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



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

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 $I^{[p^e]}:I / I^{[p^e]}$ is principal, an error is thrown, which will also happen if $R$ is not $\mathbb{Q}$-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 :

For the programmer

The object QGorensteinGenerator is a method function.