Associative Algebras / Algèbres associatives
(I. Assem and F. Huard, Organizers)
- I. ASSEM, Université de Sherbrooke
This is a preliminary report on a joint work with Flavio U. Coelho. We
define a class of algebras by homological properties, which imply in
particular that, over such an algebra, every indecomposable module,
except for at most finitely many isomorphism classes, are of projective
dimension one or of injective dimension one. We give examples of such
algebras, study their properties, give several characterisations as
well as a complete description of their Auslander-Reiten quivers and,
finally, show that the infinite radical of their module categories are
nilpotent of nilpotency index between 3 and 5.
- J. BILODEAU, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Hochschild cohomology of stable Auslander algebras
Let R be a simple singularity and CM(R) the category of maximal
Cohen-Macaulay R-modules. It is known that CM(R) is of finite
representation type. R. O. Buchweitz showed that in the Kleinian case,
the stable Auslander algebra of CM(R) has periodic Hochschild
cohomology (of period 2). Comparing the Auslander-Reiten quivers, we
explain why the stable Auslander algebra of a simple curve singularity
should also have periodic Hoschschild cohomology.
- J.C. BUSTAMANTE, Université de Sherbrooke, Sherbrooke, Québec J1k 2R1
Hochschild cohomology and simplicial cohomology of algebras
Given a finite dimensional k-algebra A, we compare its Hochschild
cohomology groups HHi(A) with its simplicial cohomology groups
SHi(A,k). In case A is schurian, it has been shown (Martins-de
la Peña) that there is a monomorphism x: SHi(A,k)® HHi(A). We give a sufficient condition for this morphism
to be an ismorphism.
- F. HUARD, Bishop's University, Lennoxville
Unit forms of tree algebras
Let q be a unit form. It is possible to associate to q a graph
Gq with two types of edges (full and dotted). We consider unit
forms q such that the full edges of Gq form a connected tree. We
give necessary and sufficient conditions for Gq to be the underlying
graph of a bound quiver tree algebra A=kQ/I.
- M. KLEINER, Syracuse University, Syracuse, New York 13244, USA
The graded preprojective algebra of a quiver
We introduce a new grading on the preprojective algebra of an arbitrary
locally finite quiver. Viewing the algebra as a left module over the
path algebra, we use the grading and the geometry of paths to construct
explicitly a canonical collection of its submodules and of exact
sequences involving these submodules. If a vertex of the quiver is a
source, the submodules behave nicely with respect to the reflection
functor associated to the vertex. If the quiver is finite and without
oriented cycles, the canonical submodules are the nonisomorphic
indecomposable preprojective modules and the canonical exact sequences
are the almost split sequences with preprojective terms. Our proof is
inspired by the argument of Gelfand and Ponomarev in the case when the
finite quiver is a tree.
- MARCELO LANZILOTTA, Universidad de la República, Uruguay
About weakly shod algebras
(Joint work with F. Coelho, IME-USP, Brasil)
In this talk we shall begin with a general review of weakly shod
algebras, which are defined by the following property: there is a
positive integer M such that any path of nonzero morphisms from an
indecomposable injective module to an indecomposable projective module
has length at most M. The familly of weakly shod algebras contains
the classes of tilted algebras, quasitilted algebras and shod
algebras. We shall continue talking about some aspects of the ordinary
quiver of a weakly shod algebra. Looking forward a classification of
such quivers we get some necessary conditions.
- J. LÉVESQUE, Universit'e de Sherbrook, Sherbrooke, Québec J1K 2R1
Module inclinant sur un produit fibré
À partir de deux K-algèbres A et B (où K est un corps
algébriquement clos) qui ont une algèbre quotient C en commun, on
peut considérer l'algèbre R qui est le produit fibré des
surjections canoniques A ® C et B ® C. Il est possible, de
cette façon, de construire une grande variété d'algèbres, et
j'étudie celles-ci sous l'angle de l'inclinaison : on voit comment
construire un module inclinant sur R à partir d'un module inclinant
sur C qui est pré-inclinant sur A et B.
- SHIPING LIU, Universite de Shebrooke, Sherbrooke, Quebec J1K 2R1
Mesh algebras without outer derivations
(joint work with Ragnar-Olaf Buchweitz)
The purpose of this talk is to give a characterisation of mesh algebras
over a commutative ring which admit no outer derivation. To be more
precise, we shall present the following result.
Theorem. Let R(G) be the mesh algebra of a finite translation quiver Gover a commutative ring with an identity R. The following conditions are equivalent are equivalent:
(1) HH1R(G) vanishes.
(2) Gis simply connected.
(3) HH1R(D) vanishes for every convex translation subquiver Dof G.
Note that although vanishing of HH1(A) of an algebra A does not in
general imply the triangularity of A, it is still interesting to
study for which classes of algebras, one has an affirmative answer. Our
above result shows that this is the case for mesh algebras.
- A. MARTSINKOVSKY, Northeastern University, 360 Huntington Ave. Boston,
Massachusetts 02115, USA
Noncommutative sheaf cohomology over Koszul quiver algebras
This is joint work with R. Martinez Villa. We show that noncommutative
sheaf cohomology of Koszul modules over a Koszul quiver algebra is
functorially isomorphic, under Koszul duality, to Vogel cohomology of
the the Koszul-dual modules over the Koszul-dual algebra. If the
original algebra is of finite global dimension, then this isomorphism
extends to all modules which can be resolved with finitely generated
projectives. If, an addition, the algebra is noetherian, then the above
statement is true for all finitely generated modules. This includes the
classical case of sheaves on projective space. If time permits, I will
mention applications to noncommutative algebraic geometry.
- R. RAPHAEL, Concordia University, Montreal, Quebec
Ring epimorphisms and C(X)
We first give some background from the theory of epimorphisms in the
category of rings.
Then we present three projects from on the topic ``epimorphisms and
rings of continuous functions''. The individual themes are as follows:
(i) When is the epimorphic hull of C(X) itself a C(Y)? (with Macoosh
(ii) If G(X) is the least von Neumann regular ring of functions on X
containing C(X), when is G(X) a C(Y)? (with Henriksen and Woods)
(iii) Let X be a topological space and let Y be a subspace of X.
Find necessary and sufficient conditions for the induced ring homomorphism
C(X)® C(Y) to be a ring epimorphism. (with Burgess).
- DAVID SMITH, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1
From simplicial complexes to partially ordered sets and back
This is a report on a joint work with J. C. Bustamante and J. Dionne.
Our objective is to simplify, in certain cases, the computation of the
fundamental groups of simplicial complexes and incidence algebras of
posets. For this purpose, we consider the functor O which
associates to a simplicial complex its non-degenerate simplices ordered
by inclusion, and the functor D which associates to each poset
its chain complex. It is known that the composition O°D preserves the fundamental group. We give combinatorial
characterisations of the images of O and O°D as well as an algorithm allowing to construct a preimage of any
element in the image of O°D.