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 )
             1   2   3   4   5   6

              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 , x , x , x , x , x a, x a, x a, x a, x a, x a]
     37  1   2   3   4   5   6   7   1    2    3    4    5    6

                            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 )
             1   2   3   4   5   6

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  1   2   3   4   5   6   7   1 8   2 8   3 8   4 8   5 8   6 8

o10 : monomial subalgebra of R