On Progenitively Koszul Commutative Rings

Abstract

This paper introduces and identifies classes of progenitively Koszul rings. A progenitively Koszul ring is a commutative ring that admits a Koszul complex A=K(R) that is formal, and such that the homology algebra H(A) generated by A is a Koszul algebra. Local complete intersections, which yield exterior algebras (an example of a Koszul algebra) for their homology algebras, serve as a prototype. It is shown that the local complete intersections occupy only a small portion of the class of progenitively Koszul rings.

The material in Chapter 1 will cover basic definitions and facts regarding free resolutions differential graded algebras, homology, lattices, Koszul algebras, and PBW basis constructions. The results of Chapter 2 consist of applications of tensor products of differential graded algebras, tensors of Koszul complexes, and tensors of formal algebras to establish that the tensor product of progenitively Koszul rings is itself a progenitively Koszul ring. Chapter 3 will classify several homology algebras based upon earlier work of Luchezar Avramov, Andrew Kustin, and Matthew Miller. Many of these rings are shown to be progenitively Koszul, and are not restriced to local complete intersections. Applying the resuls of Chapter 2 shows that any arbitrary tensor of such rings will also yield a progenitively Koszul ring.