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Asymptotic behavior of the socle of Frobenius powers. (English) Zbl 1304.13008

Author’s abstract: Let \((R,\mathfrak{m})\) be a local ring of prime characteristic \(p\) and \(q\) a varying power of \(p\). We study the asymptotic behavior of the socle of \(R/I^{[q]}\) where \(I\) is an \(\mathfrak{m}\)-primary ideal of \(R\). In the graded case, we define the notion of diagonal \(F\)-threshold of \(R\) as the limit of the top socle degree of \(R/\mathfrak{m}^{[q]}\) over \(q\) when \(q\to\infty\). Diagonal \(F\)-threshold exists as a positive number (rational number in the latter case) when:
(1) \(R\) is either a complete intersection or \(R\) is \(F\)-pure on the punctured spectrum;
(2) \(R\) is a two dimensional normal domain.
In the latter case, we also discuss its geometric interpretation and apply it to determine the strong semistability of the syzygy bundle of \((x^{d},y^{d},z^{d})\) over the smooth projective curve in \(\mathbb{P}^{2}\) defined by \(x^{n}+y^{n}+z^{n}=0\). The rest of this paper concerns a different question about how the length of the socle of \(R/I^{[q]}\) vary as \(q\) varies. We give explicit calculations of the length of the socle of \(R/\mathfrak{m}^{[q]}\) for a class of hypersurface rings which attain the minimal Hilbert-Kunz function. We finally show, under mild conditions, the growth of such length function and the growth of the second Betti numbers of \(R/\mathfrak{m}^{[q]}\) differ by at most a constant, as \(q\to\infty\).

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14H60 Vector bundles on curves and their moduli
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References:

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