Definability and approximations in triangulated categories

Laking, Rosanna; Vitória, Jorge

doi:10.48550/arxiv.1811.00340

Cited by 4 publications

(7 citation statements)

References 32 publications

(48 reference statements)

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“…For stable t-structures, all these conditions are known to be equivalent, while in the general situation this seems to be presently not known. By [La18] and [LV18], the definability of the coaisle is stronger than the two other conditions. Also, by [La18] the definability of the coaisle characterizes the situation when a left non-degenerate t-structure is induced by a pure-injective cosilting object.…”

Section: Introductionmentioning

confidence: 94%

Smashing localizations of rings of weak global dimension at most one

Bazzoni

Šťovíček

2017

Advances in Mathematics

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Abstract. We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Prüfer domains which are strongly discrete.

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Section: Introductionmentioning

confidence: 94%

Smashing localizations of rings of weak global dimension at most one

Bazzoni

Šťovíček

2017

Advances in Mathematics

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“…By [42,Corollary 6.6], there is a stable TTF triple (L, B, K) in D(Mod-A). Furthermore, by [42,Proposition 6.11], there is a pure-injective object C ∈ D(Mod-A) such that the t-structure (U ′ , V ′ ) in D(Mod-A) defined by U ′ = ⊥ ≤0 C (which exists by [42,Corollary 5.4]) satisfies 2.6. Silting and cosilting TTF triples.…”

Section: 4mentioning

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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“…A definable class in D(A) is defined as a subclass X ⊆ D(A) that is closed under taking products, directed homotopy colimits and pure subobjects. Definable classes in D(A) have been recently characterized in [26] as those classes which are closed under taking direct products, pure subobjects and pure quotients. so [25,Thm.…”

Section: Htmentioning

confidence: 99%

Locally finitely presented and coherent hearts

Parra¹,

Saorı́n²,

Virili³

2019

Preprint

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Given a torsion pair t = (T , F ) in a Grothendieck category G, we study when the heart H t of the associated Happel-Reiten-Smalø t-structure in the derived category D(G) is a locally finitely presented or a locally coherent Grothendieck category. Since H t is Grothendieck precisely when t is of finite type (i.e., F is closed under direct limits), we first study the latter torsion pairs showing that, as in modules, they are precisely the quasi-cotilting ones, that in turn coincide with the cosilting ones.We then prove that, for G chosen in a wide class of locally finitely presented Grothendieck categories that includes the locally coherent ones, the module categories and several categories of quasi-coherent sheaves over schemes, the heart H t is locally finitely presented if, and only if, t is generated by finitely presented objects. For the same class of Grothendieck categories, it is then proved that if F is a generating class in G, in which case it is known that t is given by a (1-)cotilting object Q, the heart H t is locally coherent if, and only if, it is generated by finitely presented objects and there is a set X ⊂ F ∩ fp(G) that is a set of generators of G and satisfies the following two conditions:(1) Ext 1 G (X, −) vanishes on direct limits of objects in Prod(Q), for all X ∈ X ; (2) each epimorphism p :has a kernel which is a direct summand of (1 : t)(N ), for some N ∈ fp(G). A consequence of this is that, when G = Mod-A is the module category over small pre-additive category A (e.g., over an associative unital ring) and F is generating in Mod-A, the heart H t is locally coherent if, and only if, t is generated by finitely presented modules and, for each M ∈ mod-A := fp(Mod-A), the module (1 : t)(M ) admits a projective resolution with finitely generated terms.A further consequence is that if R is a right coherent ring and t = (T , F ) is a torsion pair such that the torsion ideal t(R) is finitely generated on the right, then H t is a locally coherent Grothendieck category if, and only if, t is generated by finitely presented modules and it is given by a cosilting module which is an elementary cogenerator in Mod-R.

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Definability and approximations in triangulated categories

Cited by 4 publications

References 32 publications

Smashing localizations of rings of weak global dimension at most one

Smashing localizations of rings of weak global dimension at most one

Parametrizing torsion pairs in derived categories

Locally finitely presented and coherent hearts

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