The goal of this paper is to prove a Koszul duality result for En-operads in differential graded modules over a ring. The case of an E1-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an En-operad is identified with the n-Gerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on...

In representation theory, the existence of a Z+-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For ...

For any ideal I of finite projective dimension in a commutative noetherian local ring R, we prove that if the conormal module $$I/I^2$$ has over R/I, then must be generated by regular sequence. This resolves conjecture Vasconcelos. We similar result for first Koszul homology I. When R is localisation polynomial field K characteristic zero, Vasconcelos conjectured R/I reduced complete intersecti...