Constructing Torsion Pairs

Assem, Ibrahim; Kerner, Otto

doi:10.1006/jabr.1996.0310

Cited by 18 publications

(36 citation statements)

References 16 publications

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“…By the assumptions of the lemma, we know that there is a Z ∈ H(U) such that H 2−n U X → M [2] factors through Z [1]. There is thus a short exact sequence…”

Section: A Criterion For Derived Equivalencementioning

confidence: 96%

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Derived equivalences for hereditary Artin algebras

Stanley

Roosmalen

2016

Advances in Mathematics

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Abstract. We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let (U , V) be a t-structure on the bounded derived category D b A with heart H. We investigate when the natural embedding H → D b A can be extended to a triangle equivalence D b H → D b A. Our focus of study is the case where A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t-structure is bounded and the aisle U of the t-structure is closed under the Serre functor.

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“…By the assumptions of the lemma, we know that there is a Z ∈ H(U) such that H 2−n U X → M [2] factors through Z [1]. There is thus a short exact sequence…”

Section: A Criterion For Derived Equivalencementioning

confidence: 96%

“…The first equality follows from E ∈ H(U), and the second equality follows from Hom(X [1], τ E) ∼ = Hom(X [2], SE) = 0 together with SE ∈ H(U).…”

Section: Proofmentioning

confidence: 99%

Derived equivalences for hereditary Artin algebras

Stanley

Roosmalen

2016

Advances in Mathematics

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“…It follows from [4] and [13] that T → T ∩ {X} ⊥ ≥0 defines a bijection between the torsion classes T in Mod-H containing X as an Ext-projective module, and the torsion classes in {X} ⊥ ≥0 . Since X has n − 1 pairwise non-isomorphic indecomposable direct summands, the category X ⊥ ≥0 is equivalent to Mod-K, where K is a k-division algebra; see Section 1.…”

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

confidence: 99%

“…Before presenting our final example, we collect some facts on tilting torsion pairs in mod-H for a finite dimensional hereditary algebra H; see [21,4]. (1) Let X be an indecomposable non-projective torsion module.…”

Section: Iterated One-point Extensionsmentioning

confidence: 99%

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Constructing tilting modules

Kerner

Trlifaj

2007

Trans. Amer. Math. Soc.

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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 } ⊥ .

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Constructing Torsion Pairs

Cited by 18 publications

References 16 publications

Derived equivalences for hereditary Artin algebras

Derived equivalences for hereditary Artin algebras

Constructing tilting modules

Minimal Approximations, Orbital Elementary Modules, and Orbit Algebras of Regular Modules

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