scholarly journals Torsion pairs and filtrations in abelian categories with tilting objects

2015 ◽  
Vol 14 (08) ◽  
pp. 1550121
Author(s):  
Jason Lo
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Explicit Description ◽  
Homological Dimension ◽  
Abelian Categories ◽  
Dimension 2 ◽  
Tilting Object ◽  
Category Of Modules ◽  
Torsion Pairs ◽  
Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

scholarly journals FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO

2012 ◽  
Vol 12 (02) ◽  
pp. 1250149 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
DAG OSKAR MADSEN ◽  
XIUPING SU
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Homological Dimension ◽  
Derived Equivalence ◽  
Torsion Pair ◽  
Abelian Categories ◽  
Tilting Object ◽  
Dimension One

We consider filtrations of objects in an abelian category [Formula: see text] induced by a tilting object T of homological dimension at most two. We define three extension closed subcategories [Formula: see text] and [Formula: see text] with [Formula: see text] for j > i, such that each object in [Formula: see text] has a unique filtration with factors in these categories. In dimension one, this filtration coincides with the classical two-step filtration induced by the torsion pair. We also give a refined filtration, using the derived equivalence between the derived categories of [Formula: see text] and the module category of [Formula: see text].

scholarly journals TILTING CHAINS OF NEGATIVE CURVES ON RATIONAL SURFACES

2017 ◽  
Vol 235 ◽  
pp. 26-41 ◽  
Author(s):  
LUTZ HILLE ◽  
DAVID PLOOG
Keyword(s):  
Abelian Category ◽  
Rational Surface ◽  
Rational Curves ◽  
Endomorphism Algebra ◽  
Exceptional Sequence ◽  
Universal Extension ◽  
Tilting Object ◽  
Category Of Modules ◽  
A Chain ◽  
Tilting Objects

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of  $T$ . Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of $(-2)$ -curves, we obtain an equivalence with modules over a well-known algebra.

scholarly journals Extension of matrix pencil reduction to abelian categories

2018 ◽  
Vol 17 (04) ◽  
pp. 1850062
Author(s):  
Olivier Verdier
Keyword(s):  
Normal Form ◽  
Vector Space ◽  
Decomposition Theorem ◽  
Abelian Category ◽  
Jordan Canonical Form ◽  
Main Technique ◽  
Abelian Categories ◽  
Category Of Modules ◽  
Space Case ◽  
And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Homological Dimensions Relative to Special Subcategories

2021 ◽  
Vol 28 (01) ◽  
pp. 131-142
Author(s):  
Weiling Song ◽  
Tiwei Zhao ◽  
Zhaoyong Huang
Keyword(s):  
Abelian Category ◽  
Projective Dimension ◽  
Homological Dimensions ◽  
Category Of Modules ◽  
The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

scholarly journals FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

2019 ◽  
Vol 62 (2) ◽  
pp. 383-439 ◽  
Author(s):  
LEONID POSITSELSKI
Keyword(s):  
Associative Ring ◽  
Abelian Category ◽  
Projective Dimension ◽  
Countable Base ◽  
Linear Topology ◽  
Countable Type ◽  
Flat Ring ◽  
Abelian Categories ◽  
Induced Topology ◽  
Strongly Flat

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

scholarly journals PROPERLY STRATIFIED ALGEBRAS AND TILTING

2005 ◽  
Vol 92 (1) ◽  
pp. 29-61 ◽  
Author(s):  
ANDERS FRISK ◽  
VOLODYMYR MAZORCHUK
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Category Of Modules ◽  
Cotilting Modules ◽  
Contravariantly Finite ◽  
Properly Stratified Algebras ◽  
Stratified Algebra ◽  
Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

On the Square of a Homological Monoid

1966 ◽  
Vol 9 (1) ◽  
pp. 49-55 ◽  
Author(s):  
Rosemary Bonyun
Keyword(s):  
General Theorem ◽  
Abelian Category ◽  
Uniqueness Condition ◽  
Abelian Categories

Homological monoids, as first defined by Hilton and Ledermann [ l ], are a generalization of abelian categories. It is known that if is an abelian category, so is here we prove the more general theorem that if is a homological monoid, so is . Our definition differs from that originally given by Hilton and Ledermann by the addition of a uniqueness condition in Axiom 1.

scholarly journals Torsion Pairs and Quasi-abelian Categories

2020 ◽  
Author(s):  
Aran Tattar
Keyword(s):  
Abelian Categories ◽  
Torsion Pairs

AbstractWe 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.

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

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