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

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

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 NonPrincipalTestIdeals.m2:850:0.