Let G be a finite graph and K[G] the edge ring of G. Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers f and d with 7 ≤ f ≤ d, there exists a finite graphG on [d] = {1, . . . , d} with depthK[G] = f and with Krull-dimK[G] = d.

In this paper all coordinates in two variables over a Noetherian Q-domain of Krull dimension one are proved to be projectively tame. In order to do this, some results concerning projectively-tameness of polynomials in general are shown. Furthermore, we deduce that all automorphisms in two variables over a Noetherian reduced ring of dimension zero are tame.

It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky’s theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, Bézout domain, valuation domain, Krull domain, π-domain).

Investigators: Austin L. Brown [email protected] Tara Henderson (mentor) [email protected] Kevin Oeffinger [email protected] Kevin R. Krull [email protected] Philip J. Lupo [email protected] Robert Hayashi [email protected] Susan Hayashi [email protected] Wendy Leisenring [email protected] Yutaka Yasui [email protected] Greg Armstrong [email protected]...

Let (R, m) be a local ring of Krull dimension d and I ⊆ R be an ideal with analytic spread d. We show that the j-multiplicity of I is determined by the Rees valuations of I centered on m. We also discuss a multiplicity that is the limsup of a sequence of lengths that grow at an O(nd) rate.

We reduce the study of the Krull dimension d of the deformation ring of the functor of deformations of curves with automorphisms to the study of the tangent space of the deformation functor of a class of matrix representations of the p-part of the decomposition groups at wild ramified points, and we give a method in order to compute d.

This is the third paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. We give generalizations of the well known fact that a nontrivial commutator in a free group is not a proper power to both graphs of free groups over cyclic subgroups and freely decomposable groups. None of the paper is specifically about limit groups.

The paper considers representing bilinear forms by linear operators in the case of a Krull valuation. More precisely, after making some suitable assumptions, we prove that if φ is a non-degenerate bilinear form, then φ is representable by a unique linear operator A whose adjoint operator A∗ exists.