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TestIdeals :: frobeniusTraceOnCanonicalModule

frobeniusTraceOnCanonicalModule -- finds the u, which in a polynomail ring, determines the Frobenius trace on canonical module of a quotient of that ring

Synopsis

Description

Given R = S/I, where S is a polynomial ring, there is a map from the canonical module of R back to itself, dual to the Frobenius on R. This map comes from a p inverse linear map on S, restricted appropriately. But every p inverse linear map on S is a premultiple of the Grothendieck dual by some element u. This function finds the u, or at least finds some elements, some linear combination of them is the actual u. We note that Macaulay2 doesn’t always properly identify an ideal as principal (even though it is). Thus we need a list of u’s.

Specifically, you need to pass this two ideals. An ideal that restricts to the canonical ideal, and an ideal that defines the variety. Normally the canonical ideal should be chosen so that it contains the defining ideal (if you do not do this, there may be unexpected behavior).

i1 : S = ZZ/5[x,y,z,w];
i2 : T = ZZ/5[a,b];
i3 : f = map(T, S, {a^3, a^2*b, a*b^2, b^3});

o3 : RingMap T <--- S
i4 : defIdeal = ker f;

o4 : Ideal of S
i5 : R = S/defIdeal;
i6 : J = canonicalIdeal(R);

o6 : Ideal of R
i7 : canIdeal = sub(J, S) + defIdeal;

o7 : Ideal of S
i8 : frobeniusTraceOnCanonicalModule(canIdeal, defIdeal)

       8 4 4
o8 = {y z w }

o8 : List