62 research outputs found
On periodicity in bounded projective resolutions
Let A be a self-injective algebra over an algebraically closed field k. We
show that if an A-module M of complexity one has an open orbit in the variety
of d-dimensional A-modules, then M is periodic. As a corollary we see that any
simple A-module of complexity one must be periodic. In the course of the proof,
we also show that modules with open orbits are preserved by stable equivalences
of Morita type between self-injective algebras
Silting and Tilting for Weakly Symmetric Algebras
If A is a finite-dimensional symmetric algebra, then it is well-known that
the only silting complexes in are the tilting
complexes. In this note we investigate to what extent the same can be said for
weakly symmetric algebras. On one hand, we show that this holds for all
tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete
weakly symmetric algebra is also silting-discrete. On the other hand, we also
construct an example of a weakly symmetric algebra with silting complexes that
are not tilting.Comment: 8 page
Constructing minimal P<∞-approximations over left serial algebras
AbstractLet Λ be a finite dimensional left serial algebra over an algebraically closed field K. In this case, Burgess and Zimmermann Huisgen have shown that P<∞, the full subcategory of Λ-mod consisting of the finitely generated Λ-modules of finite projective dimension, is contravariantly finite in Λ-mod. Moreover, they show that the minimal right P<∞-approximations of the simple Λ-modules can be obtained by glueing together uniserials to form modules known as saguaros, and they state without proof an algorithm for constructing these approximations. We will review this algorithm and then demonstrate how a new notion of graphical morphisms between saguaros can be used to prove it
Tilting mutation of weakly symmetric algebras and stable equivalence
We consider tilting mutations of a weakly symmetric algebra at a subset of
simple modules, as recently introduced by T. Aihara. These mutations are
defined as the endomorphism rings of certain tilting complexes of length 1.
Starting from a weakly symmetric algebra A, presented by a quiver with
relations, we give a detailed description of the quiver and relations of the
algebra obtained by mutating at a single loopless vertex of the quiver of A. In
this form the mutation procedure appears similar to, although significantly
more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky
for quivers with potentials. By definition, weakly symmetric algebras connected
by a sequence of tilting mutations are derived equivalent, and hence stably
equivalent. The second aim of this article is to study these stable
equivalences via a result of Okuyama describing the images of the simple
modules. As an application we answer a question of Asashiba on the derived
Picard groups of a class of self-injective algebras of finite representation
type. We conclude by introducing a mutation procedure for maximal systems of
orthogonal bricks in a triangulated category, which is motivated by the effect
that a tilting mutation has on the set of simple modules in the stable
category.Comment: Description and proof of mutated algebra made more rigorous (Prop.
3.1 and 4.2). Okuyama's Lemma incorporated: Theorem 4.1 is now Corollary 5.1,
and proof is omitted. To appear in Algebras and Representation Theor
- …