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MultiGradedRationalMap :: Hm1Rees0

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

Synopsis

Description

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

Caveat

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

Ways to use Hm1Rees0 :