next | previous | forward | backward | up | top | index | toc | Macaulay2 web site

symmetricKernel -- Compute the Rees ring of the image of a matrix

Synopsis

Description

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
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.

See also

Ways to use symmetricKernel :