Torsion Pairs and Ringel Duality for Schur Algebras

Let A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Parametrizing torsion pairs in derived categories

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.

Torsion Pairs and Quasi-abelian Categories

We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.

A FRAMEWORK FOR TORSION THEORY COMPUTATIONS ON ELLIPTIC THREEFOLDS

We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold.

tCG torsion pairs

We investigate conditions when the [Formula: see text]-structure of Happel–Reiten–Smalø associated to a torsion pair is a compactly generated [Formula: see text]-structure. The concept of a [Formula: see text]CG torsion pair is introduced and for any ring [Formula: see text], we prove that [Formula: see text] is a [Formula: see text]CG torsion pair in [Formula: see text] if, and only if, there exists, [Formula: see text] a set of finitely presented [Formula: see text]-modules in [Formula: see text], such that [Formula: see text]. We also show that every [Formula: see text]CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the [Formula: see text]CG torsion pairs over Noetherian rings and von Neumman regular rings.

The Cotorsion Pair Cogenerated by the Subcategory of Stable Functors in ((R-mod)op, Ab)

Let R be an associative ring with identity. Denote by ((R-mod)op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let [Formula: see text] be the subcategory of stable functors. There are two torsion pairs [Formula: see text] and [Formula: see text], where ℱ1 is the subcategory of ((R-mod)op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair [Formula: see text] cogenerated by [Formula: see text] clearly. It is shown that [Formula: see text] if and only if G/t1(G) is a projective object in [Formula: see text], i.e., G/t1(G) = (−,M) for some R-module M; and [Formula: see text] if and only if G/t2(G) is of the form (−, E), where E is an injective R-module.