Showing 1–49 of 49 results for author: De Stefani, A

Regularity is bounded on a quasi-excellent Noetherian scheme

Authors: Alessandro De Stefani, Jack Jeffries, Nawaj KC, Luis Núñez-Betancourt

Abstract: A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitraril… ▽ More A point of a scheme has an associated tangent cone, the spectrum of a standard graded algebra encoding the local singularity. Its homological complexity can be measured by its graded Betti table: a matrix that records a part of the structure of its graded, minimal free resolution over a polynomial ring. A natural question is whether the homological complexity of the tangent cones varies arbitrarily across a scheme. In this paper, we show that this is not the case for a quasi-excellent Noetherian scheme; over such schemes, only finitely many graded Betti tables can occur. More generally, we show that a coherent sheaf over a quasi-excellent Noetherian scheme admits finitely many graded Betti tables, and that the constancy loci for the graded Betti table are constructible. As an immediate consequence, regularity is bounded on a quasi-excellent Noetherian scheme. △ Less

Submitted 9 June, 2026; originally announced June 2026.

Golod ideals in combinatorial commutative algebra

Authors: Benjamin Briggs, Trung Chau, Alessandro De Stefani

Abstract: In this article we study the Golod property of standard graded algebras. We show that determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. Next, we give a characterization of when cover ideals define Golod rings, exploiting some considerations on multidegrees of Koszul cycles and Massey products. Finally, we show that squarefree… ▽ More In this article we study the Golod property of standard graded algebras. We show that determinantal ideals, binomial edge ideals, and permanental ideals are Golod if and only if they have a linear resolution. Next, we give a characterization of when cover ideals define Golod rings, exploiting some considerations on multidegrees of Koszul cycles and Massey products. Finally, we show that squarefree strongly Golod ideals (and, more generally, lcm-strongly Golod ideals) are Golod, and not just weakly Golod. △ Less

Submitted 19 May, 2026; originally announced May 2026.

Comments: are welcome! 19 pages

MSC Class: 13D02; 13C13

Quadratic linear strands of prime ideals

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: We prove sharp estimates on the quadratic strand of the resolution of any homogeneous prime ideal in a standard graded polynomial ring over an arbitrary field. Our bounds only depend on the height of the prime ideal, and they are optimal since for every $h \geq 1$ we show that there exists a prime ideal of height $h$ achieving them. In particular, we show that a prime ideal of height $h$ can conta… ▽ More We prove sharp estimates on the quadratic strand of the resolution of any homogeneous prime ideal in a standard graded polynomial ring over an arbitrary field. Our bounds only depend on the height of the prime ideal, and they are optimal since for every $h \geq 1$ we show that there exists a prime ideal of height $h$ achieving them. In particular, we show that a prime ideal of height $h$ can contain at most $h^2$ quadratic minimal generators, and that there exists a prime ideal minimally generated by $h^2$ quadrics. △ Less

Submitted 9 May, 2026; originally announced May 2026.

Herzog ideals and $F$-singularities

Authors: Alessandro De Stefani, Linquan Ma, Matteo Varbaro

Abstract: In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gröbner degeneration) and $F$-singularities. More precisely, we show that, in positive characteristic, homogeneous Herzog ideals define $F$-anti-nilpotent rings, and we inquire, in characteristic 0, on a surprising relationship between being Herzog ideals after a change of coordinates and defining rings of… ▽ More In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gröbner degeneration) and $F$-singularities. More precisely, we show that, in positive characteristic, homogeneous Herzog ideals define $F$-anti-nilpotent rings, and we inquire, in characteristic 0, on a surprising relationship between being Herzog ideals after a change of coordinates and defining rings of dense open $F$-pure type. △ Less

Submitted 11 August, 2026; v1 submitted 10 January, 2026; originally announced January 2026.

Comments: v2: minor corrections. We added Remark 3.4 concerning Question 1.2

Bertini's theorem for $F$-rational $F$-pure singularities

Authors: Alessandro De Stefani, Thomas Polstra, Austyn Simpson

Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq\mathbb{P}^n_k$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H \subseteq \mathbb{P}^n_k$ is a general hyperplane, then $X \cap H$ is also $F$-rational and $F$-pure. Of related but independent interest, we present a relationship between the characteristic and index of a… ▽ More Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq\mathbb{P}^n_k$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H \subseteq \mathbb{P}^n_k$ is a general hyperplane, then $X \cap H$ is also $F$-rational and $F$-pure. Of related but independent interest, we present a relationship between the characteristic and index of a $\mathbb{Q}$-Gorenstein variety with isolated non-$F$-regular locus which is $F$-pure but not $F$-regular. △ Less

Submitted 29 September, 2025; v1 submitted 4 September, 2025; originally announced September 2025.

Comments: v1: 16 pages, comments welcome v2: major revision. Added Section 3; Lemma 3.1 of v1 has been replaced with Theorem 3.1 and Lemma 4.1; both main theorems are unaffected. 17 pages, comments welcome

$F$-injectivity does not imply $F$-fullness in normal domains

Authors: Alessandro De Stefani, Thomas Polstra, Austyn Simpson

Abstract: We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are $F$-injective but not $F$-full. Along the way, we find examples of two-dimensional local geometrically normal domains which are $F$-injective but not $F$-anti-nilpotent. A crucial theme of our constructions is the behavior of $F$-injectivity along a purely inseparable finite b… ▽ More We construct examples of noetherian three-dimensional local geometrically normal domains of prime characteristic which are $F$-injective but not $F$-full. Along the way, we find examples of two-dimensional local geometrically normal domains which are $F$-injective but not $F$-anti-nilpotent. A crucial theme of our constructions is the behavior of $F$-injectivity along a purely inseparable finite base change. △ Less

Submitted 9 March, 2026; v1 submitted 31 May, 2025; originally announced June 2025.

Comments: v1: 11 pages, comments welcome. v2: minor changes

Differential and symbolic powers of ideals

Authors: Alessandro De Stefani, Eloísa Grifo, Jack Jeffries

Abstract: We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability… ▽ More We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations. △ Less

Submitted 27 March, 2025; originally announced March 2025.

F-singularities of polynomials with square-free support

Authors: Aldo Conca, Alessandro De Stefani, Luis Núñez-Betancourt, Ilya Smirnov

Abstract: We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by irreducible polynomials with square-free support have F-rational singularities, positively answering a question of Bath, Mustaţă, and Walther. We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by irreducible polynomials with square-free support have F-rational singularities, positively answering a question of Bath, Mustaţă, and Walther. △ Less

Submitted 27 January, 2025; originally announced January 2025.

The defect of the F-pure threshold

Authors: Alessandro De Stefani, Luis Núñez-Betancourt, Ilya Smirnov

Abstract: Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring… ▽ More Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring. △ Less

Submitted 25 March, 2026; v1 submitted 23 January, 2025; originally announced January 2025.

Comments: v3: in v2 and in the published version of the proof of Lemma 4.6, it is claimed that A is Gorenstein. However, in our assumptions one can only say that A is quasi-Gorenstein; this correction does not affect the rest of the proof

Hilbert-Kunz multiplicity of powers of ideals in dimension two

Authors: Alessandro De Stefani, Shreedevi K. Masuti, Maria Evelina Rossi, Jugal K. Verma

Abstract: We study the behavior of the Hilbert-Kunz multiplicity of powers of an ideal in a local ring. In dimension two, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a "Ratliff-Rush version" of the Hilbert-Kunz multiplicity. We study the behavior of the Hilbert-Kunz multiplicity of powers of an ideal in a local ring. In dimension two, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a "Ratliff-Rush version" of the Hilbert-Kunz multiplicity. △ Less

Submitted 20 April, 2025; v1 submitted 10 December, 2024; originally announced December 2024.

Comments: minor fixes

Tight closure of products and F-rational singularities

Authors: Alessandro De Stefani, Ilya Smirnov

Abstract: We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier and Cutkosky characterizing rational surface singularities in terms of products of complete ideals, but are valid also in higher dimensions. We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier and Cutkosky characterizing rational surface singularities in terms of products of complete ideals, but are valid also in higher dimensions. △ Less

Submitted 13 January, 2026; v1 submitted 5 November, 2024; originally announced November 2024.

Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginni… ▽ More In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies. △ Less

Submitted 13 May, 2026; v1 submitted 19 September, 2024; originally announced September 2024.

Comments: v2: minor changes

From a local ring to its associated graded algebra

Authors: Alessandro De Stefani, Maria Evelina Rossi, Matteo Varbaro

Abstract: Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a "Gröbner-like" deformation. Using this technique we prove sharp results concerning the connectedness of $R$ and $G$. We also construct a family of local domains wh… ▽ More Let $(R,\mathfrak{m})$ be a complete local ring, and $G={\rm gr}_{\mathfrak{m}}(R)$ be its associated graded ring. We introduce a homogenization technique which allows to relate $G$ to the special fiber and $R$ to the generic fiber of a "Gröbner-like" deformation. Using this technique we prove sharp results concerning the connectedness of $R$ and $G$. We also construct a family of local domains which fail to satisfy Abhyankar's inequality for the Hilbert-Samuel multiplicity. However, we prove a version of the inequality which holds when $R$ is connected in codimension one. △ Less

Submitted 26 March, 2026; v1 submitted 24 June, 2024; originally announced June 2024.

A criterion for sequential Cohen-Macaulayness

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm gin}_{revlex}(U))$, where $F$ is a graded free $S$-module and $M \cong F/U$. This answers positively a conjecture of Lu and Yu from 2016. The purpose of this note is to show that a finitely generated graded module $M$ over $S=k[x_1,\ldots,x_n]$, $k$ a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree ${\rm adeg}(M)$ agrees with ${\rm adeg}(F/{\rm gin}_{revlex}(U))$, where $F$ is a graded free $S$-module and $M \cong F/U$. This answers positively a conjecture of Lu and Yu from 2016. △ Less

Submitted 26 July, 2023; originally announced July 2023.

Comments: 7 pages

Ladder determinantal varieties and their symbolic blowups

Authors: Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt, Lisa Seccia, Matteo Varbaro

Abstract: In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly $F$-regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is $F$-pure. The latter implies that mixed ladder determinantal rings are $F$-pure. We also show that ideals of the poset of minors of a generic matrix give rise to $F$-pure alge… ▽ More In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly $F$-regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is $F$-pure. The latter implies that mixed ladder determinantal rings are $F$-pure. We also show that ideals of the poset of minors of a generic matrix give rise to $F$-pure algebras with straightening law. △ Less

Submitted 13 January, 2026; v1 submitted 29 May, 2023; originally announced May 2023.

Comments: Minor changes from v1

On the rate of generic Gorenstein $K$-algebras

Authors: Mats Boij, Emanuela De Negri, Alessandro De Stefani, Maria Evelina Rossi

Abstract: The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and… ▽ More The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and $s \ge 3 $ has rate $ \lfloor \frac{s}{2} \rfloor. $ In the process we show that such an algebra is generated in degree $\lfloor \frac{s}{2} \rfloor +1. $ This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree. △ Less

Submitted 13 January, 2026; v1 submitted 28 April, 2023; originally announced April 2023.

Comments: 12 pages; minor changes from v1

On the notion of sequentially Cohen-Macaulay modules

Authors: Giulio Caviglia, Alessandro De Stefani, Enrico Sbarra, Francesco Strazzanti

Abstract: In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help the reader with quickly getting acquainted with this topic. We then discuss two generalizations of the notion of sequential Cohen-Macaulayness which are inspired by a theorem of Jürgen Herzog and the third author. In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help the reader with quickly getting acquainted with this topic. We then discuss two generalizations of the notion of sequential Cohen-Macaulayness which are inspired by a theorem of Jürgen Herzog and the third author. △ Less

Submitted 13 April, 2023; originally announced April 2023.

Journal ref: Research in the Mathematical Sciences 9 (2022), Article number: 40

Purity of monoids and characteristic-free splittings in semigroup rings

Authors: Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt

Abstract: Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent. Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent. △ Less

Submitted 1 September, 2023; v1 submitted 7 October, 2022; originally announced October 2022.

Comments: to appear in Mathematische Zeitschrift

Frobenius methods in combinatorics

Authors: Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt

Abstract: We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for $F$-pure rings that were inspired… ▽ More We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for $F$-pure rings that were inspired by work done for Stanley-Reisner rings. △ Less

Submitted 18 March, 2022; originally announced March 2022.

Comments: 35 pages

Blowup algebras of determinantal ideals in prime characteristic

Authors: Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt

Abstract: We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that t… ▽ More We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results we develop the notion of $F$-split filtrations and symbolic $F$-split ideals. △ Less

Submitted 18 June, 2024; v1 submitted 1 September, 2021; originally announced September 2021.

Comments: Final version. This preprint corrects, replaces, and expands the work done in preprint arXiv:2004.03831, which contains a mistake Lemma 5.4. The new version changes some terminology. We also added a new result about F-regularity for symbolic Rees algebras of ideals associated to symmetric matrices (Theorem 6.16)

MSC Class: 13A30; 13A35; 13A02; 13C15

Bounds on the number of generators of prime ideals

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: Let $S$ be a polynomial ring over any field $\Bbbk$, and let $P \subseteq S$ be a non-degenerate homogeneous prime ideal of height $h$. When $\Bbbk$ is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in $P$ which only depends on $h$. We significantly extend this result by proving that the number… ▽ More Let $S$ be a polynomial ring over any field $\Bbbk$, and let $P \subseteq S$ be a non-degenerate homogeneous prime ideal of height $h$. When $\Bbbk$ is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in $P$ which only depends on $h$. We significantly extend this result by proving that the number of minimal generators of $P$ in any degree $j$ can be bounded above by an explicit function that only depends on $j$ and $h$. In addition to providing a bound for generators in any degree $j$, not just for quadrics, our techniques allow us to drop the assumption that $\Bbbk$ is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal. △ Less

Submitted 12 August, 2021; originally announced August 2021.

Comments: 15 pages

The Eisenbud-Green-Harris Conjecture

Authors: Giulio Caviglia, Alessandro De Stefani, Enrico Sbarra

Abstract: We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature. We include a final section with some applications, and examples. We survey most of the known results concerning the Eisenbud-Green-Harris Conjecture. Our presentation includes new proofs of several theorems, as well as a unified treatment of many results which are otherwise scattered in the literature. We include a final section with some applications, and examples. △ Less

Submitted 17 January, 2022; v1 submitted 28 June, 2021; originally announced June 2021.

Comments: 25 pages, minor changes from previously posted version

Linearly presented modules and bounds on the Castelnuovo-Mumford regularity of ideals

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension. We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension. △ Less

Submitted 27 April, 2021; originally announced April 2021.

Comments: 8 pages; comments are welcome

MSC Class: Primary 13D02; Secondary 13A02; 13A15

A uniform Chevalley theorem for direct summands of polynomial rings in mixed characteristic

Authors: Alessandro De Stefani, Eloísa Grifo, Jack Jeffries

Abstract: We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand. We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand. △ Less

Submitted 11 April, 2022; v1 submitted 1 April, 2021; originally announced April 2021.

Comments: Final version

MSC Class: 13B99; 13N99; 14G45

Assessing Information Quality in IoT Forensics: Theoretical Framework and Model Implementation

Authors: Federico Costantini, Fausto Galvan, Marco Alvise de Stefani, Sebastiano Battiato

Abstract: IoT technologies pose serious challenges to digital Forensics. The acquisition of digital evidence is hindered by the number and extreme variety of IoT items, often lacking physical interfaces, connected in unprotected networks, feeding data to uncontrolled cloud services. In this paper we address "Information Quality" in IoT Forensics, taking into account different levels of complexity and includ… ▽ More IoT technologies pose serious challenges to digital Forensics. The acquisition of digital evidence is hindered by the number and extreme variety of IoT items, often lacking physical interfaces, connected in unprotected networks, feeding data to uncontrolled cloud services. In this paper we address "Information Quality" in IoT Forensics, taking into account different levels of complexity and included human factors. After drawing a theoretical framework on data quality and information quality, we focus on forensic analysis challenges in IoT environments, providing a use case of evidence collection for investigative purposes. At the end, we propose a formal framework for assessing information quality of IoT devices for Forensics analysis. △ Less

Submitted 29 December, 2020; originally announced December 2020.

Comments: accepted for publication in Journal of Applied Logics (2020)

F-stable secondary representations and deformation of F-injectivity

Authors: Alessandro De Stefani, Linquan Ma

Abstract: We prove that deformation of F-injectivity holds for local rings $(R,\mathfrak{m})$ that admit secondary representations of $H^i_{\mathfrak{m}}(R)$ which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when $(R,\mathfrak{m})$ is sequentially Cohen-Macaulay (or more generally when all the local cohomology modules $H^i_{\mathfrak{m}}(R)$ have no embedded attach… ▽ More We prove that deformation of F-injectivity holds for local rings $(R,\mathfrak{m})$ that admit secondary representations of $H^i_{\mathfrak{m}}(R)$ which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when $(R,\mathfrak{m})$ is sequentially Cohen-Macaulay (or more generally when all the local cohomology modules $H^i_{\mathfrak{m}}(R)$ have no embedded attached primes). We obtain some additional cases if $R/\mathfrak{m}$ is perfect or if $R$ is $\mathbb{N}$-graded. △ Less

Submitted 14 August, 2022; v1 submitted 18 September, 2020; originally announced September 2020.

Comments: 9 pages. Minor changes from previous version

The Eisenbud-Green-Harris conjecture for fast-growing degree sequences

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence satisfy $d_i \geqslant \sum_{j=1}^{i-1}(d_j-1)$, improving a result obtained in 2008 by the first author and Maclagan. Except for the sporadic case of a regular se… ▽ More Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence satisfy $d_i \geqslant \sum_{j=1}^{i-1}(d_j-1)$, improving a result obtained in 2008 by the first author and Maclagan. Except for the sporadic case of a regular sequence of five quadrics, recently proved by Güntürkün and Hochster, the results of this article recover all known cases of the conjecture where only the degrees of the regular sequence are fixed, and include several additional ones. △ Less

Submitted 18 November, 2020; v1 submitted 30 July, 2020; originally announced July 2020.

Comments: 8 pages; minor modifications from previous version; change of title

MSC Class: 13D02; 13A02; 13A15; 13P10

A Cayley-Bacharach theorem for points in $\mathbb{P}^n$

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections. We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections. △ Less

Submitted 3 July, 2020; v1 submitted 25 June, 2020; originally announced June 2020.

Comments: 12 pages, 1 figure. Minor modifications, citations added, acknowledged a result which was known in the literature

Decomposition of local cohomology tables of modules with large E-depth

Authors: Giulio Caviglia, Alessandro De Stefani

Abstract: We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a… ▽ More We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the second author and Smirnov. Finally, we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich. △ Less

Submitted 18 October, 2020; v1 submitted 21 March, 2020; originally announced March 2020.

Comments: 23 pages; we corrected some inaccuracies from previous version and performed minor changes

MSC Class: 13D45; Secondary: 13P10; 13D07

Stability and deformation of F-singularities

Authors: Alessandro De Stefani, Ilya Smirnov

Abstract: We study the problem of $\mathfrak{m}$-adic stability of F-singularities, that is, whether the property that a quotient of a local ring $(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for F-rationality in full generality, and for F-in… ▽ More We study the problem of $\mathfrak{m}$-adic stability of F-singularities, that is, whether the property that a quotient of a local ring $(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stability and deformation phenomena, which hold in great generality. △ Less

Submitted 17 September, 2020; v1 submitted 1 February, 2020; originally announced February 2020.

Comments: Comments are welcome

On monomial Golod ideals

Authors: Hailong Dao, Alessandro De Stefani

Abstract: We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod. We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod. △ Less

Submitted 10 February, 2019; v1 submitted 2 February, 2019; originally announced February 2019.

Comments: Comments are very welcome!

MSC Class: 13A02; 13D40

Global Frobenius Betti numbers and F-splitting ratio

Authors: Alessandro De Stefani, Thomas Polstra, Yongwei Yao

Abstract: We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair $(R,\mathscr{D})$, where $\mathscr{D}$ is a Cartier algebra, is equivalent to the positivity of the global F-signature ${\rm s}(R,\mathscr{D})$ of the pair. Th… ▽ More We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair $(R,\mathscr{D})$, where $\mathscr{D}$ is a Cartier algebra, is equivalent to the positivity of the global F-signature ${\rm s}(R,\mathscr{D})$ of the pair. This extends a result previously proved by these authors, by removing an extra assumption on the Cartier algebra. △ Less

Submitted 27 November, 2018; originally announced November 2018.

Comments: 31 pages

Decomposition of graded local cohomology tables

Authors: Alessandro De Stefani, Ilya Smirnov

Abstract: Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide alg… ▽ More Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets. △ Less

Submitted 25 February, 2020; v1 submitted 2 October, 2018; originally announced October 2018.

Comments: Slightly shortened

Cohomologically full rings

Authors: Hailong Dao, Alessandro De Stefani, Linquan Ma

Abstract: Inspired by a question raised by Eisenbud-Mustaţă-Stillman regarding the injectivity of maps from ${\rm Ext}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings which we call cohomologically full rings. This class of rings include many well-known singularities: Cohen-Macaulay rings, Stanley-Reisner rings, F-pure rings in positive character… ▽ More Inspired by a question raised by Eisenbud-Mustaţă-Stillman regarding the injectivity of maps from ${\rm Ext}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings which we call cohomologically full rings. This class of rings include many well-known singularities: Cohen-Macaulay rings, Stanley-Reisner rings, F-pure rings in positive characteristics, Du Bois singularities in characteristics $0$. We prove many basic properties of cohomologically full rings, including their behavior under flat base change. We show that ideals defining these rings satisfy many desirable properties, in particular they have small cohomological and projective dimension. When $R$ is a standard graded algebra over a field of characteristic $0$, we show under certain conditions that being cohomologically full is equivalent to the intermediate local cohomology modules being generated in degree $0$. Furthermore, we obtain Kodaira-type vanishing and strong bounds on the regularity of cohomologically full graded algebras. △ Less

Submitted 8 January, 2019; v1 submitted 1 June, 2018; originally announced June 2018.

Comments: 27 pages, fixed a mistake in Theorem 4.1 from previous version. Comments welcome

F-signature function of quotient singularities

Authors: Alessio Caminata, Alessandro De Stefani

Abstract: We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group $G\subseteq \mathrm{Gl}(d,\Bbbk)$. When $G$ is cyclic, we obtain more specific formulas for… ▽ More We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group $G\subseteq \mathrm{Gl}(d,\Bbbk)$. When $G$ is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples. △ Less

Submitted 9 May, 2018; v1 submitted 4 April, 2018; originally announced April 2018.

Comments: Corrected some typos

MSC Class: 13A35; 13A50; 13D40

Local cohomology and Lyubeznik numbers of $F$-pure rings

Authors: Alessandro De Stefani, Eloísa Grifo, Luis Núñez-Betancourt

Abstract: In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatibl… ▽ More In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard graded and, by its means, we provide some new examples of Lyubeznik tables. We study associated primes of certain Ext-modules, showing that they are all compatible ideals. Finally, we focus on properties that Lyubeznik numbers detect over a globally $F$-split projective variety. △ Less

Submitted 18 September, 2019; v1 submitted 1 October, 2017; originally announced October 2017.

Comments: Fixed errors in the proof of Lemma 3.7 and in Example 4.11 of previous version

MSC Class: 13A35 (Primary) 13D45; 14B05 (Secondary)

A Zariski-Nagata theorem for smooth $\mathbb{Z}$-algebras

Authors: Alessandro De Stefani, Eloísa Grifo, Jack Jeffries

Abstract: In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the elements that vanish up to order $n$ on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use $p$-derivatio… ▽ More In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the elements that vanish up to order $n$ on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use $p$-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of $p$-derivations to commutative algebra. △ Less

Submitted 3 December, 2018; v1 submitted 4 September, 2017; originally announced September 2017.

Comments: In the previous version, the proof of Lemma 2.1 had a gap, due to a wrong claim about the structure of certain essentially smooth algebras. We removed the wrong statement, and fixed the proof of Lemma 2.1. The rest of the paper remains unchanged

Journal ref: Journal für die reine und angewandte Mathematik, 2020 (761), 123-140

Symbolic powers of ideals

Authors: Hailong Dao, Alessandro De Stefani, Eloísa Grifo, Craig Huneke, Luis Núñez-Betancourt

Abstract: We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject. We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic powers of monomial ideals. In addition, we present some new results on these aspects of the subject. △ Less

Submitted 9 August, 2017; originally announced August 2017.

Comments: 31 pages. Comments welcome

MSC Class: 13A15; 113F20; 3N10; 13F55; 14B05

Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements

Authors: Alessandro De Stefani, Thomas Polstra, Yongwei Yao

Abstract: We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras. We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras. △ Less

Submitted 30 August, 2016; originally announced August 2016.

Comments: 15 pages, comments welcome

Globalizing F-invariants

Authors: Alessandro De Stefani, Thomas Polstra, Yongwei Yao

Abstract: In this paper we define and study the global Hilbert-Kunz multiplicity and the global F-signature of prime characteristic rings which are not necessarily local. Our techniques are made meaningful by extending many known theorems about Hilbert-Kunz multiplicity and F-signature to the non-local case. In this paper we define and study the global Hilbert-Kunz multiplicity and the global F-signature of prime characteristic rings which are not necessarily local. Our techniques are made meaningful by extending many known theorems about Hilbert-Kunz multiplicity and F-signature to the non-local case. △ Less

Submitted 30 August, 2016; originally announced August 2016.

On the existence of $F$-thresholds and related limits

Authors: Alessandro De Stefani, Luis Núñez-Betancourt, Felipe Pérez

Abstract: The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard graded algebras over a field for which $F$-pure threshold and $F$-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the $a$-i… ▽ More The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard graded algebras over a field for which $F$-pure threshold and $F$-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the $a$-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of $F$-thresholds and $F$-pure thresholds, obtaining existence of related limits in certain cases. △ Less

Submitted 11 January, 2017; v1 submitted 10 May, 2016; originally announced May 2016.

Comments: 21 pages

On the behavior of singularities at the $F$-pure threshold

Authors: Eric Canton, Daniel Hernández, Karl Schwede, Emily Witt, Alessandro De Stefani, Jack Jeffries, Zhibek Kadyrsizova, Robert Walker, George Whelan

Abstract: We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not d… ▽ More We provide a family of examples where the $F$-pure threshold and the log canonical threshold of a polynomial are different, but where $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of \mustata-Takagi-Watanabe). We then study the $F$-signature function in the case where either the $F$-pure threshold and log canonical threshold coincide or where $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide. △ Less

Submitted 10 May, 2016; v1 submitted 21 August, 2015; originally announced August 2015.

Comments: Typos corrected, other improvements to the exposition

MSC Class: 13A35; 14J17; 14B05

Journal ref: Illinois J. Math. 60 (2016), no. 3-4, 669-685

F-thresholds of graded rings

Authors: Alessandro De Stefani, Luis Núñez-Betancourt

Abstract: The a-invariant, the F-pure threshold, and the diagonal F-threshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F-regular rings. In this article, we prove that these relations hold only assuming that the algebra is F-pure. In addition, we present an interpretation of the a-invariant for F-pure Go… ▽ More The a-invariant, the F-pure threshold, and the diagonal F-threshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F-regular rings. In this article, we prove that these relations hold only assuming that the algebra is F-pure. In addition, we present an interpretation of the a-invariant for F-pure Gorenstein graded K-algebras in terms of regular sequences that preserve F-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo-Mumford regularity, and Serre's condition $S_k$. We also present analogous results and questions in characteristic zero. △ Less

Submitted 20 July, 2015; originally announced July 2015.

Comments: 20 pages

MSC Class: Primary 13A35; Secondary 13H10; 14B05

A counterexample to a conjecture of Ding

Authors: Alessandro De Stefani

Abstract: We give a counterexample to a conjecture posed by S. Ding regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and generalized Löewy length 6. We give a counterexample to a conjecture posed by S. Ding regarding the index of a Gorenstein local ring by exhibiting several examples of one dimensional local complete intersections of embedding dimension three with index 5 and generalized Löewy length 6. △ Less

Submitted 12 September, 2016; v1 submitted 30 June, 2015; originally announced June 2015.

Comments: 9 pages, a few changes from the first version

MSC Class: Primary 13H10; Secondary 13C05; 13C14

Journal ref: J. Algebra 452 (2016), 324-337

Products of ideals may not be Golod

Authors: Alessandro De Stefani

Abstract: We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for weak Golodness of monomial ideals. Along the way, we ask some related questions. We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for weak Golodness of monomial ideals. Along the way, we ask some related questions. △ Less

Submitted 12 September, 2016; v1 submitted 30 June, 2015; originally announced June 2015.

Comments: 18 pages, minor changes from first version

MSC Class: Primary 13A02; Secondary 13D40

Journal ref: J. Pure Appl. Algebra 220 (2016), no. 6, 2289-2306

An algorithm for constructing certain differential operators in positive characteristic

Authors: Alberto F. Boix, Alessandro De Stefani, Davide Vanzo

Abstract: Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $δ$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $δ$, i.e., the least integer $e$ such that $δ$ is $R^{p^e}$-linear. In partic… ▽ More Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $δ$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $δ$, i.e., the least integer $e$ such that $δ$ is $R^{p^e}$-linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator. △ Less

Submitted 4 March, 2015; originally announced March 2015.

Comments: 23 pages. Comments are welcome

MSC Class: Primary 13A35; Secondary 13N10; 14B05

Journal ref: Le Matematiche, volume 70 (2015), no. 1, 239-271

Frobenius Betti numbers and modules of finite projective dimension

Authors: Alessandro De Stefani, Craig Huneke, Luis Núñez-Betancourt

Abstract: 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 mult… ▽ More 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 $β^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods, we give conditions for the non-existence of syzygies of finite length. △ Less

Submitted 7 September, 2015; v1 submitted 13 December, 2014; originally announced December 2014.

Comments: 25 pages

MSC Class: 13

A sufficient condition for strong $F$-regularity

Authors: Alessandro De Stefani, Luis Núñez-Betancourt

Abstract: Let $(R,\mathfrak{m},K)$ be an $F$-finite Noetherian local ring which has a canonical ideal $I \subsetneq R$. We prove that if $R$ is $S_2$ and $H^{d-1}_{\mathfrak{m}}(R/I)$ is a simple $R\{F\}$-module, then $R$ is a strongly $F$-regular ring. In particular, under these assumptions, $R$ is a Cohen-Macaulay normal domain. Let $(R,\mathfrak{m},K)$ be an $F$-finite Noetherian local ring which has a canonical ideal $I \subsetneq R$. We prove that if $R$ is $S_2$ and $H^{d-1}_{\mathfrak{m}}(R/I)$ is a simple $R\{F\}$-module, then $R$ is a strongly $F$-regular ring. In particular, under these assumptions, $R$ is a Cohen-Macaulay normal domain. △ Less

Submitted 25 November, 2014; originally announced November 2014.

Comments: 9 pages, to appear in Proceedings of the American Mathematical Society

MSC Class: 13

Artinian level algebras of low socle degree

Authors: Alessandro De Stefani

Abstract: In this paper we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthe… ▽ More In this paper we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case the extremal strata have been parametrized by Cho and Iarrobino. △ Less

Submitted 16 August, 2012; originally announced August 2012.

Comments: 24 pages. To appear on "Communications in Algebra"

MSC Class: 13H10; 13H15; 14C05