Speaker: Ragnar-Olaf Buchweitz (Toronto)

Title: The Koszul Complex Blows Up a Point

Abstract: We will report on joint work with Thuy Pham (UTSC). Our key result is that the endomorphism algebra of the syzygy modules in the tautological Koszul complex is of finite global dimension, with its derived category equivalent to that of quasicoherent sheaves on affine space blown up in a point. The algebra in question admits a simple explicit description through a quiver with relations.
As a consequence, the singularity at the vertex of the cone over any Segre-Veronese embedding of projective spaces admits a noncommutative desingularization in that its canonical small desingularization has its derived category equivalent to that of an algebra of finite global dimension, with the equivalence given by an explicitly known tilting bundle. This builds on and partly generalizes work of Beilinson, Bondal, Bridgeland, Orlov and others.