Koszul Algebras of two generators and a Np property over a ruled surface

Abstract

This dissertation will discuss Koszul algebras and Koszul type Np property. It is a basic problem of homological algebra to compute cohomology algebras of various augmented algebras. One of main purpose is to find the conditions that make algebra Koszul and to find some conditions on Hilbert series of the Koszul algebras. The other purpose is to find conditions which make a line bundle satisfy Np on a projective variety over an algebraically closed field. First consider quadratic algebras with two generator. We classify the quadratic algebras with two generator and investigate conditions under which these quadratic algebras are Koszul algebras. It turns out that one can formulate these conditions in terms of the dimensions of homogeneous degree two and three parts of the algebras. Whether an algebra is a Poincar´e-Birkhoff-Witt algebra or not depends not only on the presentation of the algebra, but also on the field. Second consider the Koszul-type concepts of the syzygies of varieties. Mark L. Green invented the Np property of line bundles on a projective variety. We show that line bundles L and Lk+p satisfy the property Np if L is a globally generated ample line bundle on a projective variety such that H1(X, Ls) = 0 for all s 1. We also sharpen a result of Park's work on a smooth complex projective surface.