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TestIdeals :: QGorensteinGenerator

QGorensteinGenerator -- find an element representing the Frobenius trace map of a Q-Gorenstein ring

Synopsis

Description

Suppose that R is a ring such that (pe-1)KR is linearly equivalent to zero (this is the case, for example, if R is -Gorenstein with index not divisible by p). Then if we write R = S/I, where S is a polynomial ring, we have that I[pe]:I = uS + I[pe] for some u ∈S. By Fedder’s criterion, this element u represents the generator of the R1/pe-module Hom(R1/pe,R). For example, if I is principal, generated by f, then we may take u = f pe-1.

The function QGorensteinGenerator produces the element u described above. If the user does not specify a positive integer e, it assumes e = 1.

i1 : S = ZZ/3[x,y,z];
i2 : f = x^2*y - z^2;
i3 : I = ideal f;

o3 : Ideal of S
i4 : R = S/I;
i5 : u = QGorensteinGenerator(1, R)

      4 2    2   2    4
o5 = x y  + x y*z  + z

o5 : S
i6 : u % I^3 == f^2 % I^3

o6 = true

If Macaulay2 does not recognize that I[pe]:I / I[pe] is principal, an error is thrown, which will also happen if R is not -Gorenstein of the appropriate index. Note that in the nongraded case Macaulay2 is not guaranteed to find minimal generators of principally generated modules.

Ways to use QGorensteinGenerator :