Hm1Rees0 -- computes the module [Hm^1(Rees(I))]_0

Synopsis

Usage:

Hm1Rees0(I)
Inputs:
- I, an ideal, an ideal in a single graded polynomial ring
Outputs:
- a module, the module $[H_m^1(Rees(I))]_0$

Description

Let $R$ be the polynomial ring $R=k[x_0,...,x_n]$ and $\mathbf{m}$ be the maximal irrelevant ideal $\mathbf{m}=(x_0,...,x_n)$. Let $I \subset R$ be the ideal $I=(f_0,...,f_m)$ where $deg(f_i)=d$. The Rees algebra $\mathcal{R}(I)$ is a bigraded algebra which can be given as a quotient of the polynomial ring $\mathcal{A}=R[y_0,...,y_m]$. We denote by $S$ the polynomial ring $S=k[y_0,...,y_m]$.

The local cohomology module $H_{m}^1(\mathcal{R}(I))$ with respect to the maximal irrelevant ideal $\mathbf{m}$ is actually a bigraded $\mathcal{A}$-module. We denote by $[H_m^1(Rees(I))]_0$ the restriction to degree zero part in the $R$-grading, that is $[H_m^1(Rees(I))]_0=[H_m^1(Rees(I))]_{(0,*)}$. So we have that $[H_m^1(Rees(I))]_0$ is naturally a graded $S$-module.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : A = matrix{ {x, x^6 + y^6 + z*x^5},
                 {-y, y^6 + z*x^3*y^2},
                 {0, x^6 + x*y^4*z}
               };

             3      2
o2 : Matrix R  <-- R

i3 : I = minors(2, A) -- a birational map

             6       6    7    5       4 2    7    2 4      6       5
o3 = ideal (x y + x*y  + y  + x y*z + x y z, x  + x y z, - x y - x*y z)

o3 : Ideal of R

i4 : prune Hm1Rees0 I

o4 = 0

o4 : QQ[Z ..Z ]-module
         1   3

i5 : A = matrix{ {x^2, x^2 + y^2},
                 {-y^2, y^2 + z*x},
                 {0, x^2}
               };

             3      2
o5 : Matrix R  <-- R

i6 : I = minors(2, A) -- a non birational map

              2 2    4    3    4    2 2
o6 = ideal (2x y  + y  + x z, x , -x y )

o6 : Ideal of R

i7 : Hm1Rees0 I

                 1
o7 = (QQ[Z ..Z ])
          1   3

o7 : QQ[Z ..Z ]-module, free, degrees {2}
         1   3

Caveat

To call the method "Hm1Rees0(I)", the ideal $I$ should be in a single graded polynomial ring.

Ways to use Hm1Rees0 :

Hm1Rees0(Ideal)

For the programmer

The object Hm1Rees0 is a method function.