In representation theory, it is basic to study modules whose endomorphism algebras have finite global dimension. They appear naturally in many situations, e.g. Auslander correspondence and representation dimension, Dlab-Ringel’s approach to quasi-hereditary algebras of Cline-Parshall-Scott, Rouquier’s dimensions of triangulated categories, and cluster tilting in higher-dimensional Auslander-Reiten theory. Recently such modules are called non-commutative resolutions, and studied in commutative ring theory and algebraic geometry after Van den Bergh’s work in birational geometry. In this talk, I will show some of typical examples of non-commutative resolutions, including rings with Krull-dimension at most one, certain hypersurface singularities and Stanley-Reisner rings.
Part of this talk is a joint project with H. Dao, S. Iyengar, R. Takahashi, M. Wemyss and Y. Yoshino in American Institute of Mathematics.
This video was produced by Syracuse University Department of Mathematics as part of ICRA 2016.
