Asymptotic behavior of the socle of Frobenius powers

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 $q\to\infty$.