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...

We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension dl. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined ...

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...