Given a map between free modules f: F →G this function computes the kernel of the induced map of symmetric algebras, Sym(f): Sym(F) →Sym(G) as an ideal in Sym(F). When f defines the universal embedding of Im f, or when G is the ground ring, then (by results in the paper of Huneke-Eisenbud-Ulrich) this is equal to the defining ideal of the Rees algebra of the module Im f.
This function is the workhorse of all/most of the Rees algebra functions in the package. Most users will prefer to use one of the front end commands
reesAlgebra,
reesIdeal and others.
i1 : R = QQ[a..e]
o1 = R
o1 : PolynomialRing
|
i2 : J = monomialCurveIdeal(R, {1,2,3})
2 2
o2 = ideal (c - b*d, b*c - a*d, b - a*c)
o2 : Ideal of R
|
i3 : symmetricKernel (gens J)
o3 = ideal (b*w - c*w + d*w , a*w - b*w + c*w )
0 1 2 0 1 2
o3 : Ideal of R[w , w , w ]
0 1 2
|
Let
I be the ideal returned and let
S be the ring it lives in (also printed), then
S/I is isomorphic to the Rees algebra
R[Jt]. We can get the same information above using
reesIdeal(J), see
reesIdeal.
The following is no longer correct!. Also note that
S is multigraded allowing Macaulay2 to correctly see that the variables of
R now live in degree 0 and the new variables needed to describe
R[Jt] as a
k-algebra are in degree 1.
i4 : S = ring oo;
|
i5 : (monoid S).Options.Degrees
o5 = {{1, 2}, {1, 2}, {1, 2}}
o5 : List
|
symmetricKernel can also be computed over a quotient ring.
i6 : R = QQ[x,y,z]/ideal(x*y^2-z^9)
o6 = R
o6 : QuotientRing
|
i7 : J = ideal(x,y,z)
o7 = ideal (x, y, z)
o7 : Ideal of R
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i8 : symmetricKernel(gens J)
8 2
o8 = ideal (z*w - y*w , z*w - x*w , y*w - x*w , x*y*w - z w , x*w -
1 2 0 2 0 1 1 2 1
------------------------------------------------------------------------
7 2 2 6 3
z w , w w - z w )
2 0 1 2
o8 : Ideal of R[w , w , w ]
0 1 2
|
The many ways of working with this function allows the system to compute both the classic Rees algebra of an ideal over a ring (polynomial or quotient) and to compute the the Rees algebra of a module or ideal using a universal embedding as described in the paper of Eisenbud, Huneke and Ulrich. It also allows different ways of setting up the quotient ring.