Searcharxiv⌕ Search

arXiv subjects

Josep Àlvarez Montaner

Publications and source records attributed to Josep Àlvarez Montaner.

At least 19 recordsLinked to original sources
arXiv2025

Morse resolutions of monomial ideals and Betti splittings

We use discrete Morse theory to study free resolutions of monomial ideals in combination with splitting techniques. We establish the minimality of such pruned resolutions for several classes of ideals, including stable and linear quotient ideals. In particular, we unify classical constructions such as the Eliahou-Kervaire and Herzog-Takayama resolutions within the pruned resolution framework. Additionally, we introduce methods to reduce the minimality study of a pruned resolution for an ideal to that of a smaller subideal and present a variant of our pruned resolution for powers of monomial ideals.

math.AC↗
arXiv2024

Bernstein-Sato polynomial and related invariants for meromorphic functions

We develop a theory of Bernstein-Sato polynomials for meromorphic functions and we use it to study the analytic continuation of Archimedian local zeta functions in this setting. We also introduce both an analytic and an algebraic theory of multiplier ideals for meromorphic functions and relate the jumping numbers of these multiplier ideals to the roots of the Bernstein-Sato polynomials.

math.CV↗
arXiv2024

Lyubeznik tables of $S_r$ and $CM_r$ rings

We describe the shape of the Lyubeznik table of either rings in positive characteristic or Stanley-Reisner rings in any characteristic when they satisfy Serre's condition $S_r$ or they are Cohen-Macaulay in a given codimension, condition denoted by $CM_r$. Moreover we show that these results are sharp.

math.AC↗
arXiv2024

Limit distribution of Hodge spectral exponents of irreducible plane curve singularities

We study the distribution of the Hodge spectral exponents of an irreducible plane curve by comparing it with a continuous distribution. We provide a closed formula for this difference in terms of numerical invariants of the curve. We characterize those families of irreducible plane curves whose limit distribution of Hodge spectral exponents is the continuous distribution and we provide intervals of dominating values.

math.AG↗
arXiv2022

Bernstein's inequality and holonomicity for certain singular rings

In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. We obtain these results using a more general version of Bernstein filtrations.

math.AC↗
arXiv2021

A note on Bernstein-Sato ideals

We define the Bernstein-Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.

math.AC↗
arXiv2021

Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summands

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

math.AG↗
arXiv2021

Divisors of expected Jacobian type

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of D-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some D-module theoretic invariant given by the degree of the Kashiwara operator.

math.AC↗
arXiv2021

Poincaré series of multiplier and test ideals

We prove the rationality of the Poincaré series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carramiñana et al. Our results also hold for Poincaré series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $\mathbb{R}$ which gives an unified treatment of both cases.

math.AC↗
arXiv2019

Pruned cellular free resolutions of monomial ideals

Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in general, but we can slightly modify our algorithm in order to obtain a simplicial resolution. We also show that the Lyubeznik resolution fits into our pruning strategy. We finally use our methods to give a different approach to the theory of splitting of monomial ideals.

math.AC↗
arXiv2019

Generating functions associated to Frobenius algebras

We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this generating function explicitly as a rational function.

math.AC↗
arXiv2019

Local cohomology of binomial edge ideals and their generic ideals

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen-Macaulayness of these ideals and we describe their Castelnuovo-Mumford regularity and their Hilbert series. We also prove a conjecture of Conca, De Negri and Gorla relating the graded components of the local cohomology modules of binomial edge ideals and their generic initial ideals.

math.AC↗
arXiv2019

Bass numbers of local cohomology of cover ideals of graphs

We develop splitting techniques to study Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of the corresponding subgraphs. Indeed our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the connected components.

math.AC↗
arXiv2018

Multiplicities of jumping points for mixed multiplier ideals

In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall.

math.AG↗
arXiv2018

On some local cohomology spectral sequences

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. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster.

math.AC↗
arXiv2017

Koszul complex over skew polynomial rings

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.

math.AC↗