Budh Nashier

The prime spectra of subalgebras of affine algebras and their localizations

Abstract Prime spectra of affine domains A over a field F are known to have especially nice properties. Here we investigate Spec(R) in the cases where either (a) R is a subalgebra of AM for some maximal ideal M of A, or (b) R is a subalgebra of A, requiring in neither case that A should be an integral domain. In case (a) we show that dim R=tr.deg.FR. In case (b) it is known that R[1/ƒ] is affine over F for some ƒ ϵ R, ƒ ≠ 0, if A is an integral domain; this yields nice properties on appropriate Zariski open subsets of Spec(R), but we show that globally, Spec(R) shares few of the attractive properties of Spec(A).

When is a regular local ring a locality

Let A be a regular local ring and let R be a polynomial extension of A in a finite number of variables. Several attempts have been made to study the structure of projective modules over R and to find the minimal numbers of generators of ideals of R; to wit, [Z, V.4; 3; 61. The methods employed in these references depend very much on the structure of A; the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties. Cohen’s structure theorem tells us precisely what regular local rings are formal power series rings. We present here the following characterization for a regular local ring to be a locality over a field :

On projective modules

LetR be a commutative Noetherian ring with identity. The Hermite dimension ofR is defined to be the least integerr such that every stably freeR-module of rank greater thanr is free. In this paper we study ringsR obtained upon inversion of elements of a given ringA. We show that the Hermite dimension ofR does not depend on the Hermite dimension ofA, it depends on the Krull dimension ofA.