scholarly journals GORENSTEIN $\pi[T]$-PROJECTIVITY WITH RESPECT TO A TILTING MODULE

2020 ◽  
pp. 114-126
Author(s):  
M. Amini
Keyword(s):  
Tilting Module

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].

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.

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 Lefschetz Operators, Hodge–Riemann Forms, and Representations

2020 ◽  
Author(s):  
Peter Fiebig
Keyword(s):  
Root System ◽  
Bilinear Form ◽  
Chevalley Group ◽  
Simple Root ◽  
Complex Geometry ◽  
Linear Operators ◽  
Vector Spaces ◽  
Tilting Module ◽  
Weight Lattice ◽  
Simply Connected

Abstract For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

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.

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 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.

scholarly journals Generalized Blob Algebras and Alcove Geometry

2003 ◽  
Vol 6 ◽  
pp. 249-296 ◽  
Author(s):  
Paul P. Martin ◽  
David Woodcock
Keyword(s):  
Reflection Group ◽  
Weight Space ◽  
Space Representation ◽  
Lie Theory ◽  
Tilting Module ◽  
Finite Dimensional ◽  
Affine Hecke Algebras ◽  
Space Formalism ◽  
Affine Reflection ◽  
Labelling Scheme

AbstractA sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

scholarly journals Counterexamples to the tilting and (p,r)-filtration conjectures

2020 ◽  
Vol 2020 (767) ◽  
pp. 193-202
Author(s):  
Christopher P. Bendel ◽  
Daniel K. Nakano ◽  
Cornelius Pillen ◽  
Paul Sobaje
Keyword(s):  
Algebraic Group ◽  
Indecomposable Module ◽  
Weyl Module ◽  
Tilting Module ◽  
Simple Algebraic Group ◽  
Frobenius Kernel ◽  
Projective Indecomposable Module

AbstractIn this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the {(p,r)}-Filtration Conjecture.

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