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Steven Dale Cutkosky

Publications and source records attributed to Steven Dale Cutkosky.

At least 19 recordsLinked to original sources
arXiv2026

Limits of Amao multiplicities and Epsilon multiplicity

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and $I$ be an ideal of $R$. The epsilon multiplicity $ε(I)$ of $I$ is shown in a recent paper of Stephen Landsittel to be the limit of Amao multiplicities $$ ε(I)= \lim_{m\rightarrow\infty}\frac{a(I^m,(I^m)^{\rm sat})}{m^d} $$ if the dimension of the nilradical of $\hat R$ is less than $d$. It is shown in this paper that for an arbitrary local ring $R$ which is of finite type over a field, the limit $$ \lim_{m\rightarrow\infty}\frac{a(I^m,(I^m)^{\rm sat})}{m^d} $$ always exists.

math.AC↗
arXiv2026

Almost Mathematics, Kähler differentials and deeply ramified fields

This article discusses ramification and the structure of relative Kähler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the relative Kähler differentials of extensions of valuation rings in Artin-Schreier and Kummer extensions. We then show how this theory is applied to give a simple proof of Gabber and Ramero's characterization of deeply ramified fields. Section 4 develops the basics of almost mathematics, and should be accessible to a broad audience. Section 5 gives a simple and self contained proof of Gabber and Ramero's characterization of when the extension of a rank 1 valuation of a field to its separable closure is weakly étale. In the final section, we consider the equivalent conditions characterizing deeply ramified fields, as they are defined by Coates and Greenberg, and show that they are the same as the conditions of Gabber Ramero for local fields.

math.AC↗
arXiv2026

The Proj of the Rees algebra of a graded family of ideals

In this article we investigate the condition that the Proj of a Rees algebra of a graded family of ideals in a Noetherian local ring $R$ is Noetherian. In many cases, the Proj will be Noetherian even when the Rees algebra is not. For instance, the Proj of the Rees algebra of a graded filtration of ideals will alway be Noetherian if the analytic spread of the filtration is zero. The Proj of a Rees algebra of a divisorial filtration on a two dimensional normal excellent local ring is always Noetherian, as was proven by Russo and later with a different proof by the author. We give examples in this paper of divisorial filtrations on three dimensional normal excellent local rings whose Proj is not Noetherian, showing that this theorem does not extend to higher dimensions. A consequence of the fact that the Proj of a divisorial filtration over a two dimensional excellent normal local ring is always Noetherian is that the preimage of the maximal ideal of $R$ in the Proj has only finitely many irreducible components. As a consequence, the fiber cone of such a filtration has only finitely many minimal primes. We give an example of a graded filtration of ideals in a two dimensional regular local ring such that the preimage of the maximal ideal in the Proj of the Rees algebra of the filtration has infinitely many irreducible components, so that the Proj is not Noetherian, and the fiber cone of the filtration has infinitely many minimal primes.

math.AC↗
arXiv2026

Multiplicities of graded families of ideals on Noetherian local rings

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this article, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(\mathcal I)$ of a graded family of $m_R$-primary ideals $\mathcal I$ in $R$. This multiplicity gives the classical multiplicity $e(I)$ if $\mathcal I=\{I^n\}$ is the $I$-adic filtration, and agrees with the volume, $\displaystyle \lim_{n\rightarrow \infty}d!\frac{\ell(R/I_n) }{n^d}$ for $R$ such that the volume always exists as a limit. We will show in this paper that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity, including mixed multiplicities, the Rees theorem and the Minkowski inequality and equality. We give simple proofs which are independent of the theory of volumes and Okounkov bodies for all of our results, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of $m_R$-primary ideals as a limit of intersection products on the family of $R$-schemes which are obtained by blowing up $m_R$-primary ideals in $R$.

math.AC↗
arXiv2025

The Rees algebra and analytic spread of a divisorial filtration

In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal $I$ in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of $R/\overline{I^n}$ for some $n$. We show in Theorem 1.5 that McAdam's theorem holds for $\mathbb Q$-divisorial filtrations in an equidimensional local ring which is essentially of finite type over an excellent local ring of dimension less than or equal to 3. This generalizes an earlier result for $\mathbb Q$-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations. We consider the question of the asymptotic behavior of the function $n\mapsto λ_R(R/I_n)$ for a $\mathbb Q$-divisorial filtration $\mathcal I=\{I_n\}$ of $m_R$-primary ideals on a $d$-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity $$ e(\mathcal I)=d! \lim_{n\rightarrow\infty}\frac{λ_R(R/I_n)}{n^d} $$ can be irrational. We show in Lemma 4.1 that the limsup of the first difference function $$ \limsup_{n\rightarrow\infty}\frac{λ_R(I_n/I_{n+1})}{n^{d-1}} $$ is always finite for a $\mathbb Q$-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit. In the final section, we give an example of a symbolic filtration $\{P^{(n)}\}$ of a prime ideal $P$ in a normal two dimensional excellent local ring which has the property that the set of Rees valuations of all the symbolic powers $P^{(n)}$ of $P$ is infinite.

math.AC↗
arXiv2025

Kähler differentials of extensions of valuation rings and deeply ramified fields

Assume that $(L,v)$ is a finite Galois extension of a valued field $(K,v)$. We give an explicit construction of the valuation ring $\mathcal O_L$ of $L$ as an $\mathcal O_K$-algebra, and an explicit description of the module of relative Kähler differentials $Ω_{\mathcal O_L|\mathcal O_K}$ when $L|K$ is a Kummer extension of prime degree or an Artin-Schreier extension, in terms of invariants of the valuation and field extension. The case when this extension has nontrivial defect was solved in a recent paper by the authors with Anna Rzepka. The present paper deals with the complementary (defectless) case. The results are known classically for (rank 1) discrete valuations, but our systematic approach to non-discrete valuations (even of rank 1) is new. Using our results from the prime degree case, we characterize when $Ω_{\mathcal O_L|\mathcal O_K}=0$ holds for an arbitrary finite Galois extension of valued fields. As an application of these results, we give a simple proof of a theorem of Gabber and Ramero, which characterizes when a valued field is deeply ramified. We further give a simple characterization of deeply ramified fields with residue fields of characteristic $p>0$ in terms of the Kähler differentials of Galois extensions of degree $p$.

math.AC↗
arXiv2025

Degree functions of graded families of ideals

We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined results for divisorial filtrations. Finally, also in dimension 2, we give an example of a non-Noetherian divisorial filtration $\{I_n\}_{n\geqslant 0}$ of $\mathfrak{m}_R$-primary ideals such that the union of all the sets of Rees valuations of all the $I_n$ is a finite set, and another example of a (necessarily non-Noetherian) divisorial filtration of $\mathfrak{m}_R$-primary ideals such that the set of all Rees valuations is infinite.

math.AC↗
arXiv2025

Multiplicities and degree functions in local rings via intersection products

We prove a theorem on the intersection theory over a Noetherian local ring $R$, which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings $R$, using the theory of rational equivalence developed by Thorup, and the Snapper-Mumford-Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self-contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert-Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over $2$-dimensional excellent local rings in terms of our intersection product.

math.AC↗
arXiv2025

On the computation of Kähler differentials and characterizations of Galois extensions with independent defect

For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind differentials. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs.\ independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their rank or value groups.

math.AC↗
arXiv2024

Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings

Let $\mathcal I=\{I_n\}$ be a divisorial filtration on a two dimensional normal excellent local ring $(R,m_R)$. Let $R[\mathcal I]=\oplus_{n\ge 0}I_n$ be the Rees algebra of $\mathcal I$ and $τ:\mbox{Proj}R[\mathcal I])\rightarrow \mbox{Spec}(R)$ be the natural morphism. The reduced fiber cone of $\mathcal I$ is the $R$-algebra $R[\mathcal I]/\sqrt{m_RR[\mathcal I]}$, and the reduced exceptional fiber of $τ$ is $\mbox{Proj}(R[\mathcal I]/\sqrt{m_RR[\mathcal I]})$. We give an explicit description of the scheme structure of $\mbox{Proj}(R[\mathcal I])$. As a corollary, we obtain a new proof of a theorem of F. Russo, showing that $\mbox{Proj}(R[\mathcal I])$ is always Noetherian and that $R[\mathcal I]$ is Noetherian if and only if $\mbox{Proj}(R[\mathcal I])$ is a proper $R$-scheme. We give an explicit description of the scheme structure of the reduced exceptional fiber $\mbox{Proj}(R[\mathcal I]/\sqrt{m_RR[\mathcal I]})$ of $τ$, in terms of the possible values 0, 1 or 2 of the analytic spread $\ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I]$. In the case that $\ell(\mathcal I)=0$, $τ^{-1}(m_R)$ is the emptyset; this case can only occur if $R[\mathcal I]$ is not Noetherian.

math.AC↗
arXiv2023

Generating sequences of valuations on simple extensions of domains

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation ring of $v_0$ and such that $f(x)$ is in $A[x]$. The study of these extensions is a classical subject. This paper is devoted to the problem of describing the structure of the associated graded ring ${\rm gr}_v A[x]/(f(x))$ of $A[x]/(f(x))$ for the filtration defined by $v$ as an extension of the associated graded ring of $A$ for the filtration defined by $v_0$. We give a complete simple description of this algebra when there is unique extension of $v_0$ to $L$ and the residue characteristic of $A$ does not divide the degree of $f$. To do this, we show that the sequence of key polynomials constructed by MacLane's algorithm can be taken to lie inside $A[x]$. This result was proven using a different method in the more restrictive case that the residue fields of $A$ and of the valuation ring of $v$ are equal and algebraically closed in a recent paper by Cutkosky, Mourtada and Teissier.

math.AC↗
arXiv2023

Epsilon multiplicity and analytic spread of filtrations

We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti to epsilon multiplicity of filtrations, and show that under mild assumptions this multiplicity exists as a limit. We show that in rather general rings, the epsilon multiplicity of a Q-divisorial filtration is positive if and only if the analytic spread of the filtration is maximal (equal to the dimension of the ring). The condition that filtrations $\mathcal J\subset \mathcal I$ have the same epsilon multiplicity is considered, and we find conditions ensuring that the filtrations have the same integral closure.

math.AC↗
arXiv2022

The Asymptotic Samuel Function of a Filtration

We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties.

math.AC↗
arXiv2022

Erratic birational behavior of mappings in positive characteristic

Birational properites of generically finite morphisms $X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of $X$. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in towers of defect Artin-Schreier extensions which can have any prescribed distance.

math.AG↗
arXiv2022

Analytic Spread and Associated primes of divisorial filtrations

We prove that a classical theorem of McAdam about the analytic spread of an ideal in a Noetherian local ring is true for divisorial filtrations on an excellent local ring $R$ which is either of equicharacteristic zero or of dimension $\le 3$. In fact, the proof is valid whenever resolution of singularities holds.

math.AC↗
arXiv2022

Analytic Spread of Filtrations and Symbolic Algebras

In this paper we define and explore the analytic spread $\ell(\mathcal I)$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal $I$, we have the classical bounds $\mbox{ht}(I)\le\ell(I)\le \dim R$. The upper bound $\ell(\mathcal I)\le \dim R$ is true for filtrations $\mathcal I$, but the lower bound is not true for all filtrations. We show that for the filtration $\mathcal I$ of symbolic powers of a height two prime ideal $\mathfrak p$ in a regular local ring of dimension three (a space curve singularity), so that $\mbox{ht}(\mathcal I) =2$ and $\dim R=3$, we have that $0\le \ell(\mathcal I)\le 2$ and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if $\ell(\mathfrak p^{(n)})=3$ for all symbolic powers of $\mathfrak p$ and if and only if $\ell(\mathcal I_a)=3$ for all truncations $\mathcal I_a$ of $\mathcal I$.

math.AC↗