For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...

1.1. Algebra case. Let A be a filtered quantization of a Z-graded finitely generated Poisson algebra A. By A -mod we denote the category of finitely generated A-modules. A basic tool to study such modules is to reduce them to finitely generated A-modules that can be studied by means of Commutative algebra/ Algebraic geometry. Given an A-module M , one introduces the notion of a good filtration ...

Let V be a p-adic representation of the absolute Galois group G of Qp that becomes crystalline over a finite tame extension, and assume p 6= 2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module Vp(A) of an abelian variety A defined over Qp. These conditions are stated on the filtered (φ,G)module attached to V . 2000 Mathematics Subject Classificatio...

We prove the universal property of the infinitary exact completion of a category with weak small limits. As an application, we slightly weaken the conditions characterizing essential localizations of varieties (in particular, of module categories) and of presheaf categories. Introduction An essential localization is a reflective subcategory such that the reflector has a left adjoint. In [12], R...

We present the concept of Gröbner reduction that is a Gröbner basis technique on filtered free modules. It allows to compute the dimension of a filtered free module viewn as a K-vector space. We apply the developed technique to the computation of a generalization of Hilbert-type dimension polynomials in K[X] as well as in finitely generated differencedifferential modules. The latter allows us t...

We extend the results of Adcock, Carlsson, and Carlsson ([ACC13]) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian in [CSZ10]. The use of topology to study point cloud data has been well established ([Car09], [Car14]). Given a finite metric space (e.g., a finite set in R n), one first constructs a filter...

Let A and A! be dual Koszul algebras. By Positselski a filtered algebra U with grU = A is Koszul dual to a differential graded algebra (A!, d). We relate the module categories of this dual pair by a⊗−Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.