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# gradedReesPiece -- gets a certain degree piece of an ideal in an (extended) Rees algebra

- Usage:
    `J = gradedReesPiece(n, I)`
- Inputs:
  - n, an integer, a natural number
  - I, an ideal, an ideal in an (extended Rees algebra)
- Outputs:
  - J, an ideal, an ideal in an the base ring

## Description

Given an ideal $J$ in a Rees ring $T = \oplus T_i$ constructed either with classicalReesAlgebra or extendedReesAlgebra, this will take the $n$th graded piece of $J$ and express it as an ideal in $T_0$.

    i1 : R = QQ[x,y];

    i2 : J = ideal(x,y);

    o2 : Ideal of R

    i3 : S = classicalReesAlgebra(J);

    i4 : trim gradedReesPiece(0, ideal(1_S))

    o4 = ideal 1

    o4 : Ideal of R

    i5 : trim gradedReesPiece(1, ideal(1_S))

    o5 = ideal (y, x)

    o5 : Ideal of R

    i6 : trim gradedReesPiece(3, ideal(1_S))

                 3     2   2    3
    o6 = ideal (y , x*y , x y, x )

    o6 : Ideal of R

    i7 : T = extendedReesAlgebra(J);

    i8 : trim gradedReesPiece(-1, ideal(1_T))

    o8 = ideal 1

    o8 : Ideal of R

    i9 : trim gradedReesPiece(0, ideal(1_T))

    o9 = ideal 1

    o9 : Ideal of R

    i10 : trim gradedReesPiece(2, ideal(1_T))

                  2        2
    o10 = ideal (y , x*y, x )

    o10 : Ideal of R

Note the command basis cannot be used in the extended Rees algebra case as it cannot handle rings with both positive and negative degrees.

## Caveat

This method is peanut-butter-free.

## See also

- classicalReesAlgebra -- computes the flattened Rees algebra
- extendedReesAlgebra -- computes the flattened extended Rees algebra
- basis -- basis or generating set of all or part of a ring, ideal or module

## Ways to use gradedReesPiece:

- `gradedReesPiece(ZZ,Ideal)`

## For the programmer

The object gradedReesPiece is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NonPrincipalTestIdeals.m2:850:0.