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

- 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 associatedGradedRing.

    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

## See also

- associatedGradedRing -- The normal cone of a subscheme

## Ways to use grGr:

- `grGr(Ideal)`
- `grGr(Ideal,Ideal)`

## For the programmer

The object grGr is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiplicitySequence.m2:380:0.