Tuesday, February 12, 7.527
David Pauksztello (Lancaster), Simple-minded systems and reduction.
Module categories have two important types of generators: projective modules
and simple modules. Morita theory describes equivalences of module
categories in terms of images of projective modules. Tilting theory is the
generalisation of Morita theory to derived categories describing
equivalences of derived categories in terms of tilting objects. Tilting,
silting and cluster-tilting objects, can be thought of as
"projective-minded objects". Such objects admit a mutation procedure,
on which Iyama and Yoshino modelled an inductive technique for the
construction of cluster-tilting objects called Iyama-Yoshino reduction.
Similar ideas were employed by Iyama and Yang to provide an analogous
technique for silting objects.
The other kind of category important in representation theory is the stable
module category. For such a category an analogue of Morita theory is missing
because the projective objects become "invisible". Therefore, in this
setting it is natural to study "simple-minded objects". In this talk I will
discuss the notion of simple-minded systems and describe an inductive
technique for constructing simple-minded systems analogous to Iyama-Yoshino
reduction. This will be a report on joint work with Raquel Coelho Simões.