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whichGm -- Largest Gm satisfied by an ideal



An ideal $I$ in a ring $S$ is said to satisfy the condition $G_m$ if, for every prime ideal $P$ of codimension $0<k<m$, the ideal $I_P$ in $S_P$ can be generated by at most $k$ elements.

The command whichGm I returns the largest $m$ such that $I$ satisfies $G_m$, or infinity if $I$ satisfies $G_m$ for every $m$.

This condition arises frequently in work of Vasconcelos and Ulrich and their schools on Rees algebras and powers of ideals. See for example Morey, Susan; Ulrich, Bernd: Rees algebras of ideals with low codimension. Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653–3661.

i1 : kk=ZZ/101;
i2 : S=kk[a..c];
i3 : m=ideal vars S

o3 = ideal (a, b, c)

o3 : Ideal of S
i4 : i=(ideal"a,b")*m+ideal"c3"

             2                  2        3
o4 = ideal (a , a*b, a*c, a*b, b , b*c, c )

o4 : Ideal of S
i5 : whichGm i

o5 = 3

See also

Ways to use whichGm :

For the programmer

The object whichGm is a method function.