Tilting modules and universal localization

Hügel, Lidia Angeleri; Archetti, Maria

doi:10.1515/form.2011.080

Cited by 11 publications

(31 citation statements)

References 17 publications

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“…Let us remark that a noteworthy difference of Theorem 1.1(1) from the result [3,Proposition 1.7] is that our recollement is over derived module categories of precisely determined rings, while the recollement in [3,Proposition 1.7] involves a triangulated category. Theorem 1.1 (1) realizes this abstract triangulated category by a derived module category via describing the kernel of the functor T ⊗ L B −. Our result also distinguishes itself from the one in [40] where C is a differential graded ring instead of a usual ring, and where the consideration is restricted to ground ring being a field.…”

Section: Introductionmentioning

confidence: 80%

“…In Section 4, we discuss some homological properties of good tilting modules, and establish another crucial result, Proposition 4.6, for the proof of the main result Theorem 1.1. After these preparations, we apply the results obtained in Section 3 to prove Theorem 1.1 (1). In Section 5, we prove the second part of Theorem 1.1.…”

Section: Corollary 12 (1)mentioning

confidence: 98%

“…This may be regarded as a converse statement of the first part. In Section 6, we apply Theorem 1.1 to good tilting modules arising from ring epimorphisms, and prove Corollary 1.2 (1). In these cases the universal localization rings in Theorem 1.1 can be given by coproducts of rings.…”

Section: Corollary 12 (1)mentioning

confidence: 99%

“…, where I • and P • are chosen as in (1). Here we think of H as an induced functor between homotopy categories, and if…”

Section: Some Conventionsmentioning

confidence: 99%

“…One of the realizations of tilting modules is universal localizations. It is shown in [1] that every tilting module over a ring is associated in a canonical manner with a ring epimorphism which can be interpreted as a universal localization at a set of homomorphisms between finitely presented modules of projective dimension at most one.…”

Section: Introductionmentioning

confidence: 99%

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Good tilting modules and recollements of derived module categories

Chen

2012

Proceedings of the London Mathematical Society

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Let T be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring A, and let B be the endomorphism ring of T. We prove that if T is good, then there exists a ring C, a homological ring epimorphism B→C and a recollement among the (unbounded) derived module categories 𝒟C of C, 𝒟B of B and 𝒟A of A. In particular, the kernel of the total left‐derived functor T⊗B𝕃‐ is triangle equivalent to the derived module category 𝒟C. Conversely, if T⊗B𝕃‐ admits a fully faithful left adjoint functor, then T is good. Moreover, if T arises from an injective ring epimorphism, then C is isomorphic to the coproduct of two relevant rings. In the case of commutative rings, the ring C can be strengthened as the tensor product of two commutative rings. Consequently, we produce a large variety of examples (from Dedekind domains and p‐adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence), or different lengths. This shows that the Jordan–Hölder theorem fails even for stratifications by derived module categories.

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Section: Introductionmentioning

confidence: 80%

Section: Corollary 12 (1)mentioning

confidence: 98%

Section: Corollary 12 (1)mentioning

confidence: 99%

“…, where I • and P • are chosen as in (1). Here we think of H as an induced functor between homotopy categories, and if…”

Section: Some Conventionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Good tilting modules and recollements of derived module categories

Chen

2012

Proceedings of the London Mathematical Society

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Tilting and cotilting modules over concealed canonical algebras

Hügel

Kussin

2016

Math. Z.

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Abstract. We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is just one tilting module of that slope up to equivalence. We also provide a dual description of infinite dimensional cotilting modules and a classification result for the indecomposable pure-injective modules.

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Silting modules and ring epimorphisms

Hügel

Marks

Vitória

2016

Advances in Mathematics

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ABSTRACT. There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For hereditary rings, this assignment is used to parametrise homological ring epimorphisms by silting modules. We further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the case of finite dimensional algebras.

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Tilting modules and universal localization

Cited by 11 publications

References 17 publications

Good tilting modules and recollements of derived module categories

Good tilting modules and recollements of derived module categories

Tilting and cotilting modules over concealed canonical algebras

Silting modules and ring epimorphisms

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