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TestIdeals :: isFrational

isFrational -- whether a ring is F-rational

Synopsis

Description

Determines if a ring is F-rational. If you pass it IsLocal=>true, it will only check if the ring is F-rational at the origin (this can be slower). If you pass it AssumeCM=>true, it will not verify that the ring is Cohen-Macaulay.

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

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

o5 = true
i6 : R = ZZ/7[x,y,z]/ideal(x^3+y^3+z^3);
i7 : isFrational(R)

o7 = false

We conclude with a more interesting example of a ring that is F-rational but not F-regular. This came up in A. K. Singh’s work on deformation of F-regularity.

i8 : S = ZZ/3[a,b,c,d,t];
i9 : m = 4;
i10 : n = 3;
i11 : M = matrix{ {a^2 + t^m, b, d}, {c, a^2, b^n-d} };

              2       3
o11 : Matrix S  <--- S
i12 : I = minors(2, M);

o12 : Ideal of S
i13 : R = S/I;
i14 : isFrational(R)
testModule: Multiple trace map for omega generators (Macaulay2 failed to find the principal generator of a principal ideal).  Using them all.

o14 = true

Warning, this function assumes that Spec R is connected. Like isCohenMacaulay, if you pass it a non-equidimensional F-rational ring (for example, if Spec R has two connected components of different dimensions), this function will return false.

Ways to use isFrational :