The Legacy of Peter Gabriel – Abstracts

Cleaving diagrams, coverings and the multiplicative bases theorem
Raymundo Bautista

The techniques for giving the structure theorems in the multiplicative bases theorem are based on cleaving diagrams. For this, it is proved that some configurations are forbidden for a quotient of a quiver algebra modulo an admissible ideal, if this algebra is of finite representation type. This is done using cleaving diagrams.

In the talk we consider two examples of the use of cleaving diagrams. We will see in one of the examples its relation with coverings.

Finite representation type is open - then and now
Grzegorz Bobiński

The aim of talk is to present fundamental ideas from a seminal Gabriel's paper "Finite representation type is open". We also plan to discuss how this paper inspired studies of geometric problems in the representation theory of finite dimensional algebras in the last 50 years.

On representation finite and minimal representation infinite algebras
Klaus Bongartz

LaTeX in PHP

We discuss the following results for algebras and their modules of finite dimension over an algebraically closed field:

\[ \begin{array}{ll} \text{1.} & \text{There is no gap in the dimensions of the indecomposable } A\text{-modules.} \\ \\ \text{2.} & A \text{ is representation finite iff each indecomposable has dimension at most } \max(2 \dim A, 30). \\ \\ \text{3.} & \text{If } A \text{ is not representation finite, there is a bimodule } {}_{A}M_{k[T]}, \text{ free over } k[T]\\ & \text{of rank } r \leq \max(4 \dim A, 30), \text{ such that } X \mapsto M \otimes X \text{ is a functor from } \\ & \text{finite-dimensional } k[T]\text{-modules to those over } A, \text{ preserving indecomposability} \\ & \text{and reflecting isomorphisms.} \\ \\ \text{4.} & \text{For each } d, \text{ there is a finite set } L \text{ of } \mathbb{Z}\text{-algebras free as modules of rank } d \\ & \text{such that for any algebraically closed field } k, \text{ the algebras } k \otimes A \text{ form a list of} \\ & \text{minimal representation-infinite } k\text{-algebras of dimension } d. \\ \\ \text{5.} & \text{There is a polynomial-time algorithm to decide whether a } d\text{-dimensional algebra } A \\ & \text{as above is representation finite or not.} \end{array} \]

The ray category defined for any distributive algebra in the article on multiplicative bases and its universal covering play a central role in the proofs.

Commutative algebraic groups
Michel Brion

The talk will first present structure results for commutative algebraic groups from the book of Demazure and Gabriel, and related work of Serre and Oort. It will then discuss some recent developments and open questions.

Gabriel-Zisman homology and homotopy colimits for diagrams in chain complexes
Birgit Richter

In Appendix II.3 of "Calculus of Fractions and homotopy theory" Gabriel and Zisman define the notion of the homology of a small category with coefficients in a functor and relate it to the derived functors for colimits. This was heavily used by Quillen in his work on algebraic K-theory. Later, many authors referred back to Quillen and not to the original source. The Gabriel-Zisman approach was applied and generalized in recent work of Gálvez-Carrillo, Neumann and Tonks and plays an important role in the notion of homotopy colimis for diagrams of chain complexes. I'll sketch the development and present some sample applications.