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topComponents -- compute top dimensional component of an ideal or module

Description

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

i1 : R = ZZ/32003[a..c];
 
i2 : I = intersect(ideal(a,b), ideal(b,c), ideal(c,a), ideal(a^2,b^3,c^4));

o2 : Ideal of R
 
i3 : topComponents I

o3 = ideal (b*c, a*c, a*b)

o3 : Ideal of R

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

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

See also

Ways to use topComponents:

  • topComponents(Ideal)
  • topComponents(Module)
  • topComponents(Module,ZZ)

For the programmer

The object topComponents is a method function.

The source of this document is in PrimaryDecomposition/doc.m2:750:0.