4 research outputs found
Three results for tau-rigid modules
-rigid modules are essential in the -tilting theory introduced by
Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for
Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms
of -rigid modules. We show that every indecomposable module over iterated
tilted algebras of Dynkin type is -rigid. Finally, we give a
-tilting theorem on homological dimension which is an analog to that of
classical tilting modules.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic
Lattice theory of torsion classes
The aim of this paper is to establish a lattice theoretical framework to
study the partially ordered set of torsion
classes over a finite-dimensional algebra . We show that
is a complete lattice which enjoys very strong
properties, as bialgebraicity and complete semidistributivity. Thus its Hasse
quiver carries the important part of its structure, and we introduce the brick
labelling of its Hasse quiver and use it to study lattice congruences of
. In particular, we give a
representation-theoretical interpretation of the so-called forcing order, and
we prove that is completely congruence
uniform. When is a two-sided ideal of , is a lattice quotient of which is
called an algebraic quotient, and the corresponding lattice congruence is
called an algebraic congruence. The second part of this paper consists in
studying algebraic congruences. We characterize the arrows of the Hasse quiver
of that are contracted by an algebraic
congruence in terms of the brick labelling. In the third part, we study in
detail the case of preprojective algebras , for which
is the Weyl group endowed with the weak
order. In particular, we give a new, more representation theoretical proof of
the isomorphism between and the Cambrian
lattice when is a Dynkin quiver. We also prove that, in type , the
algebraic quotients of are exactly its
Hasse-regular lattice quotients.Comment: 65 pages. Many improvements compared to the first version (in
particular, more discussion about complete congruence uniform lattices