35 research outputs found
Tautological systems and free divisors
We introduce tautological system defined by prehomogenous actions of
reductive algebraic groups. If the complement of the open orbit is a linear
free divisor satisfying a certain finiteness condition, we show that these
systems underly mixed Hodge modules. A dimensional reduction is considered and
gives rise to one-dimensional differential systems generalizing the quantum
differential equation of projective spaces
La Geometría Algebraica: punto de encuentro de las Matemáticas
Libro completo en: http://www.rasc.es/assets/rasdc---memorias-vol.-6-(1998-2001).pd
On the modules of m-integrable derivations in non-zero characteristic
Let k be a commutative ring and A a commutative k-algebra. Given
a positive integer m, or m = ∞, we say that a k-linear derivation δ of
A is m-integrable if it extends up to a Hasse–Schmidt derivation D =
(Id, D1 = δ, D2, . . . , Dm) of A over k of length m. This condition is
automatically satisfied for any m under one of the following orthogonal
hypotheses: (1) k contains the rational numbers and A is arbitrary, since
we can take Di =
δ
i
i!
; (2) k is arbitrary and A is a smooth k-algebra.
The set of m-integrable derivations of A over k is an A-module which
will be denoted by Iderk(A; m). In this paper we prove that, if A is a
finitely presented k-algebra and m is a positive integer, then a k-linear
derivation δ of A is m-integrable if and only if the induced derivation
δp : Ap → Ap is m-integrable for each prime ideal p ⊂ A. In particular,
for any locally finitely presented morphism of schemes f : X → S
and any positive integer m, the S-derivations of X which are locally mintegrable
form a quasi-coherent submodule Ider S(OX; m) ⊂ Der S(OX)
such that, for any affine open sets U = Spec A ⊂ X and V = Spec k ⊂
S, with f(U) ⊂ V , we have Γ(U,Ider S(OX; m)) = Iderk(A; m) and
Ider S(OX; m)p = IderOS,f(p)
(OX,p; m) for each p ∈ X. We also give,
for each positive integer m, an algorithm to decide whether all derivations
are m-integrable or not.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona
The local duality theorem in D-module theory
These notes are devoted to the Local Duality Theorem for D-modules,
which asserts that the topological Grothendieck-Verdier duality exchanges the de Rham complex and the solution complex of holonomic modules over a complex analytic manifold. We give Mebkhout’s original proof and the relationship with Kashiwara-Kawai’s proof. In that way we are able to precise the commutativity of some diagrams appearing in the last one.Ce cours est consacré au théorème de du alité locale pour les D-modules, qui affirme que la dualité topologique de Grothendieck-Verdier échange le complexe de de Rham et le complexe des solutions des modules holonomes sur une variété analytique complexe. On donne la preuve originale de Mebkhout en faisant le rapport avec la preuve de Kashiwara-Kawai. Ceci nous permet de préciser la commutativité de certains diagrammes dans
cette dernière.Dirección General de Enseñanza SuperiorFondo Europeo de Desarrollo Regiona
Higher derivations of modules and the Hasse-Schmidt module
In this paper we revisit Ribenboim's notion of higher derivations of modules
and relate it to the recent work of De Fernex and Docampo on the sheaf of
differentials of the arc space. In particular, we derive their formula for the
K\"ahler differentials of the Hasse-Schmidt algebra as a consequence of the
fact that the Hasse-Schmidt algebra functors commute.Comment: 13 page
En recuerdo de Alexander Grothendieck: prólogo para una lectura de su vida y obra
En este artículo repasamos algunas de las claves de las contribuciones
matemáticas de Alexander Grothendieck y del contexto en el que
se gestaron, y nos asomamos así a la vida y obra de una de las figuras más
influyentes de las Matemáticas contemporáneas
Continuous division of linear differential operators and faithful flatness of D∞X over DX
In these notes we prove the faithful flatness of the sheaf of infinite order
linear differential operators over the sheaf of finite order linear differential operators on a complex analytic manifold. We give the Mebkhout-Narv´aez’s proof based on the continuity of the division of finite order differential operators with respect to a natural topology. We reproduce the proof of the continuity theorem given by Hauser-Narváez, which is simpler than the original proof.Dans ce cours on démontre la fidèle platitude du faisceau d’opérateurs différentiels linéaires d’ordre infini sur le faisceau d’opérateurs différentiels linéaires d’ordre fini d’une variété analytique complexe lisse. La preuve que nous donnons est celle de Mebkhout-Narváez, qui utilise la continuité de la division d’opérateurs différentiels d’ordre fini par rapport à une topologie naturelle. Nous réproduisons la preuve de Hauser-Narváez du théorème de continuité, qui est plus simple que la preuve originale.Ministerio de Ciencia y TecnologíaFondo Europeo de Desarrollo Regiona
On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities
Given two holomorphic functions and defined in two respective germs
of complex analytic manifolds and , we know thanks to M. Saito
that, as long as one of them is Euler homogeneous, the reduced (or microlocal)
Bernstein-Sato polynomial of the Thom-Sebastiani sum can be expressed in
terms of those of and . In this note we give a purely algebraic proof of
a similar relation between the whole functional equations that can be applied
to any setting (not necessarily analytic) in which Bernstein-Sato polynomials
can be defined.Comment: 8 pages, final versio
Primer Encuentro Conjunto RSME-UMA, Buenos Aires, 11 a 15 de diciembre de 2017
En la semana del lunes 11 al viernes 15 de diciembre de 2017 se llevó a cabo en Buenos Aires el Primer Encuentro Conjunto entre la Unión Matemática Argentina (UMA) y la RSME. Se celebraba el centenario de la primera visita
del Profesor Don Julio Rey Pastor a la Argentina, el punto de partida del estrecho vínculo que desde entonces ha existido entre las comunidades
matemáticas de ambos países. El encuentro, que convocó a unos 900 participantes, se organizó alrededor de tres ejes principales: Actividades Científicas, Educación, y Divulgación. Las dos primeras se desarrollaron en el predio de la Facultad de Ciencias Exactas y Naturales de la Universidad de Buenos Aires, mientras que las actividades de divulgación, dirigidas a un pú-
blico general, se llevaron a cabo en el amplio Centro Cultural de la Ciencia. Pueden consultarse los detalles, incluida la composición de los Comités Organizador y Científico, en la web del congreso: http://uma2017.dm.uba.ar