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We prove the rationality of the Poincaré series of multiplier ideals in any dimension 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 R which gives a unified treatment of both cases.

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.

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra

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., 16 pages. Final version, to appear in Journal of Algebra

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., Comment: 15 pages. Published in Linear Algebra Appl

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 and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla (J Comb Algebra 2:231–257, 2018) relating the graded components of the local cohomology modules of Cartwright–Sturmfels ideals and their generic initial ideals. We provide an alternative proof for the case of binomial edge ideals

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., 18 pages. Typos fixed. To appear in Math. Scand

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., 31 pages

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., 42 pages. A new section on Hodge ideals is included. Comments welcome

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., Comment: 63 pages, comments are welcome. To appear in International Mathematics Research Notices

The aim of this paper is to study mixed multiplier ideals associated to a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range., 26 pages

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 Poincar\'e series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall., Comment: 21 pages

We study the structure of $D$-modules over a ring $R$ which is a direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in $R$. In positive characteristic, we use this relation between $D$-modules over $R$ and $S$ to show that the set of $F$-jumping numbers of an ideal $I\subseteq R$ is contained in the set of $F$-jumping numbers of its extension in $S$. As a consequence, the $F$-jumping numbers of $I$ in $R$ form a discrete set of rational numbers. We also relate the Bernstein-Sato polynomial in $R$ with the $F$-thresholds and the $F$-jumping numbers in $R$., 24 pages. Comments welcome!

We give a purely combinatorial characterization of complete Stanley–Reisner rings having a principally generated (equivalently, finitely generated) Cartier algebra.

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals., Comment: 22 pages. Title change and major revision of the previous version. Applications to families of complete ideals included

arXiv:1412.3605v2, The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is studied extensively., All three authors were partially supported by Generalitat de Catalunya 2014 SGR-634 project and Spanish Ministerio de Economía y Competitividad MTM2012-38122-03-01/FEDER. FDC is also supported by the KU Leuven grant OT/11/069.

In this work we describe a minimal log-resolution of an ideal in a smooth complex surface from the minimal log-resolution of its generators., 21 pages. Exposition is improved. Examples added

For a polynomial ring $R=k[x_1,...,x_n]$, we present a method to compute the characteristic cycle of the localization $R_f$ for any nonzero polynomial $f\in R$ that avoids a direct computation of $R_f$ as a $D$-module. Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules $H^r_I(R)$ for any ideal $I\subseteq R$ using the \v{C}ech complex. The algorithm, in particular, is useful for answering questions regarding vanishing of local cohomology modules and computing Lyubeznik numbers. These applications are illustrated by examples of computations using our implementation of the algorithm in Macaulay~2., Comment: 15 pages

Let Ank denote the affine space of dimension n over a field k; let XCA n k be an arrangement of linear subvarieties. Set R 1⁄4 k1⁄2x1;y; xn and let ICR denote an ideal which defines X : In this paper we study the local cohomology modules H IðRÞ :1⁄4 indlimj Ext i RðR=I ;RÞ; with special regard of the case where the ideal I is generated by monomials. If k is the field of complex numbers (or, more generally, a field of characteristic zero), the module H I ðRÞ is known to have a module structure over the Weyl algebra AnðkÞ; and one can therefore consider its characteristic cycle, denoted CCðH I ðRÞÞ in this paper (see e.g. [3, I.1.8.5]). On the other hand, the arrangement X defines a partially ordered set PðX Þ whose elements correspond to the intersections of irreducible components of X and where the order is given by inclusion. Our first result is the determination of the characteristic cycles CCðH I ðRÞÞ in terms of the cohomology of some simplicial complexes attached to the poset PðXÞ: It follows from the formulas obtained that, in either the complex or the real case, these characteristic cycles determine the Betti numbers of the complement of the arrangement in Ank: In fact, it was proved by Goresky and MacPherson that the

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

Over the last 20 years there were many advances made in the computational theory of D-modules. Nowadays, the most common computer algebra systems such as Macaulay2 or Singular have important available packages for working with D-modules. In particular, the package D-modules [127] for Macaulay 2 [80] developed by A. Leykin and H. Tsai contains an implementation of the algorithms given by U. Walther [194] and T. Oaku and N.

Local cohomology was introduced by A. Grothendieck in the early 1960s and quickly became an indispensable tool in Commutative Algebra. Despite the effort of many authors in the study of these modules, their structure is still quite unknown. C

We prove that the Frobenius algebra of the injective hull of a complete Stanley-Reisner ring as well as its Matlis dual notion of Cartier algebra can be only principally generated or infinitely generated. As a consequence we are able to show that the set of F-jumping numbers of generalized test ideals associated to complete Stanley-Reisner rings form a discrete set., 16 pages

We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of the Alexander dual ideals. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support., Comment: 28 pages

LetRRbe a formal power series ring over a field of characteristic zero andI⊆RI\subseteq Rany ideal. The aim of this work is to introduce some numerical invariants of the local ringsR/IR/Iby using the theory of algebraicD\mathcal {D}-modules. More precisely, we will prove that the multiplicities of the characteristic cycle of the local cohomology modulesHIn−i(R)H_I^{n-i}(R)andHpp(HIn−i(R))H_{\mathfrak {p}}^p(H_I^{n-i}(R)), wherep⊆R\mathfrak {p} \subseteq Ris any prime ideal that containsII, are invariants ofR/IR/I.

We study, by using the theory of algebraic D -modules, the local cohomology modules supported on a monomial ideal I of the local regular ring R=k[[x 1 ,…,x n ]] , where k is a field of characteristic zero. We compute the characteristic cycle of H I r (R) and H m p (H I r (R)) , where m is the maximal ideal of R and I is a squarefree monomial ideal. As a consequence, we can decide when the local cohomology module H I r (R) vanishes and compute the cohomological dimension cd(R,I) in terms of the minimal primary decomposition of the monomial ideal I . We also give a Cohen–Macaulayness criterion for the local ring R/I and compute the Lyubeznik numbers λ p,i (R/I)=dim k Ext R p (k,H I n−i (R)) .

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