Intersection multiplicity over noncommutative algebras

In this paper, we will prove Serreʹs multiplicity conjectures for large class of connected algebras (not necessarily commutative), using the intersection multiplicity defined by Ext groups instead of Tor groups. Since this new definition of the intersection multiplicity agrees with classical Serreʹs definition for nice commutative rings, the result is a noncommutative version of that of Peskine and Szpiro. The result also applies to noncommutative filtered algebras in certain situations.