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assPrimesHeight -- The heights of all associated primes



The algorithm is based on the following result by Eisenbud-Huneke-Vasconcelos, in their 1993 Inventiones Mathematicae paper:

$\bullet$ codim $Ext^d(M,R) \geq d$ for all $d$

$\bullet$ 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 $P$

$\bullet$ If codim $Ext^d(M,R) = d$, 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 : assPrimesHeight(J)

o3 = {2, 3}

o3 : List


bigHeight works faster than using assPrimesHeight and then taking the maximum

See also

Ways to use assPrimesHeight :

For the programmer

The object assPrimesHeight is a method function.