Cohen-Macaulayness in graded rings associated to ideals

Cohen-macaulayness and Computation of Newton Graded Toric Rings

Let H ⊆ Z be a positive semigroup generated by A ⊆ H, and let K[H] be the associated semigroup ring over a field K. We investigate heredity of the Cohen–Macaulay property from K[H] to both its A -Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen–Macaulay property. On the positive side we show that for every H there exist generating sets ...

Graded Cohen-macaulayness for Commutative Rings Graded by Arbitrary Abelian Groups

We consider a commutative ring R graded by an arbitrary abelian group G, and define the grade of a G-homogeneous ideal I on R in terms of vanishing of C̆ech cohomology. By defining the dimension in terms of chains of homogeneous prime ideals and supposing R satisfies a.c.c. on G-homogeneous ideals and has a unique G-homogeneous maximal ideal, we can define graded versions of the depth of R and C...

Cohen-Macaulayness of special fiber rings

Let (R, m) be a Noetherian local ring and let I be an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring F = R/mR of I, where R denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if I is a str...

Graded Rings Associated with Contracted Ideals

The study of the ideals in a regular local ring (R,m) of dimension 2 has a long and important tradition dating back to the fundamental work of Zariski [ZS]. More recent contributions are due to several authors including Cutkosky, Huneke, Lipman, Sally and Tessier among others, see [C1, C2, H, HS, L, LT]. One of the main result in this setting is the unique factorization theorem for complete (i....

On Associated Graded Rings of Normal Ideals