We continue the study of a lattice-ordered ring G(X), associated with the ring C(X). Following [10], X called RG when G(X) = C(Xδ). An RG-space must have a dense set of very weak P-points. It must have a dense set of almost-P-points if Xδ is Lindelöf, or if the continuum hypothesis holds and C(X) has small cardinality. Space which are RG must have finite Krull dimension when taken with respect ...

Let k be a field. We show that a finitely generated simple Goldie k-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and l...

Remark 1.6. In [Mor06], this result is stated without proof. Our goal here is to indicate that the result follows without any of the machinations appearing in §3.2 of [Mor06] (where the result is stated). Ayoub has shown that if the Krull dimension of k is ≥ 2, then the unstable A-connectivity property will fail, thus the strong A-invariance property cannot hold over 2-dimensional rings. Howeve...

A result of P. Samuel ([17] p. 186, Chap.II, Théorème 5) says that in a local noetherian ring (O,M) of Krull dimension d in which the residual field k is infinite, the multiplicity of a M-primary ideal I is equal to the multiplicity of an ideal (x1, . . . , xd) generated by some parameter sequence x1, . . . , xd contained in I. By a theorem of Rees ([16] p.142 Theorem 9.44), this implies that t...

Let R be a local ring, I ⊆ R an ideal, and M and N finite Rmodules. In this paper we provide a number of results concerning the degree of the polynomial giving the lengths of the modules ExtR(N/I nN, M), when such a polynomial exists. Included among these results are a characterization of when this degree equals the Krull dimension of R, a characterization of when the degree of the polynomial a...

Proof. We will calculate the integral closure of A inside K. We begin with a more general discussion. Suppose C is an irreducible, affine curve, that is, C is some irreducible affine variety of dimension 1. Let A be the ring of regular functions on C, so that A is a noetherian domain of Krull dimension 1. Localization preserves the noetherian condition, as well as the domain condition. Thus, th...

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in Cn modulo a twisted action of the maximal torus in SL(n, C). We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst O(n). On the other hand, we show that the associated semigroup of Gelfand–Tsetlin patterns can have an essenti...

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. (See the introduction for definitions.) Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (Xi)i∈I of subsets of L with the property that i∈I Ai = 0 whene...

The previous works in [HM09], we turn out that Gersten’s conjecture is equivalent to its local version. To state local Gersten’s conjecture, we start from recalling the notation from [HM09], [MS10]. In this paper, let us fix a commutative regular local ring A of Krull dimension d. For any ideal of A generated by a regular sequence I, we denote the category of finitely generated A-modules M whos...