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.

TILTING MODULES OF FINITE PROJECTIVE DIMENSION: SEQUENTIALLY STATIC AND COSTATIC MODULES

2002 ◽  
Vol 01 (03) ◽  
pp. 295-305 ◽  
Author(s):  
ALBERTO TONOLO
Keyword(s):  
Projective Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Finite Projective Dimension

In [5], Miyashita introduced tilting modules of finite projective dimension. A tilting module AV of projective dimension less or equal than r furnishes r + 1 equivalences between subcategories of A-Mod and End V-Mod: we call static and costatic the modules in A-Mod and End V-Mod, respectively, involved in these equivalences. In this paper we characterize the modules in A-Mod and End V-Mod which have a filtration with static and costatic factors, respectively.

scholarly journals Finitistic dimensions and good filtration dimensions of stratified algegras

2004 ◽  
Vol 69 (2) ◽  
pp. 341-347
Author(s):  
Shugui Wang ◽  
Bin Zhu
Keyword(s):  
Finitistic Dimension ◽  
Good Filtration ◽  
Properly Stratified Algebras ◽  
Finitistic Dimensions ◽  
Stratified Algebra

Δ–finitistic dimensions of standardly stratified algebras are defined similarly to properly stratified algebras. It is proved that the finitistic dimension for any standardly stratified algebra is bounded by the sum of the Δ–finitistic dimension and the ∇ good filtration dimension. Finally, the ∇–good filtration dimension of standardly stratified algebras is equal to the Δ–good filtration dimension of their Ringel duals.

REPETITIVE EQUIVALENCES AND TILTING THEORY

2019 ◽  
pp. 1-40
Author(s):  
JIAQUN WEI
Keyword(s):  
Projective Dimension ◽  
Derived Categories ◽  
Tilting Module ◽  
Tilting Modules ◽  
Tilting Theory ◽  
Finite Projective Dimension ◽  
Morita Theory ◽  
Wakamatsu Tilting Module ◽  
Stable Module

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$ . We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$ ). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

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.

Wakamatsu tilting modules over ring extension

2019 ◽  
Vol 18 (10) ◽  
pp. 1950198
Author(s):  
Zhen Zhang ◽  
Jiaqun Wei
Keyword(s):  
Ring Extension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Wakamatsu Tilting Module ◽  
Extension Ring

For a ring [Formula: see text], an extension ring [Formula: see text], and a fixed right [Formula: see text]-module [Formula: see text], we prove the induced left [Formula: see text]-module [Formula: see text] is a Wakamatsu tilting module when [Formula: see text] is a Wakamatsu tilting module.

scholarly journals ON A DUALITY THEOREM OF WAKAMATSU

2008 ◽  
Vol 78 (2) ◽  
pp. 343-350
Author(s):  
ZHAOYONG HUANG
Keyword(s):  
Nonnegative Integer ◽  
Duality Theorem ◽  
Tilting Module ◽  
Coherent Ring ◽  
Finitely Presented ◽  
Generalized Tilting Module

AbstractLet R be a left coherent ring, S a right coherent ring and RU a generalized tilting module, with S=End(RU) satisfying the condition that each finitely presented left R-module X with ExtRi(X,U)=0 for any i≥1 is U-torsionless. If M is a finitely presented left R-module such that ExtRi(M,U)=0 for any i≥0 with $i \neq n$ (where n is a nonnegative integer), then $\mathrm {Ext}_S^n( \mathrm {Ext}_R^n(M, U), U)\cong M$ and ExtSi(ExtRn(M,U),U)=0 for any i≥0 with $i\neq n$. A duality is thus induced between the category of finitely presented holonomic left R-modules and the category of finitely presented holonomic right S-modules.

scholarly journals An Euler characteristic for modules of finite G-dimension

2008 ◽  
Vol 102 (2) ◽  
pp. 206 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Diana White
Keyword(s):  
Euler Characteristic ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Finite Projective Dimension ◽  
Extremal Behavior ◽  
Classical Invariant ◽  
Category Of Modules ◽  
Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

scholarly journals From Jantzen to Andersen filtration via tilting equivalence

2012 ◽  
Vol 110 (2) ◽  
pp. 161 ◽  
Author(s):  
Johannes Kübel
Keyword(s):  
Verma Module ◽  
The Other ◽  
Tilting Module ◽  
Tilting Modules ◽  
Projective Object ◽  
Other Hand

The space of homomorphisms between a projective object and a Verma module in category $\mathcal O$ inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms between a Verma module and a tilting module. Arkhipov's tilting functor, a contravariant self-equivalence of a certain subcategory of $\mathcal O$, which maps projective to tilting modules induces an isomorphism of these kinds of Hom-spaces. We show that this equivalence identifies both filtrations.

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