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

## Synopsis

• Usage:
QGorensteinGenerator(e, R)
QGorensteinGenerator(R)
• Inputs:
• e, an integer, the degree in which to search
• R, a ring, a $\mathbb{Q}$-Gorenstein ring
• Outputs:
• , representing the Frobenius trace map of R

## Description

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 :

• "QGorensteinGenerator(Ring)"
• "QGorensteinGenerator(ZZ,Ring)"

## For the programmer

The object QGorensteinGenerator is .