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Register a new userGeneralized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex
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Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${mathcal R}(mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function $HK(s)=ell({mathcal R}(mathfrak n)/(mathfrak n, mathfrak n t)^{[s]})$ is given by a polynomial for all large $s.$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).$
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For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $ell(R/I) < infty$, we introduce a new invariant $HKd(M, I)$ called the {em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {bf m}^k) - e(R, {bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $kto infty$.
We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a $beta$-density function $g_{R, {bf m}}:[0,infty)longrightarrow {mathbb R}$, where $(R, {bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$int_0^{infty}g_{R, {bf m}}(x)dx = beta(R, {bf m}),$$ where $beta(R, {bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {bf m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function $g_{R, {bf m}}:[0, infty)longrightarrow {mathbb R}$ is compactly supported and is continuous except at finitely many points, (2) the function $g_{R, {bf m}}$ is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.
We prove the existence of HK density function for a pair $(R, I)$, where $R$ is a ${mathbb N}$-graded domain of finite type over a perfect field and $Isubset R$ is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair $(R, I)$, where, in addition $R$ is standard graded. As one of the consequences we show that if $G$ is a finite group scheme acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density function $f_{R^G, {bf m}_G}$, of the pair $(R^G, {bf m}_G)$, is a piecewise polynomial function of degree $d-1$. We also compute the HK density functions for $(R^G, {bf m}_G)$, where $Gsubset SL_2(k)$ is a finite group acting linearly on the ring $k[X, Y]$.
Here we compute Hilbert-Kunz functions of any nontrivial ruled surface over ${bf P}^1_k$, with respect to all ample line bundles on it.
Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We give a recursive formulation for the lengths of the k[X]-module k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms for some other special length functions. We apply our method to give closed forms for these Hilbert-Kunz functions for cases $m le 2$.
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