Let R be the polynomial ring R=k[x0,...,xn] and m be the maximal irrelevant ideal m=(x0,...,xn). Let I ⊂R be the ideal I=(f0,...,fm) where deg(fi)=d. The Rees algebra R(I) is a bigraded algebra which can be given as a quotient of the polynomial ring A=R[y0,...,ym]. We denote by S the polynomial ring S=k[y0,...,ym].
The local cohomology module Hm1(R(I)) with respect to the maximal irrelevant ideal m is actually a bigraded A-module. We denote by [Hm1(Rees(I))]0 the restriction to degree zero part in the R-grading, that is [Hm1(Rees(I))]0=[Hm1(Rees(I))](0,*). So we have that [Hm1(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 , Z ]-module 1 2 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 , Z ]) 1 2 3 o7 : QQ[Z , Z , Z ]-module, free, degrees {2} 1 2 3 |
To call the method "Hm1Rees0(I)", the ideal I should be in a single graded polynomial ring.