One Dimensional Tilting Modules are of Finite Type

Bazzoni, Silvana; Herbera, Dolors

doi:10.1007/s10468-007-9064-3

Cited by 54 publications

(77 citation statements)

References 19 publications

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“…The proof here is in fact a part of the proof of [3,Theorem 5.1]. It is well known that for the countable direct limit above, there is an exact sequence…”

Section: Preliminariesmentioning

confidence: 91%

Sigma-cotorsion modules over valuation domains

Bazzoni¹,

Šťovíček²

2009

Forum Mathematicum

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“…The proof here is in fact a part of the proof of [3,Theorem 5.1]. It is well known that for the countable direct limit above, there is an exact sequence…”

Section: Preliminariesmentioning

confidence: 91%

Sigma-cotorsion modules over valuation domains

Bazzoni¹,

Šťovíček²

2009

Forum Mathematicum

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“…For (1) see [12,13] For the converse implication, we use that every cotilting module is pure-injective, see [11,28], or [18, 8.1.7]. So, if C is a cotilting module such that ⊥ C = A, and S consists of the pure-injective modules in B, then S is a coresolving subcategory of PI that contains C and has the stated properties, see [18, 8.1.10].…”

Section: ) (A B) Is An N-tilting Cotorsion Pair If and Only If It Imentioning

confidence: 99%

Large tilting modules and representation type

2010

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Abstract. We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in [5] for tame hereditary algebras.

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“…By [BH08, Theorem 1.6] it follows that they are both definable classes, that is, they are closed under direct products, direct limits, and pure submodules. As noted in Section 2 of [BH08], a combination of Ziegler's result [Zie84, Theorem 6.9] and Keisler-Shelah Theorem (cf. [Kei61] and [She71]) implies that Gen S R and U ⊥ coincide if and only if they contain the same pure-injective right R-modules.…”

Section: Lemma ([Gt06 Lemma 312])mentioning

confidence: 99%

Tilting Modules Arising from Ring Epimorphisms

Hügel

Sánchez

2009

Algebr Represent Theor

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Abstract. We show that a tilting module T over a ring R admits an exact sequence 0 → R → T 0 → T 1 → 0 such that T 0 , T 1 ∈ Add(T ) and Hom R (T 1 , T 0 ) = 0 if and only if T has the form S ⊕ S/R for some injective ring epimorphism λ : R → S with the property that Tor R 1 (S, S) = 0 and pdS R ≤ 1. We then study the case where λ is a universal localization in the sense of Schofield [Sch85]. Using results from [CB91], we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises from universal localization.

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One Dimensional Tilting Modules are of Finite Type

Abstract: We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one.

Cited by 54 publications

References 19 publications

Sigma-cotorsion modules over valuation domains

Sigma-cotorsion modules over valuation domains

Large tilting modules and representation type

Tilting Modules Arising from Ring Epimorphisms

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