Macaulay2 » Documentation
Packages » MultiGradedRationalMap :: Hm1Rees0
next | previous | forward | backward | up | index | toc

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

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiGradedRationalMap.m2:841:0.