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symmetricKernel -- compute the defining ideal of the rees algebra for a ma

Synopsis

Description

This function is the workhorse of all/most of the Rees algebra functions. Most users will prefer to use one of the front end commands reesAlgebra.

i1 : R = QQ[a..e]

o1 = R

o1 : PolynomialRing
i2 : J = monomialCurveIdeal(R, {1,2,3,4})

             2                               2                    2
o2 = ideal (d  - c*e, c*d - b*e, b*d - a*e, c  - a*e, b*c - a*d, b  - a*c)

o2 : Ideal of R
i3 : symmetricKernel -- (gens J)

o3 = symmetricKernel

o3 : MethodFunctionWithOptions
Let the ideal returned be I and the ring it lives in (also printed) S, then S/I is isomorphic to the Rees algebra R[Jt]We can get the same information using reesAlgebra(J), see reesAlgebra. 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.

symmetricKernel can also be computed over a quotient ring by either initially defining the ring R as a quotient ring, or by giving the quotient ideal as an optional argument.

i4 : R = QQ[x,y,z]/ideal(x*y^2-z^9)

o4 = R

o4 : QuotientRing
i5 : J = ideal(x,y,z)

o5 = ideal (x, y, z)

o5 : Ideal of R
i6 : symmetricKernel -- (gens J)

o6 = symmetricKernel

o6 : MethodFunctionWithOptions
or
i7 : R = QQ[x,y,z]

o7 = R

o7 : PolynomialRing
i8 : I = ideal(x*y^2-z^9)

              9      2
o8 = ideal(- z  + x*y )

o8 : Ideal of R
i9 : J = ideal(x,y,z)

o9 = ideal (x, y, z)

o9 : Ideal of R
i10 : symmetricKernel -- (gens J)

o10 = symmetricKernel

o10 : MethodFunctionWithOptions
These many ways of working with the 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.

See also

Ways to use symmetricKernel :