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canonicalIdeal -- produce an ideal isomorphic to the canonical module of a ring



Given a ring $R$, typically a domain, canonicalIdeal(R) produces an ideal isomorphic to the canonical module of $R$. It uses the function embedAsIdeal from Divisor.m2.

i1 : S = QQ[x,y,u,v];
i2 : T = QQ[a,b];
i3 : f = map(T, S, {a^3, a^2*b, a*b^2, b^3});

o3 : RingMap T <--- S
i4 : R = S/(ker f);
i5 : canonicalIdeal(R)

o5 = ideal (v, u)

o5 : Ideal of R

The function canonicalIdeal will not always produce the same ideal, especially in a non-domain.

i6 : R = ZZ/13[x,y,z]/(x*y, x*z, y*z);
i7 : print \ unique apply(10, i -> canonicalIdeal(R));
ideal (y - 5z, x + 4z)
ideal (y - 4z, x + 2z)
ideal (y - 3z, x + z)
ideal (y + 3z, x + 2z)
ideal (y - 4z, x - z)
ideal (y - 4z, x - 4z)
ideal (y + 5z, x - 3z)
ideal (y - 6z, x + 6z)
ideal (y + 6z, x + 2z)
ideal (y + 6z, x + 4z)

The option Attempts is passed to embedAsIdeal, and tells it how many times to try to embed the canonical module as an ideal before giving up.

Ways to use canonicalIdeal :

For the programmer

The object canonicalIdeal is a method function with options.