# grGr -- the bigraded ring Gr_m(Gr_I(R))

## Synopsis

• Usage:
grGr(I)
• Inputs:
• m, an ideal, (assumed to be the irrelevant ideal of R, if not specified)
• I, an ideal,
• Outputs:
• a ring, the bigraded ring Gr_m(Gr_I(R))

## Description

Given a (graded) ideal I in a (graded-)local ring (R,m), this function computes the bi-graded ring Gr_m(Gr_I(R)), presented as a quotient of a bigraded polynomial ring with variables names u and v. After being computed once, this ring is stored in the cache of I. This function is based on the method normalCone.

 i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : I = ideal"x2,xy" 2 o2 = ideal (x , x*y) o2 : Ideal of R i3 : A = grGr I o3 = A o3 : QuotientRing i4 : describe A QQ[u ..v ] 0 1 o4 = ----------------------- 2 (u v - u v , u , u u ) 0 1 1 0 1 0 1 i5 : hilbertSeries A 2 3 2 1 - 2T - T T + T + T T 0 0 1 0 0 1 o5 = -------------------------- 2 2 (1 - T ) (1 - T ) 1 0 o5 : Expression of class Divide