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.Comment: 15 pages. Published in Linear Algebra App

Computing the support of local cohomology modules

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 page

DD-modules, Bernstein-Sato polynomials and FF-invariants of direct summands

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 $R$.Comment: 24 pages. Comments welcome

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

El nou mecanisme de dosificació

Premis Pharmanews-Fedefarma 2016Des de temps immemorials, l'estudi de la cinètica dels fàrmacs dins de l'organisme ha estat i és un dels problemes principals ja que cada substància té un perfil concret i és complicat concloure patrons de comportament. Si no s'estudia correctament aquesta cinètica, pot haver-hi infradosificació o sobredosificació. En el cas de la infradosificació, el resultat majoritàriament serà la ineficàcia del fàrmac ja que el pacient, en principi, no desenvoluparà cap resposta. En un cas de sobredosificació, es pot comprometre la salut del pacient