arXiv subjects

V. H. Jorge Pérez

Publications and source records attributed to V. H. Jorge Pérez.

7 recordsLinked to original sources

arXiv2026

Generalized mixed sequence multiplicity

Let $(R,\mathfrak m,k)$ be a Noetherian local ring, let $M$ be a nonzero finitely generated $R$-module, and let $J_1,\ldots,J_s$ be arbitrary proper ideals of positive height on $M$. We introduce the notion of directional mixed multiplicities, obtained from the multigraded quotients $J^{\mathbf u}M/J^{\mathbf u+\mathbf e_j}M$ by retaining one distinguished direction $j$. These invariants refine the mixed multiplicity sequences of arbitrary ideals while remaining compatible with the two classical extremes: for one ideal they recover the positive components of the Achilles--Manaresi multiplicity sequence, and for $\mathfrak m$-primary ideals their top-degree terms recover the usual mixed multiplicities individually. We establish additivity, an associativity formula, superficial reduction, and an exact decomposition of the diagonal coefficients into directional components. Over an infinite residue field, every positive directional coefficient is realized as an eventual length by a sufficiently general directional $(FC)$-system. We then identify an additional synchronization condition under which the directional coefficient becomes the Hilbert--Samuel multiplicity of a parameter ideal, yielding a Rees-type theorem and recovering the classical theorem in the $\mathfrak m$-primary case. In addition, we show a multihomogeneous product formula for $J_1^{n_1}\cdots J_s^{n_s}$, which specializes to the classical Bhattacharya--Rees--Teissier formula.

math.AC↗

arXiv2026

Asymptotic multiplicity sequences of graded families

We study asymptotic multiplicity sequences of graded families of arbitrary ideals in Noetherian local rings. We characterize convergence of the zeroth component and prove that the first positive component not forced to vanish always converges under natural equidimensionality and catenarity assumptions. In contrast, higher components may exhibit arbitrary asymptotic behavior: even in $k[[x,y]]$, integrally closed filtrations with fixed radical, height, and analytic spread can realize any prescribed lower and upper limits. For Noetherian graded filtrations, however, all asymptotic components exist, are finite and rational, and satisfy Rees-type and valuative criteria for integral dependence. In the reduced pure-dimensional complex analytic setting, a common principalizing modification provides a second convergence mechanism without finite-generation assumptions. We also develop a diagonal mixed theory: for graded $\mathfrak m$-primary families, the top diagonal components are sums of Cutkosky's mixed multiplicities, while on a common analytic model all diagonal mixed components converge. The latter result extends the one-family common-model theorem.

math.AC↗

arXiv2026

Mixed multiplicities and Rees Theorems for graded families and their converses

Let $(R,\mathfrak m)$ be a Noetherian local ring and let $\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$ be graded $\mathfrak m$-primary families. We introduce weak asymptotic and strong homogeneous joint reductions and prove corresponding extensions of Rees's mixed multiplicity theorem relating joint reductions and mixed multiplicities. Actually, for weak asymptotic joint reductions, mixed multiplicities are realized as normalized limits of Hilbert--Samuel multiplicities, yielding an asymptotic Rees theorem. For multigraded admissible collections, a strong homogeneous joint reduction $\{x_{ij}\}$ of type $\mathbf d=(d_1,\ldots,d_s)$, with $x_{ij}\in I_{a_{ij}}^{(i)}$, satisfies the formula \[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \] We also prove a localized converse under formal equidimensionality, recovering the classical converse of Rees in the adic case. As an application, we derive a Böger-type theorem for graded families, whose adic specialization recovers Böger's classical theorem. In contrast, the converse to the weak asymptotic Rees theorem fails even for an adic family on a one-dimensional regular local ring.

math.AC↗

arXiv2024

On the gluing of formal schemes

The main focus of this paper is to show that the gluing of formal schemes is also a formal scheme. The algebraic approach established here also leads us to conclude when the gluing of $k$-formal schemes is a $k$-formal scheme. In addition, we derive that the gluing of formal schemes is always singular, regardless of whether we know the structure of the schemes involved.

math.AC↗

arXiv2015

Artinianness and Finiteness of Formal Local Cohomology Modules with Respect to a Pair of Ideals

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$. We investigate the structure of formal local cohomology modules of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},I,J}(M)$ with respect to a pair of ideals, for all $i\geq 0$. The main subject of the paper is to study the finiteness properties and Artinianness of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$. We study the maximum and minimum integer $i\in \N$ such that $\mathfrak{F}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ are not Artinian. We obtain some results involving cossuport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Čech-formal local cohomology modules with respect to a pair of ideals.

math.AC↗

arXiv2011

Mixed multiplicities of arbitrary modules

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in \cite{Kleiman-Thorup2} and \cite{Kirby-Rees1} (see also \cite{Bedregal-Perez}), to an arbitrary family $E,E_1,..., E_q$ of $R$-submodules of $R^p$ with $E$ of finite colength. We prove that these mixed multiplicities coincide with the Buchsbaum-Rim multiplicity of some suitable $R$-module. In particular, we recover the fundamental Rees's mixed multiplicity theorem for modules, which was proved first by Kirby and Rees in \cite{Kirby-Rees1} and recently also proved by the authors in \cite{Bedregal-Perez}. Our work is based on, and extend to this new context, the results on mixed multiplicities of ideals obtained by Viêt in \cite{Viet8} and Manh and Viêt in \cite{Manh-Viet}. We also extend to this new setting some of the main results of Trung in \cite{Trung} and Trung and Verma in \cite{Trung-Verma1}. As in \cite{Kleiman-Thorup2}, \cite{Kirby-Rees1} and \cite{Bedregal-Perez}, we actually work in the more general context of standard graded $R$-algebras.

math.AC↗

arXiv2011

(FC)-Sequences, Mixed Multiplicities and Reductions of Modules

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we use the notion of (FC)-sequences, as defined in \cite{perez-bedregal1}, to present some results concerning reductions and the positivity of mixed multiplicities of a finite collection of arbitrary submodules of $R^p.$ We also investigate the length of maximal (FC)-sequences. We actually work in the more general context of standard graded $R$-algebras.

math.AC↗