The level of a witt ring

We show that the level of the Witt ring W(F) of symmetric bilinear forms over a field F can only take the values 1, 3, 4, or x and that the level of F determines the level of W(F).     

The level of a witt ring

We show that the level of the Witt ring W(F) of symmetric bilinear forms over a field F can only take the values 1, 3, 4, or x and that the level of F determines the level of W(F).     

The spherical spectrum of a graded ring

We introduce a functor Sph, the spherical spectrum, which assigns to a graded ringG a space Sph(G) of homogeneous orderings ofG. It combines ideas of concrete geometry in theN-sphere defined by positively homogeneous polynomial equations and inequalities with the abstract notion of the real spectrum of a ring to give a counterpart for real semialgebraic geometry of the functor Proj.     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

On when a graded ring is graded equivalent to a crossed product

Let be a ring graded by a group . We are concerned with describing those -graded rings that are graded equivalent to -crossed products. We give necessary and sufficient conditions for when a strongly graded ring is graded equivalent to a crossed product, provided that the 1-component is either Azumaya or semiperfect. Our result uses the torsion product theorem of Bass and Guralnick. We also construct various examples of such rings.

     

A strongly graded ring that is

We construct a ring that is strongly graded by the integers such that it is not graded equivalent to a skew group ring. This is in contrast to the finite case and the results of Cohen and Montgomery, in which every strongly graded ring is graded equivalent to a skew group ring ^{2 2}1991 mathematics subject classification: 16D90, 16S35, 16S40, 16S50, 16W50      

The Castelnuovo regularity of the Rees algebra and the associated graded ring

It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of -sequences.