Eivind Eriksen

BI Norwegian Business School | BINBS · Department of Economics

We consider the algebra O(M) of observables and the (formally) versal morphism η:A→O(M) defined by the noncommutative deformation functor DefM of a family M={M1,…,Mr} of right modules over an associative k-algebra A. By the Generalized Burnside Theorem, due to Laudal, η is an isomorphism when A is finite dimensional, M is the family of simple A-mod...

In this paper, we study length categories using iterated extensions. We consider the problem of classifying all indecomposable objects in a length category, and the problem of characterizing uniserial length categories. We solve the last problem, and obtain a necessary and sufficient criterion under weak assumptions; this criterion turns out to be...

In this paper, we study the holonomic $D$-modules when $D$ is the ring of $k$-linear differential operators on $A = k[\Gamma]$, the coordinate ring of an affine monomial curve over the complex numbers $k = \mathbb C$. In particular, we consider the graded case, and give a complete classification of the simple graded $D$-modules. These are the simpl...

Noncommutative Deformation Theory is aimed at mathematicians and physicists studying the local structure of moduli spaces in algebraic geometry. This book introduces a general theory of noncommutative deformations, with applications to the study of moduli spaces of representations of associative algebras and to quantum theory in physics. An essenti...

The Generalized Burnside Theorem is due to Laudal. It generalizes the classical Burnside Theorem, and is obtained using noncommutative deformations of the family of simple right $A$-modules when $A$ is a finite dimensional associative algebra over an algebraically closed field. In this paper, we prove a form of the Generalized Burnside Theorem that...

Let M be a right module over an associative k-algebra A, where k is a field. We show how to compute noncommutative deformations of M in concrete terms, using an obstruction calculus based on free resolutions.

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheav...

In this paper, we show that the Weyl algebra $A_n(k)$ is a coherent ring for any field $k$ and any integer $n \ge 1$. Comment: AMS-LaTeX, 3pages

Let A be an associative algebra over a field, and let M be a finite family of right A-modules. Study of the noncommutative deformation functor of the family M leads to the construction of the algebra of observables and the Generalized Burnside Theorem, due to Laudal. In this paper, we give an overview of aspects of noncommutative deformations close...

In this paper, we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules.

Let R be the local ring of a singular point of a complex analytic space, and let M be an R-module. Under what conditions on R and M is it possible to find a connection on M? To approach this question, we consider maximal Cohen—Macaulay (MCM) modules over CM algebras that are isolated singularities, and review an obstruction theory implemented in th...

Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable...

We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.

Let (X,D) be a D-scheme in the sense of Beilinson and Bernstein, given by an algebraic variety X and a morphism O script> X →D of sheaves of rings on X. We consider noncommutative deformations of quasi-coherent sheaves of left D-modules on X, and show how to compute their pro-representing hulls. As an application, we compute the noncommutative defo...

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen–Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assum...

Let k be an algebraically closed field of characteristic 0, and let $A = k[x,y]/(f)$ be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e. a graded A-linear homomorphism $\nabla: \operatorname{Der}_k(A) \to \operatorname{End}_k(M)$ that satisfy the derivati...

We consider the notion of a connection on a module over a commutative ring, and recall the obstruction calculus for such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Grobner bases, we need the conversion to a description using free resolutions. We describe our implementation in Sing...

Let k be an algebraically closed field, let R be an associative k-algebra, and let F = {Mα : α ∈ I} be a family of orthogonal points in Mod(R) such that End R (Mα) ∼ = k for all α ∈ I. Then Mod(F), the minimal full sub-category of Mod(R) which contains F and is closed under extensions, is a full exact Abelian sub-category of Mod(R) and a length cat...

Let k be an algebraically closed field, let R be an associative k-algebra, and let F = {M_a: a in I} be a family of orthogonal points in R-Mod such that End_R(M_a) = k for all a in I. Then Mod(F), the minimal full sub-category of R-Mod which contains F and is closed under extensions, is a full exact Abelian subcategory of R-Mod and a length categor...

We work over an algebraically closed field k, and consider the noncommutative deformation functor Def_F of a finite family F of presheaves of modules defined over a presheaf of k-algebras A on a small category c. We develop an obstruction theory for Def_F, with certain global Hochschild cohomology groups as the natural cohomology. In particular, we...

Let k be an algebraically closed field of characteristic 0, let Gamma subset of or equal to N-0 be a numerical semigroup, and let A = k[Gamma] be the corresponding semigroup algebra. We give an explicit description of the ring D = D(A) of k-linear differential operators on A, the associated graded ring Gr D(A) and the module of derivations Der(k) (...

This paper gives an elementary introduction to noncommutative deformations of modules. The main results of this deformation theory are due to Laudal. Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M_1, ..., M_p} be a finite family of left A-modules. We study the simultaneous formal deformations...

We consider algebraic varieties X defined over C which are smooth, affine and irreducible. We study the ring D = D(X) of C-linear differential operators on X, and we explain Bernstein's theory of holonomic D-modules in this case. This is a generalization of Bernstein's original work, which covers the case when X is affine n-space and D is the n'th...

Let k be an algebraically closed eld of characteristic 0, A a Noetherian k-algebra and M a nitely generated A-module. In this paper, we investigate the possible D-module structures on M lifting the A-module structure. Existence of regular Derk(A)-connections on M (or covariant derivations on M) is a rst obstruction for lifting the structure on M. S...