Tilting Wild Algebras

Kerner, Otto

doi:10.1112/jlms/s2-39.1.29

Cited by 115 publications

(79 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such algebras were called in [10] concealed-canonical algebras and almost concealedcanonical algebras, respectively. Concealed-canonical algebras and almost concealed-canonical algebras are important classes of quasi-tilted algebras in the sense of Happel, Reiten and Smalø [3].Our result generalizes theorems of Kerner [7] and [8] studying the case of tilted algebras and of Lenzing and de la Peña [11] considering the case of canonical algebras.The representation type of Σ depends on the weight type, or equivalently, on the virtual genus g of the weighted projective line X. If g < 1 then Σ is a tame concealed algebra and the Auslander-Reiten quiver is well known.…”

supporting

confidence: 63%

Auslander-Reiten components for concealed-canonical algebras

Meltzer¹

1996

Colloq. Math.

View full text Add to dashboard Cite

1. Introduction. We describe the structure of the Auslander-Reiten components of finite-dimensional modules for endomorphism rings Σ of tilting bundles and tilting sheaves on a weighted projective line. Such algebras were called in [10] concealed-canonical algebras and almost concealedcanonical algebras, respectively. Concealed-canonical algebras and almost concealed-canonical algebras are important classes of quasi-tilted algebras in the sense of Happel, Reiten and Smalø [3].Our result generalizes theorems of Kerner [7] and [8] studying the case of tilted algebras and of Lenzing and de la Peña [11] considering the case of canonical algebras.The representation type of Σ depends on the weight type, or equivalently, on the virtual genus g of the weighted projective line X. If g < 1 then Σ is a tame concealed algebra and the Auslander-Reiten quiver is well known. If g = 1, the algebra Σ is a tubular algebra and the structure of the Auslander-Reiten components was described by Ringel in [13] (see also [9] for a classification using the geometrical approach).Here we are interested in the case g > 1, i.e. X is wild. In this case Σ is strictly wild as was shown in [10]. In that paper we also gave a global view of the category mod(Σ) of finite-dimensional modules over Σ. Identifying the derived category of mod(Σ) with the derived category of coherent sheaves coh(X) and transporting the notions of rank and degree of sheaves to modules we have four typical parts of Σ-modules, mod + (Σ), mod

show abstract

supporting

confidence: 63%

Auslander-Reiten components for concealed-canonical algebras

Meltzer¹

1996

Colloq. Math.

View full text Add to dashboard Cite

show abstract

“…We know that Z 1 is a regular C-module. By [20,28] there exists a natural number m with the following properties: (1) and (2) for X. Lemma 2.6. Let H be a hereditary artin algebra and X be a finitely generated indecomposable H-module without self-extensions.…”

Section: Proposition 23 Let a Be An Artin Algebra And T A Tiltingmentioning

confidence: 99%

Constructing tilting modules

Kerner

Trlifaj

2007

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We investigate the structure of (infinite dimensional) tilting modules over hereditary artin algebras. For connected algebras of infinite representation type with Grothendieck group of rank n, we prove that for each 0 ≤ i < n − 1, there is an infinite dimensional tilting module T i with exactly i pairwise non-isomorphic indecomposable finite dimensional direct summands. We also show that any stone is a direct summand in a tilting module. In the final section, we give explicit constructions of infinite dimensional tilting modules over iterated one-point extensions.The study of finite dimensional tilting modules of projective dimension at most one over finite dimensional algebras was initiated by Brenner and Butler [11] and continued by Happel and Ringel [17]. Since then, many variations of this concept have been introduced and used successfully, for example: Tilting modules of higher projective dimension, tilting modules over rings, tilting complexes in derived categories, tilting objects in hereditary categories or in cluster categories. In this paper, we use the term tilting module as follows: Let R be a ring and T be a right R-module. Then T is a tilting module provided that (T1) p.dimT ≤ 1, (T2) Ext 1 R (T, T (I) ) = 0 for any set I, and (T3) there is a short exact sequence 0 → R → T 0 → T 1 → 0 where T 0 and T 1 are direct summands in a direct sum of (possibly infinitely many) copies of T . Equivalently, T is tilting if and only if Gen(T ) = {T } ⊥ , [13].Here, Gen(T ) denotes the class of all homomorphic images of direct sums of copies of T , and, for a class of modules C,If T is a tilting module, then {T } ⊥ is a torsion class in Mod-R, the tilting class generated by T . If T is another tilting module, then T is said to be equivalent toThough our definition of a tilting module allows infinitely generated modules, there is an implicit finiteness property connected with tilting, recently proved by Bazzoni and Herbera in [9, Theorem 2.4]. Namely, any tilting module T is of finite type, that is, there exists a set S consisting of finitely presented modules of projective dimension at most 1 such that S ⊥ = {T } ⊥ .

show abstract

“…Then C = End H (T ) is said to be a concealed algebra of type ∆. It is known that such a C is a wild triangular algebra of global dimension at most 2, and its Tits form q C is not weakly nonnegative (see [18], [28]). The concealed algebras of types A m , T 5 , D n , E 6 , E 7 , E 8 have been classified by quivers and relations in [20], [36], [38].…”

Section: Consider the Extended Euclidean Graphsmentioning

confidence: 99%

Tame tensor products of algebras

Leszczyński

Skowroński

2003

Colloq. Math.

View full text Add to dashboard Cite

Abstract. With the help of Galois coverings, we describe the tame tensor products A ⊗ K B of basic, connected, nonsimple, finite-dimensional algebras A and B over an algebraically closed field K. In particular, the description of all tame group algebras AG of finite groups G over finite-dimensional algebras A is completed.Introduction. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite-dimensional K-algebra (associative, with an identity) which we moreover assume to be basic and connected. An algebra A can be written as a bound quiver algebra A ∼ = KQ/I, where Q = Q A is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q.By Drozd's Tame and Wild Theorem [9] the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all finite-dimensional algebras over K. Accordingly, we may realistically hope to classify the indecomposable finitedimensional modules only for the tame algebras. The representation theory of arbitrary tame algebras is still only emerging.We are concerned with the problem of describing when the tensor product A ⊗ K B of two nonsimple algebras A and B is tame. The class of tensor product algebras contains several important classes of algebras, including:

show abstract

Tilting Wild Algebras

Cited by 115 publications

References 12 publications

Auslander-Reiten components for concealed-canonical algebras

Auslander-Reiten components for concealed-canonical algebras

Constructing tilting modules

Tame tensor products of algebras

Contact Info

Product

Resources

About