On partial tilting modules and bounded complexes

We investigate reasonably large modules with a tilting-type behaviour, regarded as bounded complexes of projective modules.     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.     

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.     

Constructing tilting modules

We investigate the structure of (infinite dimensional) tilting modules over hereditary artin algebras. For connected algebras of infinite representation type with Grothendieck group of rank , we prove that for each , there is an infinite dimensional tilting module with exactly pairwise non-isomorphic indecomposable finite dimensional direct summands. We also show that any stone is a direct summand in a tilting module. In the final section, we give explicit constructions of infinite dimensional tilting modules over iterated one-point extensions.

     

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.     

Comparisons of recollements and tilting modules

The purpose of this paper is to provide the TTF-theories and investigate the comparisons of recollements(one-sided recollements) both induced by the BB-tilting modules.     

Characteristic tilting modules and Ringel duals

The characteristic tilting modules of quasi-hereditary algebras which are dual extensions of directed monomial algebras are explicitly constructed; and it is shown that the Ringel dual of the dual extension of an arbitrary hereditary algebra has triangular decomposition and bipartite quiver.     

Tensor products of tilting modules

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T ? T′: The main result is that T ? T′ turns out to be an (n + m)-tilting A ? B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.     

On right S-Noetherian rings and S-Noetherian modules

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.     

A note on relative tilting modules

Bazzoni had given a simple characterization of infinitely generated n-tilting modules. Though her method is even inapplicable to classical n-tilting modules over Artin algebras, we show in this note that a similar characterization does hold for (finitely generated) relative n-tilting modules introduced by Auslander and Solberg for Artin algebras, by using a different method. We also present some applications.