Big height of an ideal: the largest height of an associated prime. The algorithm is based on the following result by Eisenbud-Huneke-Vasconcelos, in their 1993 Inventiones Mathematicae paper:
• 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.
i1 : R = QQ[x,y,z,a,b] o1 = R o1 : PolynomialRing |
i2 : J = intersect(ideal(x,y,z),ideal(a,b)) o2 = ideal (z*b, y*b, x*b, z*a, y*a, x*a) o2 : Ideal of R |
i3 : bigHeight(J) o3 = 3 |
bigHeight works faster than assPrimesHeight