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TestIdeals :: isFinjective

isFinjective -- whether a ring is F-injective

Synopsis

Description

This verifies if a ring of finite type over a prime field is F-injective or not. Over a more general field this checks the F-injectivity of the relative Frobenius. We begin with an example of an F-injective ring that is not F-pure (taken from the work of Anurag Singh).

i1 : S = ZZ/3[a,b,c,d,t];
i2 : m = 4;
i3 : n = 3;
i4 : M = matrix{ {a^2 + t^m, b, d}, {c, a^2, b^n-d} };

             2       3
o4 : Matrix S  <--- S
i5 : I = minors(2, M);

o5 : Ideal of S
i6 : R = S/I;
i7 : isFinjective(R)

o7 = true
i8 : isFpure(R)

o8 = false

Next let’s form the cone over P1 ×E where E is an elliptic curve. We begin with a supersingular elliptic curve. This should be F-injective and only if it is F-pure.

i9 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
i10 : EP1 = ZZ/3[x,y,z,s,t]/ideal(x^3+y^2*z-x*z^2); --supersingular elliptic curve
i11 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});

o11 : RingMap EP1 <--- S
i12 : R = S/(ker f);
i13 : isFinjective(R)

o13 = false
i14 : isFpure(R)

o14 = false

Now we do a similar computation this time with an ordinary elliptic curve.

i15 : S = ZZ/3[xs, ys, zs, xt, yt, zt];
i16 : EP1 = ZZ/3[x,y,z,s,t]/ideal(y^2*z-x^3+x*y*z); --ordinary elliptic curve
i17 : f = map(EP1, S, {x*s, y*s, z*s, x*t, y*t, z*t});

o17 : RingMap EP1 <--- S
i18 : R = S/(ker f);
i19 : isFinjective(R)

o19 = true
i20 : isFpure(R)

o20 = true

See also

Ways to use isFinjective :