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stablePrimes -- compute the set of stable primes of a monomial ideal

Synopsis

Description

By a classical result of Brodmann, $\text{Ass}(I^k)=\text{Ass}(I^{k+1})$ for all $k\gg0$. We denote the common set $\text{Ass}(I^k)$ for $k\gg0$ by $\text{Ass}^\infty(I)$. A prime ideal $\mathfrak{p}\subset S$ such that $\mathfrak{p}\in \text{Ass}(I^k)$ for all $k \gg 0$ is called a stable prime of $I$. This method computes $\text{Ass}^\infty(I)$, the set of the stable primes of $I$.

i1 : S = QQ[x_1..x_3];
i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3);

o2 : Ideal of S
i3 : stablePrimes I

o3 = {ideal (x , x ), ideal (x , x ), ideal (x , x ), ideal (x , x , x )}
              2   1           3   1           3   2           1   2   3

o3 : List
i4 : S = QQ[a..e];
i5 : I = ideal(a*d,a*e,b*d,c*d,c*e);

o5 : Ideal of S
i6 : stablePrimes I

o6 = {ideal (c, b, a), ideal (d, c, a), ideal (e, d)}

o6 : List

See also

Ways to use stablePrimes :

For the programmer

The object stablePrimes is a method function.