David Pauksztello, Lancaster University
Simple-mindedness: reduction and mutation
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’.
‘Simple-minded objects’ are generalisations of simple modules. They satisfy Schur’s lemma and a version of the Jordan-Holder theorem, depending on context. Although the theory of simple-minded objects shows many parallels with that of projective-minded objects, it remains relatively undeveloped and is technically more challenging. It remains important to develop this theory because many natural classes of examples, for instance, stable module categories, have no projective-minded objects but do have simple-minded objects. In this talk, I will explain aspects of the theory of simple-minded objects, including mutation and reduction. This talk will be based on joint work with Nathan Broomhead, Raquel Coelho Simoes, David Ploog and Jon Woolf.