arXiv subjects

Rémi Arcadias

Publications and source records attributed to Rémi Arcadias.

4 recordsLinked to original sources

arXiv2012

Filtered restriction of geometric D-modules

In D-module theory, we have the notion of the restriction of a module along a smooth variety. T. Oaku and N. Takayama have described a process to compute the restriction, which starts from a free resolution adapted to the V-filtration of Kashiwara-Malgrange. By using the theory of free resolutions adapted to the (order and V)-bifiltration (from T. Oaku , N. Takayama and M. Granger), we endow the restriction with an order filtration and we show that the process of Oaku and Takayama respects the order filtration. In consequence, we compute a minimal presentation of the algebraic local cohomology module associated with a quasi-homogeneous polynomial with an isolated singularity, viewed as a D-module and endowed with a kind of a pole order filtration.

math.AG↗

arXiv2011

Multidegree for bifiltered $D$-modules and hypergeometric systems

In that paper, we recall the notion of the multidegree for $D$-modules, as exposed in a previous paper, with a slight simplification. A particular emphasis is given on hypergeometric systems, used to provide interesting and computable examples.

math.RA↗

arXiv2010

Multidegree for bifiltered D-modules

In commutative algebra, E. Miller and B. Sturmfels defined the notion of multidegree for multigraded modules over a multigraded polynomial ring. We apply this theory to bifiltered modules over the Weyl algebra D. The bifiltration is a combination of the standard filtration by the order of differential operators and of the so-called V-filtration along a coordinate subvariety of the ambient space defined by M. Kashiwara. The multidegree we define provides a new invariant for D-modules. We investigate its relation with the L-characteristic cycles considered by Y. Laurent. We give examples from the theory of A-hypergeometric systems defined by I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky. We consider the V-filtration along the origin. When the toric projective variety defined from the matrix A is Cohen-Macaulay, we have an explicit formula for the multidegree of the hypergeometric system.

math.RA↗

arXiv2009

Minimal resolutions of geometric D-modules

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V-filtrations. We are interested in the module D_{x,t}f^s associated with germs of functions f_1,...,f_p, which we call a geometric module, and it is endowed with the V-filtration along t_1=...=t_p=0. The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f_1,...,f_p. For p=1, we show that under some natural conditions on f, the computation of the Betti numbers is reduced to a commutative algebra problem. This includes the case of an isolated quasi homogeneous singularity, for which we give explicitely the Betti numbers. Moreover, for an isolated singularity, we characterize the quasi-homogeneity in terms of the minimal resolution.

math.AG↗