Hilbert Kunz Multiplicity Research Articles

-SIGNATURE UNDER BIRATIONAL MORPHISMS

We study$F$-signature under proper birational morphisms$\unicode[STIX]{x1D70B}:Y\rightarrow X$, showing that$F$-signature strictly increases for small morphisms or if$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$. In certain cases, we can even show that the$F$-signature of$Y$is at least twice as that of $X$. We also provide examples of$F$-signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.

CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

Hilbert–Kunz multiplicities and F-thresholds

Motivated by relations between Hilbert–Samuel multiplicity and F-thresholds, we conjecture an inequality that relates F-thresholds with Hilbert–Kunz multiplicity. In this article, we present several results that support the conjecture. In particular, we prove it for hypersurfaces and we give several consequences of this inequality. In addition, we extend previous results for the Hilbert–Samuel multiplicity.

Hilbert–Kunz density function and asymptotic Hilbert–Kunz multiplicity for projective toric varieties

For a toric pair (X,D), where X is a projective toric variety of dimension d−1≥1 and D is a very ample T-Cartier divisor, we show that the Hilbert–Kunz density function HKd(X,D)(λ) is the d−1 dimensional volume of P‾D∩{z=λ}, where P‾D⊂Rd is a compact d-dimensional set (which is a finite union of convex polytopes).We also show that, for k≥1, the function HKd(X,kD) can be replaced by another compactly supported continuous function φkD which is ‘linear in k’. This gives the formula for the associated coordinate ring (R,m):limk→∞⁡eHK(R,mk)−e0(R,mk)/d!kd−1=e0(R,m)(d−1)!∫0∞φD(λ)dλ, where φD (see Proposition 1.2) is solely determined by the shape of the polytope PD, associated to the toric pair (X,D). Moreover φD is a multiplicative function for Segre products.This yields explicit computation of φD (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope PD, one can explicitly compute the limit for two dimensional toric pairs and their Segre products.We further show that (Theorem 6.3) the renormalized limit takes the minimum value if and only if the polytope PDtiles the spaceMR=Rd−1 (with the lattice M=Zd−1). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.

Equimultiplicity in Hilbert–Kunz theory

This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A number of applications follows, in particular we show that Hilbert-Kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed.

Interpolating between Hilbert–Samuel and Hilbert–Kunz multiplicity

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 have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real space.

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

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.

Local Okounkov bodies and limits in prime characteristic

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert–Kunz multiplicity, and F-signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton–Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert–Kunz multiplicity and F-signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn–Minkowski theorem for Hilbert–Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.

Uniform bounds in F-finite rings and lower semi-continuity of the F-signature

This paper establishes uniform bounds in characteristic p 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. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.

A Function Determined by a Hypersurface of Positive Characteristic

Let $R=k [[X_1, \dots ,X_{n+1}]]$ be a formal power series ring over a perfect field $k$ of characteristic $p>0$, and let $\mathfrak{m} = (X_1 , \dots , X_{n+1})$ be the maximal ideal of $R$. Suppose $0\neq f \in\mathfrak{m}$. In this paper, we introduce a function $\xi_{f}(x)$ associated with a hypersurface $R/(f)$ defined on the closed interval $[0,1]$ in $\mathbb{R}$. The Hilbert-Kunz multiplicity and the F-signature of $R/(f)$ appear as the values of our function $\xi_{f}(x)$ on the interval's endpoints. The F-signature of the pair, denoted by $s(R,f^{t})$, was defined by Blickle, Schwede and Tucker. Our function $\xi_{f}(x)$ is integrable, and the integral $\int_{t}^{1}\xi_{f}(x)dx$ is just $s(R,f^{t})$ for any $t\in[0,1]$.

Asymptotic Hilbert–Kunz multiplicity

For a pair (M,I), where M is finitely generated graded module over a standard graded ring R of dimension d≥2, and I is a graded ideal with ℓ(R/I)<∞ and generated by elements of the same degree, we prove that limq→∞⁡e1(M,I[q])/qd exists, where e1(M,I[q]) denotes the first coefficient of the Hilbert–Samuel polynomial of (M,I[q]).We use this to get an expression for limk→∞⁡[eHK(M,Ik)−e0(M,Ik)/d!]/kd−1, where eHK denotes the Hilbert–Kunz multiplicity. In particular, if dim⁡ M=d then we deduce that the difference eHK(M,Ik)−e(M,Ik)/d! grows at least as a fixed positive multiple of kd−1 as k→∞.This is proved using ‘renormalized’ HK density functions.

Frobenius Betti numbers and modules of finite projective dimension

Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods we give conditions for the non-existence of syzygies of finite length.

Hilbert-Kunz density function and Hilbert-Kunz multiplicity

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 $k\to \infty$.

Nonfiniteness of Hilbert–Kunz functions

Here we answer a question of C. Huneke, by giving an example of a family (parametrized by \({\mathbb {A}}^{1}_{\bar {\mathbb {F}}_{p}}\)) of one-dimensional reduced Cohen–Macaulay rings such that every member has the same Hilbert–Kunz multiplicity but there are infinitely many Hilbert–Kunz functions in the family.

GENERALIZED HILBERT–KUNZ FUNCTION IN GRADED DIMENSION 2

We prove that the generalized Hilbert–Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$, with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$. Moreover, we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow +\infty$ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.

Graph connectivity and binomial edge ideals

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.

Some extensions of Hilbert–Kunz multiplicity

Let R be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do not assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert–Kunz multiplicity.

An observation on generalized Hilbert-Kunz functions

We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.

Upper semi-continuity of the Hilbert–Kunz multiplicity

We prove that the Hilbert–Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

Hilbert–Kunz Multiplicities in Families of Certain Two-Dimensional, ℕ-Graded Rings

We generalize a theorem of Monsky to compute the Hilbert–Kunz multiplicity of ℕ-graded rings k[X, Y, Z]/(F), where F is a quasi-homogeneous, regular trinomial. This result will be used to study a limit of the Hilbert–Kunz multiplicity in families of such rings, where one exponent of F is dominating the others.