Related papers: Localization, local cohomology, and the b-function…

We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.

We describe algorithms for computing various functors for algebraic D-modules, i.e. systems of linear partial differential equations with polynomial coefficients. We will give algorithms for restriction, tensor product, localization, and…

In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic…

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…

We define an indicial polynomial of a $D$-module along an arbitrary subvariety as a generalization of both the classical indicial polynomial for a single linear differential equation and the Bernstein-Sato polynomial of a variety defined by…

We present a method to compute the holonomic extension of a $D$-module from a Zariski open set in affine space to the whole space. A particular application is the localization of coherent $D$-modules which are holonomic on the complement of…

The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function,…

A holonomic D-module on a complex analytic manifoldadmits always a b-function along any submanifold. If the module is regular, itadmits also a regular b-function, that is a b-function with a condition on the order of the lower terms of the…

A famous theorem of Harish-Chandra shows that all invariant eigendistributions on a semisimple Lie group are locally integrable functions. We give here an algebraic version of this theorem in terms of polynomials associated with a holonomic…

We study binomial D-modules, which generalize A-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these states that a binomial D-module is holonomic…

Let $X$ be an $F$-finite smooth scheme of essentially finite type over a perfect field. This article proves the existence of $b$-functions for locally finitely generated unit $F$-modules when equipped with their induced…

Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This…

Let $X$ be an algebraic variety, $f$ a regular function, $j:U\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\otimes "f^s"$. The goal of this note is to…

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…

We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of…

We present a biequivariant version of Kremnizer-Tanisaki localization theorem for quantum D-modules. We also obtain an equivalence between a category of finitely generated equivariant modules over a quantum group and a category of finitely…

We introduce all six operations for D-cap-modules on smooth rigid analytic spaces by considering the derived category of complete bornological D-cap-modules. We then focus on a full subcategory which should be thought of as consisting of…

We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for…

In this paper we study local cohomology of finitely generated bigraded modules over a standard bigraded ring with respect to the irrelevant bigraded ideals and establish a duality theorem. Several applications are considered.

For any dg algebra $A$, not necessarily commutative, and a subset $S$ in $H(A)$, the homology of $A$, we construct its derived localisation $L_S(A)$ together with a map $A\to L_S(A)$, well-defined in the homotopy category of dg algebras,…