We generalize the notion of Hilbert-Kunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit
$$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j= -\infty}^{\infty}λ\left( (\frac{M}{I^{[p^n]}M})_j\right)e^{-iyj/p^n}$$ exists. We prove that the limiting function in the complex variable $y$ is entire and name this function the \textit{Frobenius-Poincaré function}. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal $I$; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of $\frac{R}{I^{[p^n]}} $ as $n$ varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.
We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its …
We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.
In the paper, by the singular Riemann-Roch theorem, it is proved that the class of the e-th Frobenius power can be described using the class of the canonical module for a normal local ring of positive characteristic. As a corollary, we prove that the coefficient of the second term of the Hilbert-Kunz function of a finitely generated A-module M vanishes if A is a Q-Gorenstein ring and M is of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the direct image of the structure sheaf via e-th Fr…
We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an $\mathfrak{m}$-primary ideal exists in a Noetherian local ring $(R,\mathfrak{m})$ with prime characteristic $p>0.$ This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring $R$ of a simplicial complex and an ideal $J$ generated by pure powers of the variables, the generalized Hilbert-Kunz function $\ell(R/(J^{[q]})^k)$ is a polynomial for all $q,k$ and also give an expression of the generalized Hilb…
Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in te…
This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert-Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundl…
We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.
We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.
Let $(R,\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $β^F_i(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m},R)=β^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $β^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $β^F_i(M,R)$…
We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they ha…
In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating 0-1 vectors with as many component as the Frobenius number.
We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article provide a unified approach to the study of these $F$-invariants under localization and as measurements of singularities.
This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was …
We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one irreducible and one homogeneous numerical semigroup at each class in the relation, and from those two elements we reconstruct the whole class. An alternative more efficient method is proposed based on the use of the Kunz-coordinates vectors of the elements in S(F).
The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from …
Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $ϕ:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^ϕ\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^ϕ\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^ϕ\!M <\infty$. This result supplies …
Let $(R, m)$ 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 $m$ -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 $R/m^{[q]}$ 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 al…
A finitely generated additive submonoid $Λ$ of ${\mathbb N}^d$ has the partial order defined by $λ\le λ+ μ$ for $λ, μ\in Λ$. The Frobenius complex is the order complex of an open interval of $Λ$. In this paper, we express the homotopy type of the Frobenius complex of $Λ[ρ/ r]$ in terms of those of $Λ$, where $Λ[ρ/ r]$ is the additive monoid which is added the $r$-th part of $ρ$ to $Λ$. As applications, we determine the homotopy type of the Frobenius complex of some submonoids of $\mathbb N$, for example, the submonoid generated by a finite geometric sequence. We also state an application to th…
A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius complex of A is determined when A is the submonoid of N generated by two relatively prime integers, or the submonoid of N^2 generated by three elements of which any two are linearly independent. As an application, the multigraded Poincare series of the quotient algebra K[x, y, z] / (x^p y^q - z^r) over a field K is determined and proved to be rational.
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $Λ$ has the partial order defined by $λ\le λ+ μ$. The Frobenius complex is the order complex of an open interval of $Λ$ with respect to this partial order. The reduced homology of the Frobenius complex is related to the torsion group of the monoid algebra $K[Λ]$. In this paper, we pay attention to homotopy types of Frobenius complexes, and we express the homotopy types of the Frobenius complexes of $Λ$ in terms of those of $Λ_1$ and $Λ_2$ when $Λ$ is an affine monoid obtained by gluin…
