-Louis Verdier, Alexandre Grothendieck, and applied to algebras by Bernhard Keller and Jeremy Rickard, classifies algebras up to equivalence of their bounded derived categories D^b(mod-Λ). Tilting theory and Rickard's Morita theory for derived categories provide practical criteria for derived equivalence, linking to topics studied by Dmitri Orlov, Amnon Neeman, and Raphaël Rouquier.
Applications and connections to other areas
Artin algebras interface with algebraic geometry through noncommutative projective techniques as in work of M. Artin and Michel Van den Bergh, with mathematical physics via category-theoretic methods in Konstevich-style mirror symmetry, and with combinatorics through quiver mutation studied by Fomin–Zelevinsky. Connections to the representation theory of finite groups tie to classical theory of Isaac Schur and modern modular representation theory by Richard Brauer and Jon Alperin. Theories of cluster algebras, categorical invariants, and computational approaches implemented in systems influenced by Donald Knuth and Stephen Wolfram further broaden applicability. Contemporary research links Artin algebras to higher category theory investigated by Jacob Lurie and to homological conjectures addressed by Karen Smith and Paul Balmer.