homogeneous system of parameters


Let k be a field, let R be an m-graded (http://planetmath.org/GradedAlgebra) k-algebra, and let M be a m-graded R-module.

Let (R+) be the homogeneous union of the irrelevant ideal of R.

A partial homogeneous system of parameters for M is a finite sequencePlanetmathPlanetmath of elements θ1,θ2,,θr(R+) such that

dim(M/(i=1rθiM))=dim(M)-r,

where dim gives the Krull dimensionMathworldPlanetmath.

A () homogeneous system of parameters is a partial homogeneous system of parameters such that r=dim(M).

A sequence θ1,,θr(R+) is a M-sequence if for all i with 0i<r, we have that θi+1 is not a zero-divisor in

M/(j=1iθiM).

Finally, view M as being -graded by using any specialization of the above m-grading. Then we define the depth of M to be the length of the longest homogeneousPlanetmathPlanetmathPlanetmathPlanetmath M-sequence.

References

  • 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
Title homogeneous system of parameters
Canonical name HomogeneousSystemOfParameters
Date of creation 2013-03-22 14:14:55
Last modified on 2013-03-22 14:14:55
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Definition
Classification msc 13A02
Defines partial homogeneous system of parameters
Defines complete homogeneous system of parameters
Defines homogeneous M-sequence
Defines depth
Defines depth of a module