next | previous | forward | backward | up | top | index | toc | home

topComponents -- compute top dimensional component

Description

The method used is that of Eisenbud-Huneke-Vasconcelos, in their 1993 Inventiones Mathematicae paper.

If M is a module in a polynomial ring R, then the implementations of topComponents and removeLowestDimension are based on the following observations:

  • codim Extd(M,R) ≥d for all d
  • If P is an associated prime of M of codimension d := codim P > codim M, then codim Extd(M,R) = d and the annihilator of Extd(M,R) is contained in P
  • If codim Extd(M,R) = d, then there really is an associated prime of codimension d.
  • If M is R/I, then topComponents(I) = ann Extc(R/I,R), where c = codim I

See also

Ways to use topComponents :