Representation Theory of Artin Algebras
This book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is suitable for any mathematicians (especially graduate students) wanting an introduction to this active field.'...written in a clear comprehensive style with full proofs. It can very well serve as an excellent reference as well as a textbook for graduate students.' EMS Newletter
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A-module abelian group algebras of finite AR-quiver arrow Assume c-basis Cartan matrix cG-module commutative diagram component composition factors composition series consequence Corollary corresponding decomposition define denote DTrC duality Dynkin diagram equivalence of categories exact sequence finite dimensional finite length finite representation type following are equivalent functor give given gl.dim Grothendieck group group algebra Hence hereditary algebras hereditary artin algebra HomA(/l HomA(X idempotents indecomposable injective indecomposable modules indecomposable projective modules induced injective envelope injective module integer irreducible morphisms isomorphism left almost split left artin ring Lemma minimal projective presentation minimal right mod(Aop monomorphism morphism in mod Nakayama algebra phism preinjective preprojective projective cover Proof Let Proposition prove R-algebra R-functor result right almost split semisimple short cycle simple modules socle split epimorphism split monomorphism split morphism split sequence stable equivalence subadditive function submodule Suppose Theorem translation quiver uniserial modules vector space vertex zero