Cluster tilting for higher Auslander algebras

By Osamu Iyama. 2008

242 citations recorded by the source corpus.

Abstract The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander–Reiten translation functor τ n plays an important role in the study of n-cluster tilting subcategories. We study the category M n of preinjective-like modules obtained by applying τ n to injective modules repeatedly. We call a finite-dimensional algebra Λ n-complete if M n = add M for an n-cluster tilting object M. Our main result asserts that the endomorphism algebra End Λ ( M ) is ( n + 1 ) -complete. This gives an inductive construction of n-complete algebras. For example, any representation-finite hereditary algebra Λ ( 1 ) is 1-complete. Hence the Auslander algebra Λ ( 2 ) of Λ ( 1 ) is 2-complete. Moreover, for any n ⩾ 1 , we have an n-complete algebra Λ ( n ) which has an n-cluster tilting object M ( n ) such that Λ ( n + 1 ) = End Λ ( n ) ( M ( n ) ) . We give the presentation of Λ ( n ) by a quiver with relations. We apply our results to construct n-cluster tilting subcategories of derived categories of n-complete algebras.

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

View Paper: https://www.semanticscholar.org/paper/e38e870df54c90289ad8885e7fe1bae89db91943

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Citations in the corpus

Auslander–Reiten theory revisited (2008, methodology): “In this subsection we give a series of examples of finite-dimensional algebras with n-cluster tilting objects from [ 87 ].”
Higher-dimensional cluster combinatorics and representation theory (2010, background): “A1 n Pi its projective cover, and by A1 n Ii its injective envelope. Then the numbering of the vertices is chosen in such a way that HomA1 n (A1 n Pi,A1 n Pj) 6= 0 ⇐⇒i ≤j. Theorem / Construction 3.3 ([Iya]). •A1 n is 1-representation ﬁnite. We denote its basic 1- cluster tilting module by A1 n M, an”
An introduction to higher Auslander–Reiten theory (2016, background): “egory are split. Nevertheless, the following simple observation motivates the investigation of the properties of exact sequences of length d+2 in such subcategories. Proposition 2.2 (cf. Lemma 3.5 in [Iya11]). Let M be a d-rigid subcategory of modΛ and 0 L M1 ··· Md N 0 an exact sequence in modΛ who”
n-Representation infinite algebras (2012, background): “i=0 ν−i n P is an n-cluster tilting Λ-module [1].”
Representations of Algebras and Related Topics (2011, background): “ p ∈Z}⊂Db(Λ) provides a Z-covering of the graded algebra Π3(Λ). Hence the study of the category UΛ = π−1(Λ) is one of the fundamental tools for the study of the grading of Π3(Λ). Theorem 5.2 (Thm 1.22[61], Prop5.4.2[2]). If Λ is τ2-ﬁnite, then the subcategory UΛ is a cluster-tilting subcategory of D”
n-angulated categories (2010, result): “rm of the higher octahedra described in Remark 1.1.14 of [4]. Cluster tilting theory (as developed in [2, 12, 20] and many other articles) and in particular Iyama’s higher Auslander-Reitentheory (see [16, 17]) have lead to the surprising discovery that there is a large class of categories which are ”
n-Abelian and n-exact categories (2014, background): “From a different perspective, n-cluster-tilting subcategories of certain exact categories were introduced by Iyama in [29] and further investigated in [30, 28] from the viewpoint of higher Auslander-Reiten theory.”
Selfinjective quivers with potential and 2-representation-finite algebras (2010, background): “ebra A is 2-representation-ﬁnite if there exists a cluster tilting A-module and gl.dimA ≤2. In this case it is known that Π3(A) regarded as an A-module gives a unique basic 2-cluster tilting A-module [I4]. We have the following criterion for 2-representation-ﬁniteness. Proposition 3.9. Let A be a K-”
Higher Nakayama algebras I: Construction (2016, background): “modA), which is a Serre functor on Db(modA), see [Hap88, Sec. 1.4] and [RvdB02, Thm. I.2.4]. Notation 1.5.5. Let Abe a nite dimensional algebra of nite global dimension and set d := [ d]. Following [Iya11], we dene the subcategory U(A) of Db(modA) as follows: U(A) := add i d(DA) i2Z : If Ais a d-rep”
Torsion Classes and t-Structures in Higher Homological Algebra (2014, methodology): “Higher homological algebra was introduced and developed by Iyama in [10], [11], [12].”
Maximal rigid subcategories in 2-Calabi-Yau triangulated categories (2010, background): “It is well-known that the derived category of Q contains cluster tilting subcategories, see for example the example in Section 5 of [KZ], or [I2].”