In this paper, we study length categories using iterated extensions. We fix a field k, and for any family S of orthogonal k-rational points in an Abelian k-category A, we consider the category Ext(S) of iterated extensions of S in A, equipped with the natural forgetful functor Ext(S) → A(S) into the length category A(S). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects in A(S) when it is a uniserial length
category. In particular, we prove that there is an bstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family S in A. As an application, we classify all graded holonomic D-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when D is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions
and show that they do not vanish.
Eriksen, Eivind & Siqveland, Arvid (2019)
The algebra of observables in noncommutative deformation theory
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-modules, and k is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field k. Secondly, we prove that the O-construction is a closure operation when A is any finitely generated k-algebra and M is any family of finite dimensional A-modules, in the sense that ηB : B → OB(M) is an isomorphism when B = O(M) and M is considered as a family of B-modules.
Eriksen, Eivind; Laudal, Olav Arnfinn & Siqveland, Arvid (2017)
There has been several attempts to generalize commutative algebraic geometry to the noncommutative situation. Localizations with good properties rarely exists for noncommutative algebras, and this makes a direct generalization di cult. Our point of view, following Laudal, is that the points of the noncommutative geometry should be represented as simple modules, and that noncommutative deformations should be used to obtain a suitable localization in the noncommutative situation. Let A be an algebra over an algebraically closed eld k. If A is commutative and nitely generated over k, then any simple A-module has the form M = A=m, the residue eld, for a maximal ideal m A, and the commutative deformation functor DefM has formal moduli A^m. In the general case, we may replace the A-module A=m with the simple A-module M, and use the formal moduli of the commutative deformation functor DefM as a replacement for the complete local ring A^m. We recall the construction of the commutative scheme simp(A), with points in bijective correspondence with the simple A-modules of nite dimension over k, and with complete local ring at a point M isomorphic to the formal moduli of the corresponding simple module M. The scheme simp(A) has good properties, in particular when there are no in nitesimal relations between di erent points, i.e. when Ext1 A(M;M0) = 0 for all pairs of non-isomorphic simple A-modules M;M0. It does not , however, characterize A. We use noncommutative deformation theory to de ne localizations, in general, and we nd a presheaf O, of noncommutative algebras, de ned on the Jacobson topology of simp(A) which re nes the commutative scheme, simp(A), by accounting for the in nitesimal relations in simp(A). We consider the quantum plane, given by A = khx; yi=(xy qyx), as an example. This is an Artin-Schelter algebra of dimension two.
Eriksen, Eivind (2011)
The Generalized Burnside Theorem in Noncommutative Deformation Theory
Let A be an associative algebra over a field k, and let be a finite family of right A-modules. A study of the noncommutative deformation functor Def of the family leads to the construction of the algebra A() of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.
Eriksen, Eivind (2010)
Computing Noncommutative Deformations of Presheaves and Sheaves of Modules
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 presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases.
In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space (X,A) . We show that for any good A -affine open cover U of X , the forgetful functor QCohA→PreSh(U,A) induces an isomorphism of noncommutative deformation functors.
Applications. We consider noncommutative deformations of quasi-coherent A -modules on X when (X,A)=(X,O X ) is a scheme or (X,A)=(X,D) is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of X closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left D X -module D X when X is an elliptic curve as an example.
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.
Eriksen, Eivind & Gustavsen, Trond Stølen (2009)
Lie-Rinehart cohomology and integrable connections on modules of rank one
Journal of Algebra, 322(12) , s. 4283-4294.
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 . We consider the Lie-Rinehart cohomology with values in EndR(M) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d2, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.
Eriksen, Eivind & Gustavsen, Trond (2009)
Lie-Rinehart cohomology and integrable connections on modules of rank one
Computing obstructions for existence of connections on modules
Journal of symbolic computation, 42, s. 313-323.
Sørhus, Vidar; Eriksen, Eivind, Grønningsæter, Nils, Halbwachs, Yvon, Hvidsten, Per Øyvind, Strøm, Kyrre, Westgaard, Geir & Røtnes, Jan Sigurd (2005)
A new platform for laparoscopic training and education
Studies in Health Technology and Informatics, 111, s. 502-507.
Eriksen, Eivind (2019)
Noncommutative deformations and iterated extensions
[Conference Lecture]. Event
Eriksen, Eivind (2017)
Matematikk for økonomi og finans. Oppgaver og løsningsforslag.
[Textbook]. Cappelen Damm Akademisk
Eriksen, Eivind (2016)
Matematikk for økonomi og finans
[Textbook]. Cappelen Damm Akademisk
Eriksen, Eivind & Fausk, Halvard (2014)
Mattenøkkelen
[Textbook]. Gyldendal Akademisk
Eriksen, Eivind (2013)
Noncommutative deformations and geometry of simple modules
[Conference Lecture]. Event
Eriksen, Eivind (2013)
Simple modules over matric algebras and their geometry
[Conference Lecture]. Event
Eriksen, Eivind (2012)
A-infinity algebras and noncommutative deformations
[Conference Lecture]. Event
Eriksen, Eivind (2010)
The Weyl algebra is coherent
[Report Research]. Cornell University
Eriksen, Eivind (2009)
The Generalized Burnside Theorem in noncommutative deformation theory
[Conference Lecture]. Event
Eriksen, Eivind & Gustavsen, Trond S. (2008)
Lie-Rinehart cohomology and integrable connections on modules of rank one
[Report Research]. Arxiv http://arxiv.org
Eriksen, Eivind (2008)
Integrable connections on modules of rank one
[Conference Lecture]. Event
Eriksen, Eivind & Gustavsen, Trond S. (2008)
Equivariant Lie-Rinehart cohomology
[Report Research]. Arxiv http://arxiv.org
Eriksen, Eivind (2007)
Kryptografi og elliptiske kurver
[Conference Lecture]. Event
Eriksen, Eivind (2007)
Examples in noncommutative deformation theory
[Conference Lecture]. Event
Eriksen, Eivind (2006)
Noncommutative deformations of differential structures
[Conference Lecture]. Event
Eriksen, Eivind (2006)
Connections on modules over simple curve singularities
[Report Research]. ArXiv
Eriksen, Eivind (2006)
Konneksjoner på moduler over singulariteter
[Conference Lecture]. Event
Eriksen, Eivind (2006)
Non-commutative deformations of differential structures
[Conference Lecture]. Event
Eriksen, Eivind (2006)
Computing noncommutative global deformations of D-modules
[Report Research]. ArXiv
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Computing obstructions for existence of connections on modules
[Report Research]. ArXiv
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Connections on modules over singularities of finite CM representation type
[Report Research]. ArXiv
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Connections on modules over singularities of finite CM representation type
[Conference Lecture]. Event
Eriksen, Eivind (2004)
Noncommutative deformations of sheaves of modules
[Conference Lecture]. Event
Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F_p a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def_F: a_p -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F_i is a quasi-coherent left A-module for all i. Then Def_F has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F_i) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F_i is coherent for all i, then H^n(U,F_j,F_i) is isomorphic to Ext^n_A(F_j,F_i) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.
Eriksen, Eivind (2004)
A_{\infty} algebras and A_{\infty} modules
[Conference Lecture]. Event
We introduced the notions of A_{\infty} algebras and A_{\infty} modules, and discussed some relations to Massey products and deformation theory.
Eriksen, Eivind (2004)
Noncommutative deformations of sheaves of modules
[Conference Lecture]. Event
Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def: a -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F is a quasi-coherent left A-module for all i. Then Def has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F is coherent for all i, then H^n(U,F_j,F) is isomorphic to Ext^n(F_j,F) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.
Eriksen, Eivind (2004)
Noncommutative deformations of sheaves of modules
[Report Research]. Institut Mittag-Leffler
We introduce noncommutative deformation theory for finite families of quasi-coherent sheaves of modules on certain noncommutative ringed spaces (X,A), and describe the corresponding obstruction theory.
A comprehensive platform for laparoscopic education
[Conference Lecture]. Event
Eriksen, Eivind (2004)
Iterated extensions in module categories
[Report Research]. Cornell University
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 Fand is closed under extensions, is a full exact Abelian subcategory of R-Mod and a length category in the sense of Gabriel. In this paper, we use iterated extensions to relate the length category Mod(F) to noncommutative deformations of modules, and use some new methods to study Mod(F) via iterated extensions. In particular, we give a new proof of the characterization of uniserial length categories, which is constructive. As an application, we give an explicit description of some categories of holonomic and regular holonomic D-modules on curves which are uniserial length categories.
