Welcome to the Session on Homological Methods

and Representation Theory.

CONTENTS

· Front page

· Participants

· Titles and abstracts

· Afternoon schedule

· Morning schedule

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XVI COLOQUIO LATINOAMERICANO

DE ÁLGEBRA

SESSION ON HOMOLOGICAL METHODS AND REPRESENTATION THEORY.

August 1-6, 2005

G. CORTIÑAS M. I. PLATZECK

COORDINATORS

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PARTICIPANTS

· Natalia Vanesa Abad Santos nasantos at uns.edu.ar · Eli Aljadeff aljadeff at techunix.technion.ac.il · Iván Angiono ivanangiono at hotmail.com · Ibrahim Assem ibrahim.assem at usherbrooke.ca · Diana Avella Alaminos avella at matem.unam.mx · Jonathan Ariel Barmak jonathanbarmak at hotmail.com · Raymundo Bautista raymundo at matmor.unam.mx · Juan Carlos Bustamante juan.carlos.bustamante at usherbrooke.ca · Leandro Cagliero cagliero at famaf.unc.edu.ar · Francisco Javier Calderón Moreno calderon at algebra.us.es · José Luis Castiglioni jlc at mate.unlp.edu.ar · Francisco J. Castro-Jiménez castro at us.es · Claudia Chaio algonzal at mdp.edu.ar · Claude Cibils Claude.Cibils at math.univ-montp2.fr · Flávio U. Coelho fucoelho at ime.usp.br · José Antonio De la Peña herlop at matem.unam.mx

· Max Karoubi karoubi at math.jussieu.fr · Eduardo Marcos enmarcos at ime.usp.br · Roberto Martinez-Villa mvilla at matmor.unam.mx · Octavio Mendoza Hernandez omendoza at matem.unam.mx · Gabriel Minian gminian at dm.uba.ar · Agustín Moreno Cañadas amorenoca at unal.edu.co · Adrian Ocneanu adrian at math.psu.edu · María Inés Peña mapena at mdp.edu.ar · Nilda Isabel Pratti nilprat at mdp.edu.ar · Pedro Real Jurado real at us.es · Matias Luis Del Hoyo matiasdelhoyo at hotmail.com · Eugenia Ellis eugenia at cmat.edu.uy · Marco Andrés Farinati mfarinat at dm.uba.ar · Franco Grimoldi fgrimo at gmx.net · Juan Jose Guccione jjgucci at dm.uba.ar · Maria Julia Redondo mredondo at criba.edu.ar

· Manuel Saorín msaorinc at um.es · Marco Schlichting mschlich at math.lsu.edu · David Smith David.Smith at USherbrooke.ca · Maria Jose Souto Salorio mariaj at udc.es · Corina Sáenz ecsv at lya.fciencias.unam.mx · Andrea Solotar asolotar at dm.uba.ar · Gordana Todorov todorov at neu.edu · Cecilia Tosar Escuder cecilia at ime.usp.br · Sonia Elisabet Trepode strepode at mdp.edu.ar · Boris Tsygan tsygan at math.northwestern.edu · Christian Valqui cvalqui at pucp.edu.pe · Melina Verdecchia mverdec at uns.edu.ar · Sarah Witherspoon sjw at math.tamu.edu · Mariusz Wodzicki wodzicki at Math.Berkeley.edu · Dan Zacharia zacharia at syr.edu · Rita Zuazua zuazua at matmor.unam.mx

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TITLES AND ABSTRACTS

• Eli Aljadeff
  Elementary Abelian subgroup induction and its application to infinite groups
  Elementary abelian subgroup induction plays an important role in cohomology and representation theory of finite groups. Roughly speaking, the results say that many cohomological properties are determined by their restriction to the family of elementary abelian subgroups. In general, similar statements are false if one replaces elementary abelian by cyclic subgroups. In the lecture I'll explain some of these results and exhibit a method which allows us to "pass" from finite to infinite groups. Finally we apply this method to a conjecture of Moore.

• Ibrahim Assem
  Categorias de racimos y algebras duplicadas
  Sea A un algebra hereditaria. En esta charla, construimos un dominio fundamental para la categoria de racimos ("cluster category") C(A) dentro de la categoria de modulos del algebra duplicada A' de A. Probamos que existe una biyeccion entre los objetos inclinantes de la categoria C(A) y los modulos inclinantes dentro de la parte de izquierda de A'.

• Diana Avella Alaminos
  Gentle algebras
  The family of gentle algebras is closed under derived equivalence (by a result of Schröer and Zimmermann). The classification of this family according to derived equivalence is an interesting problem. This is well understood in the case of gentle algebras whose associated quiver has one cycle. We study the case of two cycles. We introduce certain numerical invariants of the quiver with relations which turn out to be stable under derived equivalence. We can show that except in a degenerate case our invariants distinguish different derived equivalence classes.
  
  
• Raymundo Bautista
  Complejos sin autoextensiones no triviales.
  (Trabajo conjunto con Efrén Pérez Terrazas) Sea A un álgebra de Artin, veremos relaciones entre la categoría de complejos de A-módulos proyectivos acotados superiormente con homología acotada y la categoria C(m,projA) que consiste de los complejos de proyectivos $X=(X^{i},d_{X}^{i})$, con $X^{i}=0$ para i fuera del intervalo [1,m]. Como una aplicación probamos que si A es un álgebra de dimension finita sobre un campo algebraicamente cerrado el conjunto de complejos X, inescindibles en D(A), la categoría derivada acotada de A, con dimensiones homologicas dadas y sin morfismos no triviales de X a X[1], en D(A), tiene únicamente un número finito de clases de isomorfía.

• Juan Carlos Bustamante
  Cohomología de Hochschild de productos fibrados.
  Este es un trabajo conjunto con J. Dionne y D. Smith, de la Universidad de Sherbrooke, Quebec, Canada. Sean k un cuerpo algebráicamente cerrado, y, para i = 1,2, A_i=k Q_i/I_i dos k-álgebras de dimensión finita, que consideraremos también como k-categorías. Sea además C una sub-categoría plena y convexa común a A_1 y A_2. Tenemos de este modo dos proyecciones de k-álgebras p_i: A_i --> C, y denotaremos por R al productio fibrado de éstas. Bajo ciertas hipótesis adicionales, obtenemos sucesiones exactas largas (de Mayer-Vietoris) que permiten calcular los grupos de cohomología de Hochschild, y simplicial de R en función de los grupos correspondientes para A_1, A_2 y C. Más allá de eso, los morfismos obtenidos inducen morfismos de anillos de cohomología. Veremos como la sucesión para la cohomología de Hochschild difiere de las bien conocidas sucesiones análogas de Happel, y de Michelena - Platzeck.

• Leandro Cagliero
  Cohomología de álgebras de Lie nilpotentes y representaciones: un teorema
  En el año 1961 B. Kostant descubre que el Teorema de Bott, Borel y Weil (que calcula la cohomología de haces de ciertas variedades proyectivas compactas con coeficientes en el haz de secciones locales de un fibrado vectorial) es equivalente a determinar la estructura de la cohomología de álgebras de Lie de cierta familia de álgebras de Lie nilpotentes como módulo sobre su álgebra de derivaciones. El Teorema de Kostant no sólo estableció una conexión importantísima entre la geometría y el álgebra, sino que además es aún hoy uno de los teoremas más destacados en la teoría de cohomología de álgebras de Lie. El objetivo del curso es estudiar este teorema de Kostant y considerar algunas instancias particulares del mismo, haciendo hincapié en la teoría de representaciones de álgebras de Lie semisimples como herramienta para calcular la cohomología de álgebras de Lie.

• Francisco Javier Calderón Moreno
  D-módulos logarítmicos y complejos de intersección
  On a complex analytic manifold, intersection complexes associated with irreducible local systems on a dense open regular subset of a closed analytic subspace are the simple pieces which form any perverse sheaf. The Riemann-Hilbert correspondence allows us to consider the regular holonomic D-modules which correspond to these intersection complexes, that we can call "intersection D-modules". They are the simple pieces which form any regular holonomic D-module. Whereas intersection complexes are topological objects, intersection D-modules are algebraic, specially when we are working over an algebraic manifold. Intersection complexes can be constructed by an important operation: the intermediate direct image. Its description in terms of Verdier duality and usual derived direct images can be algebraically interpreted in the category of holonomic regular D-modules by using the deep properties of the de Rham functor. We need to compute localizations and D-duals. We exploit the logarithmic point of view to compute the intersection D-modules associated with integrable logarithmic connections along a locally quasi-homogeneous free divisor. Quasi-homogeneous plane curves and free hyperplane arrangements are covered by this case. The main ingredients are a duality theorem and the generalization of the comparison theorem for arbitrary integrable logarithmic connections, proved in [2]. In the talk we recall the basic notions and notations and we review our previous results about logarithmic D-modules with respect to free divisors proved in [1] and we give the theorem describing the intersection D-module associated with an integrable logarithmic connections along a locally quasi-homogeneous free divisor. This general results are explicitly written down in the case of quasi-homogeneous plane curves. Finally, we study some explicit examples of integrable logarithmic connections with respect to a cusp $x^2-y^t = 0$. [1] Calderón-Moreno. Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor. Ann. Sci. École Norm. Sup. (4), 32(5) (1999), 701--714. [2] Calderón~Moreno and Narváez~Macarro. Dualité et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres. Ann. Inst. Fourier (Grenoble), 55(1), (2005), 1251-1260.

• José Luis Castiglioni
  Dold-Kan correspondence for N-complexes
  N-complexes were introduced by M. M. Kapranov [arXiv:q-alg/9611005] and extensively studied by M. Dubois-Violette [K-theory 14: 371-404]. Several ways of associate an N-(co)chain complex to a (co)simplicial module were proposed in the latter. However, no analog of the Dold-Kan correspondence can be found in the literature. In this talk we shall establish a correspondence between the category of N-(co)chain complexes and an appropriate subcategory of (co)simplicial modules. Some applications of this result will be developed.

• Francisco J. Castro-Jiménez
  Teorema de dualidad para D-módulos logarítmicos/A duality theorem for logarithmic D-modules
  (Joint work with J.M. Ucha-Enríquez) Following previous work by L. Narváez-Macarro and F.J. Calderón-Moreno we study the so called logarithmic D-modules associated with a complex hypersurface Y in dimension n. The logarithmic D-modules are built from the logarithmic derivations --in the sense of K. Saito, associated to the hypersurface Y. If Y belongs to a certain class we describe explicit free resolutions of such logarithmic D-modules and we prove, in the D-module category, a duality formula. We compare some logarithmic D-modules to the one of meromorphic functions with poles along Y. We also treat the comparison between the complex of meromorphic differential forms and the one of logarithmic differential forms both with respect to the hypersurface Y.

• Claudia Chaio
  On the behaivor of irreducible morphisms
  The notion of irreducible morphism has played an important role in the study of the category mod A of finitely generated modules over an artin algebra A. The connection with the radical of this category is well known. In particular is natural to look at the composition of n irreducible morphisms. An interesting question is when such composition falls into the (n+1)th power of the radical. In this talk we consider the composition of n-irreducible morphisms between indecomposable modules. We give characterizations, in several particular cases, when this composition belongs to the (n+1)th-power of the radical. In particular we consider morphisms that belong to directed components, stable tubes, or components of type ZAinfinity.

• Claude Cibils
  Galois coverings, Morita equivalence and smash extensions of categories.
  We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category. This work is joint with Andrea Solotar.

• Flávio U. Coelho
  On minimal non-tilted algebras
  This is a joint work with J. A. de la Peña and S. Trepode. A non-tilted triangular algebra such that any proper semiconvex subcategory is tilted is called "tilt-critical". In this talk we will descibe the tilt-critical algebras which are quasitilted or one-point extensions of tilted algebras of tame hereditary type. We will also describe inductive procedures to decide whether of not a given algebra is tilted.

• José Antonio De la Peña
  Algebras Derivadas Hereditarias
  Sea $A$ un algebra de dimension finita sobre un campo algebraicamente cerrado $k$. Si $H = kQ$ es un algebra de caminos con $Q$ un carcaj finito sin ciclos orientados, entonces $H$ es un algebra hereditaria. Consideramos algebras $A$ cuya categoria derivada de complejos acotados $D(A)$ es triangularmente equivalente a $D(H)$. Estudiamos algunas de propiedades de la catgoria de modulos de $A$ y de la estructura de $A$. En particular, obtenemos criterios para decidir si un algebra dada es inclinada.

• Matias Luis Del Hoyo
  classifying spaces and groupoid atlases
  (join work with Gabriel Minian) I will start the talk by reviewing the basic theory of groupoid atlases and the relationship with simplicial complexes and K-Theory. We introduce two notions for the classifying space of a groupoid atlas as generalizations of the classifying space of a category and apply this constructions to relate the homology theory of groupoid atlases with the classical homology theory of simplicial complexes and spaces.
  
  
• Marco Andrés Farinati
  Corchetes de Lie en la cohomologia de bialgebras (de Lie y asociativas)
  Recientemente, M.Markl (math.AT/0411456) definio una estructura $L_{\infty}$ en la cohomologia de bialgebras (PROPs). En esta charla, discutire la especiakizacion al caso de bialgebras de Lie, su relacion con el doble de Drinfeld, y algunos calculos de esta estructura de Lie en bialgebras de Lie semisimples. Tambien comentare cuales resultados generales son ciertos mutatis mutandis en bialgebras asociativas.
  
  
• Juan Jose Guccione
  Relative homology of square zero extensions
  Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler that the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. We also give new proofs of two theorems of Goodwillie

• Max Karoubi
  Quasi-commutative cochains in Algebraic Topology
  We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or QDGA). We associate to any space a QDGA and show how we can recover the homotopy type of the space from this algebraic structure. The construction by itself is interesting since it uses the difference calculus (instead of the differential calculus) and a new type of tensor product, called "reduced tensor product".

• Eduardo Marcos
  $\delta$ Koszul Algebras
  This is a talk about a joint work with E. L. Green. Let $A=A_0\oplus A_1\oplus A_1 \oplus \cdots$ be a graded $K$-algebra such that $A_0$ is a finite product of copies of the field $K$, $A$ is generated in degrees $0$ and $1$, and $\dim_K{A_1}<\infty$. We study those graded algebras $A$ with the property that $A_0$, viewed as a graded $A$-module, has a graded projective resolution, $\cdots \to P^t \to \cdots \to P^1\to P^0\to A_0\to 0$, such that each $P^i$ can be generated in a single degree. Our work describes necessary and sufficient conditions for the Ext-algebra of $A$, $\bigoplus_{n\ge 0}\Ext^n_A(A_0,A_0)$, to be finitely generated. We also investigate classes of modules over such algebras and Veronese subrings of the Ext-algebra.

• Roberto Martinez-Villa
  Categorias Graduadas y de Koszul
  Reportaremos un trabajo conjunto con Oeyvind Solberg en el que generalizamos las nociones de algebras graduadas y Koszul a las categorias aditivas. Se aplican estas ideas al estudio de las representaciones de las algebras de dimension finita, obteniendose resultados sobre las componentes de Auslander-Reiten en una linea similar al trabajo sobre categorias herediarias con dualidad de Serre obtenidos por: D. Happel, I. Reien, H. Lenzing y M. van den Bergh.

• Octavio Mendoza Hernandez
  Tilting Categories (joint work with C. Saenz)
  In this talk we introduce the notion of Tilting Category. Our purpose is to get a good understanding of stratifying systems and their relationship with generalized tilting modules and some homological dimensions.

• Gabriel Minian
  Spectra of small categories and Segal's machine
  $\Gamma$-spaces were introduced by Segal in order to construct spectra, and hence cohomology theories of spaces, out of simpler space level data. He showed also that $\Gamma$-spaces arise naturally from $\Gamma$-categories, via the classifying space functor. In this talk I will show how to construct spectra (and so, cohomology theories) of small categories, using a generalization of Segal´s machine. In order to define the right notion of spectra of small categories, we need a countable family of loop and suspension functors, instead of just one loop and suspension (as in the case of topological spaces).

• Agustín Moreno Cañadas
  On certain reduction functors for representations of posets with additional structures.
  During the last three decades, the differentiation technique is being the main investigation tool in the algebraic representation theory of posets (partially ordered sets), both ordinary and with some additional structures (with involution, with an equivalence relation, dyadic, equipped, etc.). The differentiation procedure consists in applying certain reduction functors (algorithms) acting between representation categories and allowing to prove results by induction on dimensions of indecomposable representations. In [1] and [3] there were constructed systems of the reduction functors II,...,V and VII,...,XVII respectively, with help of which, in particular, the criteria of tameness and of finite-growthness for posets with involution [2] and for equipped posets [3] were proved. Though in [1,3] all necessary reduction functors were defined, the substantiation of the differentiation procedure was described largely using the matrix language. Thus, the constructions were adapted well to a description of objects only, without paying much attention to morphisms. In the present work, we select for a more deep categorical investigation several (of the constructed in [1] and [3]) reduction functors and give for them a complete categorical description, establishing, in particular, an equivalence between the quotient categories of the categories of representations of the initial poset and the derived one, modulo some evidently described ideals. 1. A.G. Zavadskij, An algorithm for posets with an equivalence relation, Can. Mat. Soc. Conf. Proc., 11, AMS (1991), 299-322. 2. V.M. Bondarenko, A.G. Zavadskij, Posets with an equivalence relation of tame type and of finite growth, Can. Math. Soc. Conf. Proc., 11 (1991), 67-88. 3. A.G. Zavadskij, Tame equipped posets, Linear Algebra Appl., 365 (2003), 389-465.

• Adrian Ocneanu
  TBA
  
  
  
• Pedro Real Jurado
  Algebraic Topological Techniques in 3D and 4D digital Imagery
  In this talk, we define the notion of an Algebraic Topological Model (AT-Model, for short) of a 4D digital binary-valued image I. To do this, we need first to associate to I in a natural way a simplicial complex K(I). Secondly, an AT-model of I means to add to I a chain homotopy equivalence connecting the chain complex cannonically associated to K(I) and its own homology (assuming a field as ground ring). There are several algorithms for computing AT-models in polynomial time. In this context, some problems concerning topological interrogation, simplification and control are treated.

• Maria Julia Redondo
  Cohomology of the Grothendieck construction
  Joint work with Teimuraz Pirashvili. Given a small category $K$ and a functor $L: K \to CAT$, there is a well defined category $\int_{K} L$ called the Grothendieck construction of $L$. We consider the cohomology of the Grothendieck construction of $L$ with coefficients in a natural system $D$, in the sense of Baues and Wirsching. We show that there exists a spectral sequence abutting to the cohomology of the Grothendieck construction of $L$ in terms of the cohomology of $K$ and of $L(k)$, for $k$ an object in $K$.

• Manuel Saorín
  A classification of split TTF-triples in module and derived categories
  This is the result of joint works with I. Assem and P. Nicol\'as. A TTF-triple in an abelian category is a triple $(\mathcal{C},\mathcal{T},\mathcal{F})$ of full subcategories such that both $(\mathcal{C},\mathcal{T})$ and $(\mathcal{T},\mathcal{F})$ are torsion pairs. They will be called left (resp. right) split when $(\mathcal{C},\mathcal{T})$ (resp. $(\mathcal{T},\mathcal{F})$) is a split torsion pair. In the case when $A$ is a ring and $Mod_A$ is the category of all right $A$-modules, it is well-known that there is a one-to-one correspondence between TTF-triples in $Mod_A$ and idempotent (two-sided) ideals of $A$. It is also known that the correspondence induces another one between centrally split TTF-triples in $Mod_A$ and central idempotentes of $A$. However, although algebrists were aware of the existence of one-sided split TTF-triples which are not split on the other side, a classification of one-sided split TTF-triples in $Mod_A$ was missing. The goal of this talk is to present such a classification. Since in our talk in the last ICRA meeting of P\'atzcuaro we presented a complete classification of left split TTF-triples, but that of right split was still incomplete, we shall concentrate on the classification of these latter ones which has been obtained recently. We will then look at the corresponding problem for derived categories, using t-structures in the 'triangulated world' as substitute of torsion pairs in the 'abelian world'. We will see that, unlike the abelian case, one-sided split TTF-triples in a triangulated category are always centrally split and, in the case of the derived category $D^*(\mathcal{A})$ of an abelian category $\mathcal{A}$, it restricts to a centrally split TTF-triple in $\mathcal{A}$. Therefore TTF-triples in $D^*(Mod_A)$ are in one-to-one correspondence with central idempotents of $A$.

• Marco Schlichting
  Higher Grothendieck-Witt groups of schemes and derived categories
  In the late 1950s, Raoul Bott proved a theorem which is now commonly called ``Bott periodicity''. In the 1970/80s, Max Karoubi proved an algebraic version of this theorem, now called ``Karoubi periodicity''. In its current form, the proof of Karoubi periodicity relies on Bott periodicity, and it is only valid under the additional assumption of ``2 being invertible''. In my talk I will explain a generalization from ``algebraic K-theory'' to ``hermitian K-theory'' of a local-to-global principle of Robert W. Thomason. This generalization does not rely on Bott-periodicity, yet it gives a new proof of Karoubi- and Bott- periodicity. The work to be presented does not explicitly use the assumption of ``2 being invertible''; however, it uses the existence of a topological space, the existence of which is not yet proved in case ``2 is not invertible''.

• David Smith
  Laura algebras and quasi-directed Auslander-Reiten components
  This is a report on a joint work with M. Lanzilotta. First, we define and recall the known characterizations of the laura algebras (which generalize the quasi-tilted algebras) and the quasi-directed AR-components (which generalize the connecting components of tilted algebras). Then we give a "unified characterization" of those two objects and show that they always come in pair. Finally, we give a necessary condition for a convex AR-component to be quasi-directed.

• Maria Jose Souto Salorio
  Ext-proyectivos en categorias suspendidas
  Este trabajo es conjunto con Ibrahim Assem and Sonia Trepode. En el estudiamos propiedades generales relativas al concepto de Ext-proyectivo en subcategorias suspendidas. Recordemos que una subcategoria plena U de una categoria triangulada se dice suspendida si es cerrada bajo extensiones y traslaciones positivas. En particular, estamos interesados en el caso de categorias derivadas hereditarias y subcategorias suspendidas generadas por objetos. Referencias: 1. Auslander, M., Reiten, I. and Smal\o, S.: Representation theory of artin algebras. Cambridg Univ. Press (1995) 443-518. 2. Auslander, M. and Smal\o, S.: Almost split sequences in subcategories. J. Algebra 69 (1981) 426-454, "addendum" J. Algebra 71 (1981) 592-594. 3. Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Note Series. 119 Cambridge University Press (1988). 4. Keller B. and Vossieck D.:Aisles in derived categories. Bull. Soc. Math. Belg. 40, (2) (1988), 239-253. 5. Reiten, I. and van der Bergh, M.: Grothendieck groups and tilting objects, Algebras and Representations Theory 4 (2001) 257-272.

• Corina Sáenz
  Applications of Tilting Categories to Stratifying Systems
  Let $(\theta,\leq)$ be a stratifying system of size $t$ and $F(\theta)$ the category of all the $R$-modules that are filtered by modules in $\theta)$. Assuming that the category $I(\theta)$, of the Ext-injective $R$-modules relative to the category $F(\theta), is coresolving we prove that the category $\F(\theta)$ is a partial tilting category. Moreover, using the theory developed for Tilting Categories, exposed in the talk given by Octavio Mendoza, we prove a theorem that generalizes some of the results obtained by Bin Zhu and S.Caenepeel in [1] and another one of V.Mazorchuk and S.Ovsienko in [2]. [1] Bin Zhu and S.Caenepeel, On Good Filtration Dimension for Standardly Stratified Alegbras. Communications in Algebra, Vol.32, No.4, pp. 1603-1614, 2004. [2] V. Mazorchuk and S.Ovsienko, Finitistic dimension of properly stratified algebras. Advances in Mathematics, Vol. 186, pp.251-265, 2004.

• Gordana Todorov
  Cluster categories and related topics
  Cluster categories were introduced as a natural model for the combinatorics of cluster algebras. They are defined as certain quotients of bounded derived categories of the module categories of finite dimensional hereditary algebras of Dynkin type. Using the cluster category, definition and theorems of semiinvarants were extended to the –mixed” dimension vectors. Relations to the cohomology of nilpotent torsion-free groups will also be discussed.

• Cecilia Tosar Escuder
  Sobre categorías derivadas y álgebras quasi-tilted
  En esta presentación mostraremos algunos resultados sobre categorías derivadas de álgebras de Artin (es decir, k-álgebras de dimensión finita), y en particular sobre los objetos indescomponibles en estas categorías. Para estos resultados utilizaremos propiedades homológicas. Definimos una aplicación sobre los complejos en la categoría derivada de una álgebra de Artin que mide, en cierta forma, la longitud de los complejos, y damos relaciones entre las propiedades homológicas del álgebra y la respectiva aplicación. También determinamos dos subcategorías da la categoría derivada de una álgebra, que resultan ser las categorías derivadas de álgebras quasi-tilted. Las álgebras quasi-tilted fueron caracterizadas por propiedades homológicas y sus categorías derivadas son bien conocidas. Como consecuencia de los resultados que presentaremos, obtenemos nuevas caracterizaciones de las álgebras quasi-tilted.

• Sonia Elisabet Trepode
  Some Characterisations of supported algebras
  This is a joint work with I. Assem, J. Cappa, and M. I. Platzeck. We give several equivalent characterisations of left supported algebras.

• Boris Tsygan
  TBA

• Christian Valqui
  Equivalencia débil de pro-complejos
  Damos primero una breve introducción a pro-categorías. Damos una descripción del funtor derivado Ext del funtor Hom en la categoría de pro-módulos. Esta misma descripción funciona en el caso topológico y allí vemos distintas caracterizaciones de la equivalencia débil. Luego vemos la relación que hay entre equivalencia débil, equivalencia homotópica, existencia de inversos locales y quasi-isomorfismos; en el caso general, el caso constante y el caso de pro-supercomplejos. En particular, el caso constante la equivalencia débil se reduce a equivalencia homotópica, lo cual nos da un teorema de escisión en términos de homotopía. Finalmente vemos que el funtor X^n preserva equivalencias débiles.
  
  
• Sarah Witherspoon
  Graded Hecke Algebras and Hochschild Cohomology
  A crossed product of an algebra with a group of automorphisms encodes the group action in a larger algebra. Drinfel'd's graded Hecke algebras are deformations of crossed products of polynomial rings with finite groups. These have been studied for particular types of groups by many people, such as Lusztig (real reflection groups), Ram-Shepler (complex reflection groups), and Etingof-Ginzburg (symplectic reflection groups). We will discuss these examples and put them in a broader context using Hochschild cohomology. This will lead to generalizations and new examples.

• Mariusz Wodzicki
  TBA
  
• Rita Zuazua
  Sobre los complejos de módulos proyectivos (About the complex of projective modules)
  En esta charla veremos algunas propiedades de los complejos de módulos proyectivos y ejemplos. We will see some properties and examples of complex of projective modules.

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AFTERNOON SCHEDULE

(Includes Course and Plenary talk by session participants)

	MONDAY	TUESDAY	WEDNESDAY		THURSDAY		FRIDAY	SATURDAY
15:00 a 15:50	Plenary Talk J.A. DE LA PEÑA	Course B. TYSGAN	Course B. TYSGAN	Course L. CAGLIERO	Course B. TSYGAN	Course L. CAGLIERO	F R E E A F T E R N O O N
								O. MENDOZA (15.30 a 15.50)
16:00 a 16:40	R. BAUTISTA	M. SAORÍN	F. CALDERÓN