Representation dimension of artin algebras

By Steffen Oppermann. 2010

324 citations recorded by the source corpus.

Auslander has shown that an algebra is of finite representation type, that is, it admits only finitely many indecomposable modules up to isomorphism, if and only if its representation dimension is at most 2. We will give a proof of this fact in Section 1 as Corollary 1.9. This led Auslander to the expectation, “that this notion gives a reasonable way of measuring how far an artin algebra is from being of finite representation type.” [1, III.5, lines 2, 3]

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9ee9e5cba3c7beca0d9358d14dd21fb3cd54b522

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Citations in the corpus

The classification of special Cohen–Macaulay modules (2008, methodology): “Auslander classified the indecomposable CM modules over quotient singularities in terms of the irreducible representations of the corresponding group and furthermore showed that when the group is small the AR and McKay quivers coincide [Aus86].”
On Artin Algebras Arising from Morita Contexts (2013, background): “Then the result follows from [2] since LL ( Λ Λ Λ Λ ) (0,0) = 2 LL (Λ).”
Krull-Schmidt categories and projective covers (2014, background): “The abstract concept of a Krull-Schmidt category can be found, for example, in expositions of Auslander [2, 3] and Gabriel-Roiter [5].”
Cohomology of twisted tensor products (2008, methodology): “that is, a codimension nquantum complete intersection where all the deﬁning exponents are 2), the global dimension of the endomorphism ring of the graded generator-cogenerator ⊕Γ/(RadΓ)i is n+ 1 (cf. [1]). Using this and Proposition 3.8, we obtain the following improvement of Remark 5.7. 5.8. Theore”
Derived equivalences and stable equivalences of Morita type, I (2008, methodology): “D(B), the morphism bf is a homomorphism of modules. If we have another A-module homomorphism b′ f such that πXb ′ f =F(f)πY , then πX(bf −b′f )=F(f)πY −F(f)πY =0 and bf −b′ f factorizes through Q¯+ X [1]. By Lemma 3.3, the B-module homomorphismbf −b′ f factorizes through a projective B-module. Thus,”
Recollement of Homotopy Categories and Cohen-Macaulay Modules (Representation Theory of Finite Groups and Algebras, and Related Topics) (2009, methodology): “ in B. •A morphism from α : X →T to β : X′ →T′ is a pair (f,g) of morphisms f : X →X′ and g : T →T′ such that gα = βf. When B= ModR for a ring R, this category was studied systematically by Auslander [A1] in the representation theory of algebras. For an object α : X →T in Mor(ModR), we have a T2(R)-”
Finitistic dimension and Igusa-Todorov algebras (2009, background): “(1) Assume that A is an algebra of representation dimension at most 3 with an Auslander generator AV .”
n-Representation infinite algebras (2012, background): “s The notion of representation dimension was introduced by Auslander to measure how far an algebra being from representation ﬁnite. Let us start with recalling the deﬁnition. Deﬁnition 7.1 (Auslander [A2]). The representation dimension of Λ is deﬁned by repdimΛ := inf{gl.dimEndΛ(Λ ⊕M ⊕DΛ) | M ∈ modΛ”
Categorical resolutions of irrational singularities (2012, background): “To our best knowledge, the first example of a categorical resolution was a theorem of Auslander [2].”
Auslander–Reiten theory revisited (2008, methodology): “Let us recall another approach given by Auslander [7].”