### REDCOM - Second Workshop

Abstracts

** ** ** 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.