t-structures on hereditary categories

Stanley, Donald; Roosmalen, Adam-Christiaan van

doi:10.1007/s00209-018-2190-2

Cited by 6 publications

(12 citation statements)

References 26 publications

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“…Remark 3.9. A similar condition to (ii) of Proposition 3.7 appears in a slightly different formulation in [46], where sequences of subcategories of the module category determining a coaisle of a t-structure over a hereditary ring are called "reflective co-narrow sequences". In our setting, the reflectivity is ensured by the definability of the members of the sequence.…”

Section: Definable Coaislesmentioning

confidence: 87%

Definable coaisles over rings of weak global dimension at most one

Bazzoni¹,

Hrbek²

2021

Publicacions Matemàtiques

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In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that, unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the Telescope Conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.

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Section: Definable Coaislesmentioning

confidence: 87%

Definable coaisles over rings of weak global dimension at most one

Bazzoni¹,

Hrbek²

2021

Publicacions Matemàtiques

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“…• In [60], coaisles of t-structures in the case of hereditary categories were studied using a notion of reflective co-narrow sequences of subcategories. These sequences satisfy essentially the same closure condition as the one considered in Proposition 5.1.…”

Section: Constructing T-structures From Chains Of Epimorphismsmentioning

confidence: 99%

Parametrizing torsion pairs in derived categories

Hügel

Hrbek

2021

Represent. Theory

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We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A , which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) . This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.

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“…It is then shown in Corollary 7.4 that the existence of such a t-structure (X , Y) implies that (U, V) induces a derived equivalence. The construction of the t-structure (X , Y) is heavily based on the description of t-structures on D b mod Λ given in [49]; we will recall the relevant results in §3. 5.…”

Section: ≥0mentioning

confidence: 99%

“…We know ([9, Proposition 2.4], see also [49] based on [41, Proposition 1.4]) that the embedding U → C admits a right adjoint and that the embedding V → C admits a left adjoint.…”

Section: Weight Structuresmentioning

confidence: 99%

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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t-structures on hereditary categories

Cited by 6 publications

References 26 publications

Definable coaisles over rings of weak global dimension at most one

Definable coaisles over rings of weak global dimension at most one

Parametrizing torsion pairs in derived categories

Derived equivalences for hereditary Artin algebras

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