# 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:
• , 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 .