Pre-Projective Parts of Tilting Quivers Over Certain Path Algebras

Happel and Unger defined a partial order on the set of basic tilting modules. We study the poset of basic preprojective tilting modules over path algebras of representation-infinite type. First we will give a criterion for Ext-vanishing for preprojective modules. With the using of this result, we will give combinatorial characterizations of the poset of basic preprojective tilting modules. Finally, we will see the structure of a preprojective part of tilting quivers.     

倾斜RnA-模及其导出的挠理论

本文通过函子Ｔ＝－ＡＲｎＡ讨论了倾斜Ａ 模与倾斜ＲｎＡ 模的重要联系，推广了［１］的主要结果；讨论了倾斜ＲｎＡ 模ＴＸ与倾斜Ａ 模导出的挠理论在相同性和分裂性等方面的关系．     

Gorenstein Cotilting and Tilting Modules

Auslander and Solberg introduced the concepts of finitely generated cotilting and tilting modules in relative homological algebra considering subfunctors of the Ext-functor. In this article we generalize Auslander–Solberg relative notions by giving the definitions of infinitely generated Gorenstein cotilting and tilting modules by means of Gorenstein exact sequences. Using the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we prove a characterization of relative Gorenstein cotilting and tilting modules, which is a generalization of the beautiful characterization of relative cotilting and tilting modules given by Bazzoni.     

All tilting modules are of finite type

We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Ext-orthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules.

     

Tilting Preenvelopes and Cotilting Precovers

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.     

On Modules over Endomorphism Algebras of Maximal Rigid Objects in 2-Calabi-Yau Triangulated Categories

This note investigates the modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. We study the possible complements for almost complete tilting modules. Combining with Happel's theorem, we show that the possible exchange sequences for tilting modules over such algebras are induced by the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated categories. For the modules of infinite projective dimension, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.     

Symmetric Group Modules with Specht and Dual Specht Filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL _n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology.

Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL _n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL _n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.     

Tilting and cluster tilting for quotient singularities

We shall show that the stable categories of graded Cohen–Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander–Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen–Macaulay modules.     

Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.     

When are definable classes tilting and cotilting classes?

It is known that tilting classes are of finite type, while cotilting classes are not always of cofinite type. We investigate this phenomenon. By using a bijection between definable classes of left modules and definable classes of right modules, we prove that it reflects the asymmetry existing between the notions of covers and envelopes or, otherwise stated, right and left approximations.In particular we show that there exist definable torsion classes containing the injective modules which are not tilting classes.     

On the support varieties of tilting modules

Let G be a reductive algebraic group scheme defined over the finite field F_p, with Frobenius kernel G₁. The tilting modules of G are defined as rational G-modules for which both the module itself and its dual have good filtrations. In 1997, J.E. Humphreys conjectured that the support varieties of certain tilting modules for regular weights should be given by the Lusztig bijection between cells of the affine Weyl group and nilpotent orbits of G, when p>h, where h is the Coxeter number. We present a conjecture for the support varieties of tilting modules when G=GL_n. Our conjecture is equivalent to Humphreys’ conjecture for p≥h and regular weights, but our formulation allows us to consider small p or singular weights as well. We obtain results for several infinite classes of tilting modules, including the case p=2, and tilting modules whose support variety corresponds to a hook partition. In the case p=2, we prove the conjecture by S. Donkin for the support varieties of tilting modules.     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Complements to Projective Almost Complete Tilting Modules

ABSTRACT

An equivalent version of the Generalized Nakayama Conjecture states that any projective almost complete tilting module admits a finite number of non-isomorphic indecomposable complements. Motivated by this connection, we investigate the number of possible complements of projective almost complete tilting modules for some particular classes of Artin algebras, namely monomial algebras and algebras with exactly two simple modules.     

偏倾斜模和倾斜模的分次与非分次性质

This paper gives the relationships among partial tilting objects （tilting objects） of categories of graded left A-modules of type G, left A-modules, left Ae-modules and A#-modules, and then proves that for graded partial tilting modules, there exist the Bongartz complements in the category of graded A-modules.     

A Bongartz-type lemma for silting complexes over a hereditary algebra

Let A be a finite dimensional algebra, the Bongartz lemma for classic tilting modules says that any partial tilting module is a direct summand of a tilting module. In this paper, we prove that a Bongartz-type lemma for silting complexes in the bounded derived category $$D^b(A)$$ holds if A is a hereditary algebra.     

Gorenstein tilting pairs

Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.     

One Dimensional Tilting Modules are of Finite Type

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one. Presented by Claus Michael Ringel.     

Canonical Tilting Modules Over Shod Algebras are Regular in Codimension One

We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

On Certain Blocks of Schur Algebras

In this paper, what is already known about defect 2 blocks ofsymmetric groups is used to deduce information about the correspondingblocks of Schur algebras. This information includes Ext-quiversand decomposition numbers, as well as Loewy structures of theWeyl modules, principal indecomposable modules and tilting modules.