Alberto F. Boix

Universitat Politècnica de Catalunya | UPC

The purpose of this paper is to study rank metric codes over Galois rings. In particular, we prove MacWilliams identities for finite chain rings relating the sequences of $q$-binomial moments of a code and its dual and for Gabidulin analogues of free codes over Galois rings we prove the corresponding MacWilliams identity as a functional expression...

In this paper we study stronger forms of Goldbach's conjecture enriched with the linear representations of prime numbers given by classic Dirichlet's theorem and extensions of it. We call such a representation a Goldbach-Dirichlet representation (GD-representation). Among other results , we show that Dirichlet's Theorem on Arithmetic Progressions i...

Let R be a commutative Noetherian ring of prime characteristic p. The main goal of this paper is to study in some detail when is an open set in the Zariski topology, where \(\mathcal {F}^{E_{\mathfrak {p}}}\) denotes the Frobenius algebra attached to the injective hull of the residue field of \(R_{\mathfrak {p}}.\) We show that this is true when R...

The goal of this paper is twofold: on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne–Lichtenbaum Vanishing theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules....

The goal of this paper is twofold; on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne--Lichtenbaum Vanishing Theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules....

In this paper, first, we generalize a result of Peskine–Szpiro on the relation between the cohomological dimension and projective dimension. Then, we give conditions for the vanishing of local cohomology from local to global and vice versa. Our final goal in the present paper is to examine the set-theoretically Cohen–Macaulay ideals to find some co...

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action a...

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. The main goal of this paper is to study in some detail when \[ \overline{W^R}:=\{\mathfrak{p}\in\operatorname{Spec} (R):\ \mathcal{F}^{E_{\mathfrak{p}}}\text{ is finitely generated as a ring over its degree zero piece}\} \] is an open set in the Zariski topology, where $\mathcal{...

Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises 1/f to its pth power. We first discuss a relation between the “level” of this differential operator and the notion of “stratification” in the case of hyperelliptic curves. Next, we extend the notion of level...

Let $\nu$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaqui\'e key polynomials for valuations $\mu\le\nu$ and abstract key polynomials for $\nu$. Also, some results on invariants attached to limit key polynomials are obtained. In particular, if $\opera...

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, {Ij}, and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to Cr-rings, for 2≤r≤∞. As an auxiliary step...

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this short note we extend several approximation results both to rather general filtrations and to $C^r$-rings, for $2\le r\le\inft...

The classical lemma of Borel reads: any power series with real coefficients is the Taylor series of a smooth function. Algebraically this means the surjectivity of the completion map at a point, $C^\infty(\Bbb{R}^n) \twoheadrightarrow \Bbb{R}[[\underline{x}]]$. Similarly, Whitney extension theorem implies the surjectivity of the completion at close...

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion o...

This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given $p^{-e}$-linear map.

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action a...

The author would like to correct the errors in the publication of the original article. The corrected details are given below for your reading.

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module....

We construct a Koszul complex in the category of left skew polynomial rings associated to a flat endomorphism that provides a finite free resolution of an ideal generated by a Koszul regular sequence.

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number...

The goal of this paper is to shed light on the determinacy question that arises in New Keynesian models as result of a combination of several monetary policy rules; in these models, we provide conditions to guarantee existence and uniqueness of equilibrium by means of results that are obtained from theoretical analysis. In particular, we show that...

The purpose of this report is to introduce a formalism to produce two collections of spectral sequences. On one hand, a collection is made up by spectral sequences which involve in their second page the left derived functors of the colimit on a certain finite poset. On the other hand, the other is made up by spectral sequences which involve in thei...

It is known that the Cartier algebra of a Stanley-Reisner ring can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1-1 correspondence between new generators appearing on each graded piece and fr...

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some n...

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 $\delta$ 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 $\delta$, i.e., the le...

This paper describes a method for computing all F-pure ideals for a given Cartier map of a polynomial ring over a finite field.

Bayer and Blanco-Chac\'on have recently defined quadratic modular symbols for the Shimura curves X(D,N) attached to Eichler orders of level N of an indefinite quaternion rational algebra of discriminant D. In this paper, we give a cohomological interpretation of these quadratic modular symbols. Explicit computations for the homology of some Shimura...

We study the generation of the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring over a field with positive characteristic. In particular, by extending the ideas used by M. Katzman to give a counterexample to a question raised by G. Lyubeznik and ME. Smith about the finite generation of Frobenius algebras, we prove that the...