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}
};
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o2 : Matrix R <--- R
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i3 : I = minors(2, A) -- a birational map
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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
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i4 : prune Hm1Rees0 I
o4 = 0
o4 : QQ[Z ..Z ]-module
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i5 : A = matrix{ {x^2, x^2 + y^2},
{-y^2, y^2 + z*x},
{0, x^2}
};
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o5 : Matrix R <--- R
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i6 : I = minors(2, A) -- a non birational map
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o6 = ideal (2x y + y + x z, x , -x y )
o6 : Ideal of R
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i7 : Hm1Rees0 I
1
o7 = (QQ[Z ..Z ])
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o7 : QQ[Z ..Z ]-module, free, degrees {2}
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To call the method "Hm1Rees0(I)", the ideal $I$ should be in a single graded polynomial ring.
The object Hm1Rees0 is a method function.