Reducibility of parameter ideals in low powers of the maximal ideal

A commutative noetherian local ring ( R , m ) (R,\mathfrak {m}) is Gorenstein if and only if every parameter ideal of R R is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer ℓ \ell (depending on R R ) such that R R is Gorenstein if and only if there exists an irreducible parameter ideal contained in m ℓ \mathfrak {m}^\ell . We give upper bounds for ℓ \ell that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.

The nice m-system of parameters for Artinian modules

This paper restates the definition of the nice m-system of parameters for Artinian modules. It also shows its effects on the differences between lengths and multiplicities of certain systems of parameters for Artinian modules: In particular, if is a nice m-system of parameters then the function is a polynomial having very nice form. Moreover, we will prove some properties of the nice m-system of parameters for Artinian modules. Especially, its effect on the annihilation of local homology modules of Artinian module A.

Hilbert polynomials of j-transforms

We study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients.

Mora’s holy graal: Algorithms for computing in localizations at prime ideals

This paper discusses a computational treatment of the localization [Formula: see text] of an affine coordinate ring [Formula: see text] at a prime ideal [Formula: see text] and its associated graded algebra [Formula: see text] with the means of computer algebra. Building on Mora’s paper [T. Mora, La queste del Saint [Formula: see text]: A computational approach to local algebra, Discrete Appl. Math. 33 (1991) 161–190], we present shorter proofs on two of the central statements and expand on the applications touched by Mora: resolutions of ideals, systems of parameters and Hilbert polynomials, as well as dimension and regularity of [Formula: see text]. All algorithms are implemented in the library graal.lib for the computer algebra system Singular.