Let I denote a homogeneous R_+-primary ideal in a two-dimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I^* if and only if e_{HK}(I) = e_{HK}((I,f)), where e_{HK} denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the well-known Hilbert-Kunz criterion for tight closure in positive characteristic.
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 \emph{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 g…
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…
We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.
We show that the Hilbert-Kunz multiplicity of a $d$-dimensional nonregular complete intersection over the algebraic closure of $F_p$, $p>2$ prime, is bounded by below by the Hilbert-Kunz multiplicity of the hypersurface $\sum _{i=0}^{d} x_i^2=0$, answering positively a conjecture of Watanabe and Yoshida in the case of complete intersections.
In arXiv:1109.6438v1 [math.AG] we introduced and studied a notion of algebraic entropy. In this paper we will give an application of algebraic entropy in proving Kunz Regularity Criterion for all contracting self-maps of finite length of Noetherian local rings in arbitrary characteristic. Some conditions of Kunz Criterion have already been extended to the general case by Avramov, Iyengar and Miller in arXiv:math/0312412v2 [math.AC], using different methods.
Let R denote a two-dimensional normal standard-graded domain over the algebraic closure K of a finite field of characteristic p, and let I denote a homogeneous primary ideal. We prove that the Hilbert-Kunz function of I has the form = e_{HK}(I) q^{2} + g (q) with rational Hilbert-Kunz multiplicity e_{HK}(I) and an eventually periodic function g (q).
Hara and Yoshida introduced a notion of $\aaa$-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from the point of view of test ideals. We also consider variants with positive real exponents.
In this paper, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach a Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish in this paper a Volume = Multiplicity formula for $p$-families of $\mathfrak{m}_{R}$-prima…
Building on work of Brenner and Monsky from 2010 and on a Hilbert-Kunz calculation of Monsky from 1998, we exhibit a novel example of a hypersurface over $\overline{\mathbb{F}_2}$ in which tight closure does not commute with localization. Our methods involve a surprising tiling argument using Sierpiński triangles, as well as an inspection of a certain dynamical system in characteristic two.
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 mult…
Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G. Faltings that was used by Taylor and Wiles in \cite{TW} for a simplification of the proof of the minimal deformation problem.
A Noetherian local ring $(R,\mathfrak{m})$ is called Buchsbaum if the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q})$, where $\mathfrak{q}$ is an ideal generated by a system of parameters, is a constant independent of $\mathfrak{q}$. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$ and dimension at least one, the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q}^*)$ is independent of $\mathfrak{q}$ if and only if the parameter test ideal $τ_{\text{par}}(R)$ contains $\ma…
We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a geometric interpretation of the Reid-Roberts-Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal $I$ of a ring $A$ an ideal $I_>$, which consists of all elements of $A$ such that $v(a)>v(I)$, for …
We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a question of Bhatt, Iyengar, and Ma. We use almost mathematics over $R^{+}$ to deduce properties of the noetherian ring $R$ and rational surface singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is $m$-adically ideal(wise) separated, a condition which appears in the proof of local criterion for flatness. In dimension $2$ it is Ohm-Rush and i…
Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the Frobenius skew polynomial ring over $R$, then $R$ has a tight closure test element (for modules) and is $F$-pure, and to relate the test ideal of $R$ to the smallest '$E$-special' ideal of $R$ of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where $R$ is an $F$-pure homomorphic image of an $F$-finite reg…
Let R be a regular, local and F-finite ring defined over a field of finite characteristic. Let I be an ideal of height c with normal quotient $A=R/I$. It is shown that the local cohomology module H^c_I(R) contains a unique simple D_R--submodule L(A,R). This should be viewed as a finite characteristic analog of the Kashiwara--Brylinski D_R--module in characteristic zero which corresponds to the intersection cohomology complex via the Riemann--Hilbert correspondence. Besides the existence of L(A,R), more importantly, we give its construction as a certain dual of the tight closure of zero in $H^d…
We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.
