Description
TestIdeals is a package for basic computations of $F$-singularities. It is focused on computing test ideals and related objects. It does this via
frobeniusRoot, which computes $I^{[1/p^e]}$, as introduced by Blickle-Mustata-Smith (this is equivalent to the image of an ideal under the Cartier operator in a polynomial ring).
Notable functions:
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testIdeal computes the test ideal of a normal \mathbb{Q}-Gorenstein ring or pair.
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testModule computes the parameter test module of a ring or pair.
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parameterTestIdeal computes the parameter test ideal of a Cohen-Macaulay ring.
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FPureModule computes the stable image of the trace of Frobenius on the canonical module.
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isFRegular checks if a normal \mathbb{Q}-Gorenstein ring or pair is $F$-regular.
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isFPure checks if a ring is $F$-pure.
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isFRational checks if a ring is $F$-rational.
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isFInjective checks if a ring is $F$-injective.
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compatibleIdeals finds the compatibly $F$-split ideals with a (near) $F$-splitting.
Consider, for instance, the test ideal of the cone over an elliptic curve.
i1 : R = ZZ/5[x,y,z]/(z*y^2 - x*(x - z)*(x + z));
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i2 : testIdeal(R)
o2 = ideal (z, y, x)
o2 : Ideal of R
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The following example was studied by Anurag Singh when showing that $F$-regularity does not deform.
i3 : S = ZZ/3[A,B,C,D,T];
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i4 : M = matrix{{A^2 + T^4, B, D}, {C, A^2, B^3 - D}};
2 3
o4 : Matrix S <-- S
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i5 : I = ideal(T) + minors(2, M);
o5 : Ideal of S
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i6 : isFRegular(S/I)
o6 = true
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Acknowledgements:The authors would like to thank David Eisenbud, Daniel Grayson, Anurag Singh, Greg Smith, and Mike Stillman for useful conversations and comments on the development of this package.