isFRegular(R)
isFRegular(t, f)
isFRegular(tList, fList)
Given a normal $\mathbb{Q}$-Gorenstein ring $R$, the function isFRegular checks whether the ring is strongly $F$-regular. It can also prove that a non-$\mathbb{Q}$-Gorenstein ring is $F$-regular (but cannot show it is not); see below for how to access this functionality.
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The function isFRegular can also test strong $F$-regularity of pairs.
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When checking whether a ring or pair is strongly $F$-regular, the option AtOrigin determines if this is to be checked at the origin or everywhere. The default value for AtOrigin is false, which corresponds to checking $F$-regularity everywhere; setting AtOrigin => true, $F$-regularity is checked only at the origin.
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Here is an example of AtOrigin behavior with a pair.
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The option AssumeDomain (default value false) is used when finding a test element. The option FrobeniusRootStrategy (default value Substitution) is passed to internal frobeniusRoot calls.
When working in a $\mathbb{Q}$-Gorenstein ring $R$, isFRegular looks for a positive integer $N$ such that $N K_R$ is Cartier. The option MaxCartierIndex (default value $10$) controls the maximum value of $N$ to consider in this search. If the smallest such $N$ turns out to be greater than the value passed to MaxCartierIndex, then testIdeal returns an error.
The $\mathbb{Q}$-Gorenstein index can be specified by the user through the option QGorensteinIndex; when this option is used, the search for $N$ is bypassed, and the option MaxCartierIndex ignored.
The function isFRegular can show that rings that are not $\mathbb{Q}$-Gorenstein are $F$-regular (it cannot, however, show that such a ring is not $F$-regular). To do this, set the option QGorensteinIndex => infinity. One may also use the option DepthOfSearch to increase the depth of search.
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The object isFRegular is a method function with options.