281 research outputs found
Quiver Grassmannians and Auslander varieties for wild algebras
Let k be an algebraically closed field and A a finite-dimensional k-algebra.
Given an A-module M, the set G_e(M) of all submodules of M with dimension
vector e is called a quiver Grassmannian. If D,Y are A-modules, then we
consider Hom(D,Y) as a B-module, where B is the opposite of the endomorphism
ring of D, and the Auslander varieties for A are the quiver Grassmannians of
the form G_e Hom(D,Y). Quiver Grassmannians, thus also Auslander varieties are
projective varieties and it is known that every projective variety occurs in
this way. There is a tendency to relate this fact to the wildness of quiver
representations and the aim of this note is to clarify these thoughts: We show
that for an algebra A which is (controlled) wild, any projective variety can be
realized as an Auslander variety, but not necessarily as a quiver Grassmannian.Comment: On the basis of vivid feedback, the references to the literature were
adjusted and correcte
The Self-injective Cluster Tilted Algebras
We are going to determine all the self-injective cluster tilted algebras. All
are of finite representation type and special biserial
On the representation dimension of artin algebras
The representation dimension of an artin algebra as introduced by M.Auslander
in his Queen Mary Notes is the minimal possible global dimension of the
endomorphism ring of a generator-cogenerator. The paper is based on two texts
written in 2008 in connection with a workshop at Bielefeld. The first part
presents a full proof that any torsionless-finite artin algebra has
representation dimension at most 3, and provides a long list of classes of
algebras which are torsionless-finite. In the second part we show that the
representation dimension is adjusted very well to forming tensor products of
algebras. In this way one obtains a wealth of examples of artin algebras with
large representation dimension. In particular, we show: The tensor product of n
representation-infinite path algebras of bipartite quivers has representation
dimension precisely n+2
Gabriel-Roiter inclusions and Auslander-Reiten theory
Let be an artin algebra. The aim of this paper is to outline a
strong relationship between the Gabriel-Roiter inclusions and the
Auslander-Reiten theory. If is a Gabriel-Roiter submodule of then
is shown to be a factor module of an indecomposable module such that there
exists an irreducible monomorphism . We also will prove that the
monomorphisms in a homogeneous tube are Gabriel-Roiter inclusions, provided the
the tube contains a module whose endomorphism ring is a division ring
The Gorenstein projective modules for the Nakayama algebras. I
The aim of this note is to outline the structure of the category of the
Gorenstein projective modules for a Nakayama algebra. We are going to introduce
the resolution quiver of such an algebra. It provides a fast algorithm in order
to obtain the Gorenstein projective modules and to decide whether the algebra
is a Gorenstein algebra or not, and whether it is CM-free or not
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