REDCOM  - Second Workshop


Ingo Blechschmidt, Universität Augsburg
Without loss of generality, any reduced ring is a field

Abstract: We present a reduction method which allows us, in a certain precise  technical sense, to pretend that any reduced ring is a field. This  technique has been used to give an almost trivial and even constructive proof of Grothendieck's generic freeness lemma, an important theorem in algebraic geometry, which substantially improved on the longer, somewhat convoluted and unconstructive previously-known proofs. The technique is driven by logical methods, imparting new logical depth to the established spectrum construction in algebraic geometry.

Francesca Fedele, University of Padova
Ext-projectives in subcategories of triangulated categories

Abstract: Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext^1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b—>c in C, then the triangle x—>b—>c—> extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C’ of T and see how this operation is related to the classic concept of mutation of C with respect to a rigid subcategory.

Gloria Tabarelli, University of Verona
Graphs with large palette index

Abstract: Given an edge-coloring of a graph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. We define the palette index of a graph as the minimum number of distinct palettes, taken over all edge-colorings, occurring among the vertices of the graph. Several results about the palette index of some specific classes of graphs are known. In this talk we propose a different approach that leads to new and more general results on the palette index. Our main theorem gives a sufficient condition for a graph to have palette index larger than its minimum degree. This result allows us to construct two families of graphs with large palette index. In the talk we present, for every odd r, a family of r-regular graphs with palette index reaching the maximum admissible value and the first known family of simple graphs whose palette index grows quadratically with respect to their maximum degree.