David Pauksztello (Lancaster), Simple-minded systems and reduction.

Abstract:

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.