next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Normaliz :: The generators of the integral closure of the Rees algebra of a monomial ideal.

The generators of the integral closure of the Rees algebra of a monomial ideal.

We use intclMonIdeal to compute the integral closure of a monomial ideal and of its Rees algebra.

i1 : R=ZZ/37[x_1..x_7];
i2 : I=ideal(x_1..x_6, x_1*x_2*x_3*x_7, x_1*x_2*x_4*x_7, x_1*x_3*x_5*x_7, x_1*x_4*x_6*x_7, x_1*x_5*x_6*x_7, x_2*x_3*x_6*x_7, x_2*x_4*x_5*x_7, x_2*x_5*x_6*x_7,x_3*x_4*x_5*x_7,x_3*x_4*x_6*x_7);

o2 : Ideal of R
i3 : (intcl,rees)=intclMonIdeal I;
i4 : intcl

o4 = ideal (x , x , x , x , x , x )
             6   5   4   3   2   1

              ZZ
o4 : Ideal of --[x , x , x , x , x , x , x , a]
              37  1   2   3   4   5   6   7
i5 : rees

     ZZ
o5 = --[x , x , x a, x , x a, x , x a, x , x a, x , x a, x , x a]
     37  7   6   6    5   5    4   4    3   3    2   2    1   1

                            ZZ
o5 : monomial subalgebra of --[x , x , x , x , x , x , x , a]
                            37  1   2   3   4   5   6   7

The first entry is an ideal, the integral closure of the original ideal, the second one a monomial subalgebra. Each variable in the example appears in a generator of the ideal. Therefore an auxiliary variable a is added to the ring. If there were a free variable in the ring, say x8, then one can give this variable as a second argument to the function, which then is used as auxiliary variable.

i6 : R=ZZ/37[x_1..x_8];
i7 : I=ideal(x_1..x_6, x_1*x_2*x_3*x_7, x_1*x_2*x_4*x_7, x_1*x_3*x_5*x_7, x_1*x_4*x_6*x_7, x_1*x_5*x_6*x_7, x_2*x_3*x_6*x_7, x_2*x_4*x_5*x_7, x_2*x_5*x_6*x_7,x_3*x_4*x_5*x_7,x_3*x_4*x_6*x_7);

o7 : Ideal of R
i8 : (intcl,rees)=intclMonIdeal(I,x_8);
i9 : intcl

o9 = ideal (x , x , x , x , x , x )
             6   5   4   3   2   1

o9 : Ideal of R
i10 : rees

      ZZ
o10 = --[x , x , x x , x , x x , x , x x , x , x x , x , x x , x , x x ]
      37  7   6   6 8   5   5 8   4   4 8   3   3 8   2   2 8   1   1 8

o10 : monomial subalgebra of R