Employee Profile

Eivind Eriksen

Professor

Department of Economics

Room number B3Y-085
Phone +4747408701
E-mail eivind.eriksen@bi.no

Biography

I am a norwegian mathematician. My research is in algebra and algebraic geometry, and I am particularly interested in the following topics:

Noncommutative algebraic geometry.
Moduli problems and (noncommutative) deformation theory.
Differential structures in algebraic geometry (rings of differential operators, connections, D-modules).
Representation theory.

For further information, please see my personal homepage.

Area of Expertise

Applied mathematics
Mathematics

Publications

Shows 5 of 19 publication(s)

Article Eivind Eriksen (2020)

Iterated Extensions and Uniserial Length Categories

Algebras and Representation Theory 24 p. 273-286 Doi: https://doi.org/10.1007/s10468-020-09946-0

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.

Article Eivind Eriksen, Arvid Siqveland (2019)

The algebra of observables in noncommutative deformation theory

Journal of Algebra 547 p. 162-172 Doi: https://doi.org/10.1016/j.jalgebra.2019.10.057

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.

Anthology Eivind Eriksen, Olav Arnfinn Laudal, Arvid Siqveland (2017)

Noncommutative Deformation Theory

CRC Press

Chapter Eivind Eriksen (2014)

Computing Noncommutative Deformations

Algebra, Geometry and Mathematical Physics p. 285-291 Doi: https://doi.org/10.1007/978-3-642-55361-5_17

Article Eivind Eriksen, Arvid Siqveland (2011)

Geometry of Noncommutative Algebras

Banach Center Publications 93 p. 69-82 Doi: https://doi.org/10.4064/bc93-0-6

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.

Article Eivind Eriksen (2011)

The Generalized Burnside Theorem in Noncommutative Deformation Theory

Journal of Generalized Lie Theory and Applications 5 Doi: https://doi.org/10.4303/jglta/G110109

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.

Article Eivind Eriksen (2010)

Computing Noncommutative Deformations of Presheaves and Sheaves of Modules

Canadian Journal of Mathematics (CJM) 62(3) p. 520-542 Doi: https://doi.org/10.4153/CJM-2010-015-6

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.

Article Eivind Eriksen, Trond Stølen Gustavsen (2010)

Equivariant Lie-Rinehart cohomology

Proceedings of the Estonian Academy of Sciences : Physics, Mathematics 59(4) p. 294-300 Doi: https://doi.org/10.3176/proc.2010.4.07

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.

Article Eivind Eriksen, Trond Stølen Gustavsen (2009)

Lie-Rinehart cohomology and integrable connections on modules of rank one

Journal of Algebra 322(12) p. 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.

Article Eivind Eriksen, Trond Gustavsen (2009)

Lie-Rinehart cohomology and integrable connections on modules of rank one

Journal of Algebra 322(12) p. 4283-4294 Doi: https://doi.org/10.1016/j.jalgebra.2009.09.015

Article Eivind Eriksen, Trond S. Gustavsen (2008)

Connections on modules over singularities of finite CM representation type

Journal of Pure and Applied Algebra 212(7) p. 1561-1574 Doi: https://doi.org/10.1016/j.jpaa.2007.10.008

Chapter Eivind Eriksen, Trond S. Gustavsen (2008)

Connections on modules over singularities of finite and tame CM representation type

Generalized Lie Theory in Mathematics, Physics and Beyond p. 99-108

Article Eivind Eriksen (2008)

An example of noncommutative deformations

Journal of Generalized Lie Theory and Applications 2(3) p. 52-56

Article Eivind Eriksen (2008)

Connections on Modules Over Quasi-Homogeneous Plane Curves

Communications in Algebra 36(8) p. 3032-3041

Academic book Sergei Silvestrov, Eugen Paal, Viktor Abramov, Alexander Stolin, Eivind Eriksen, Trond S. Gustavsen (2008)

Generalized Lie Theory in Mathematics, Physics and Beyond

Springer

Chapter Eivind Eriksen (2008)

Computing noncommutative global deformations of D-modules

Generalized Lie Theory in Mathematics, Physics and Beyond p. 109-117

Article Eivind Eriksen, TS Gustavsen (2007)

Computing obstructions for existence of connections on modules

Journal of symbolic computation 42 Doi: https://doi.org/10.1016/j.jsc.2006.10.001

Article Eivind Eriksen, Trond Stølen Gustavsen (2007)

Computing obstructions for existence of connections on modules

Journal of symbolic computation 42 p. 313-323

Article Vidar Sørhus, Eivind Eriksen, Nils Grønningsæter, Yvon Halbwachs, Per Øyvind Hvidsten, Kyrre Strøm, Geir Westgaard, Jan Sigurd Røtnes (2005)

A new platform for laparoscopic training and education

Studies in Health Technology and Informatics 111 p. 502-507

Shows 5 of 30 publication(s)

Conference lecture Eivind Eriksen (2019)

Noncommutative deformations and iterated extensions

Deformation theory, Hopf algebras and mathematical physics

Textbook Eivind Eriksen (2017)

Matematikk for økonomi og finans. Oppgaver og løsningsforslag.

Cappelen Damm Akademisk

Textbook Eivind Eriksen (2016)

Matematikk for økonomi og finans

Cappelen Damm Akademisk

Textbook Eivind Eriksen, Halvard Fausk (2014)

Mattenøkkelen

Gyldendal Akademisk

Conference lecture Eivind Eriksen (2013)

Noncommutative deformations and geometry of simple modules

Algebra, Combinatorics, Dynamics and Applications

Conference lecture Eivind Eriksen (2013)

Simple modules over matric algebras and their geometry

MODULI OPERADS DYNAMICS

Conference lecture Eivind Eriksen (2012)

A-infinity algebras and noncommutative deformations

Conference AGMP-8 Brno'12

Report Eivind Eriksen (2010)

The Weyl algebra is coherent

Cornell University

Conference lecture Eivind Eriksen (2009)

The Generalized Burnside Theorem in noncommutative deformation theory

Algebra, Geometry, and Mathematical Physics, 5th Baltic-Nordic AGMP Workshop

Report Eivind Eriksen, Trond S. Gustavsen (2008)

Lie-Rinehart cohomology and integrable connections on modules of rank one

Arxiv http://arxiv.org

Conference lecture Eivind Eriksen (2008)

Integrable connections on modules of rank one

Algebra, Geometry, and Mathematical Physics. 4th Baltic-Nordic Workshop: Tartu, 09-11 October, 2008

Report Eivind Eriksen, Trond S. Gustavsen (2008)

Equivariant Lie-Rinehart cohomology

Arxiv http://arxiv.org

Conference lecture Eivind Eriksen (2007)

Examples in noncommutative deformation theory

Algebra, Geometry, and Mathematical Physics, Baltic-Nordic Workshop

Conference lecture Eivind Eriksen (2007)

Kryptografi og elliptiske kurver

Principles of Computer Security

Conference lecture Eivind Eriksen (2006)

Noncommutative deformations of differential structures

Algebra-seminaret ved HiBu

Report Eivind Eriksen (2006)

Connections on modules over simple curve singularities

ArXiv

Conference lecture Eivind Eriksen (2006)

Konneksjoner på moduler over singulariteter

Algebra-seminaret

Conference lecture Eivind Eriksen (2006)

Non-commutative deformations of differential structures

Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop

Report Eivind Eriksen, Trond S. Gustavsen (2006)

Computing obstructions for existence of connections on modules

ArXiv

Report Eivind Eriksen, Trond S. Gustavsen (2006)

Connections on modules over singularities of finite CM representation type

ArXiv

Report Eivind Eriksen (2006)

Computing noncommutative global deformations of D-modules

ArXiv

Conference lecture Eivind Eriksen, Trond S. Gustavsen (2006)

Connections on modules over singularities of finite CM representation type

Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop

Working paper Eivind Eriksen (2006)

Lineær Algebra og Vektorrom

Eivind Eriksen

Kompendiet er skrevet for og brukt i undervisningen ved kurset FO-008 Lineær Algebra (HiO/IU V2006).

Introduction Eivind Eriksen (2005)

An introduction to noncommutative deformations of modules

Noncommutative Algebra and Geometry p. 90-126

In this paper, we give an introduction to the theory of noncommutative deformations of modules. This theory was introduced by O.A. Laudal.

Conference lecture Eivind Eriksen (2004)

Noncommutative deformations of sheaves of modules

utenTitteltekst

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.

Conference lecture Eivind Eriksen (2004)

A_{\infty} algebras and A_{\infty} modules

utenTitteltekst

We introduced the notions of A_{\infty} algebras and A_{\infty} modules, and discussed some relations to Massey products and deformation theory.

Conference lecture Eivind Eriksen (2004)

Noncommutative deformations of sheaves of modules

utenTitteltekst

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.

Report Eivind Eriksen (2004)

Noncommutative deformations of sheaves of modules

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.

Conference lecture Vidar Sørhus, Eivind Eriksen, Nils Grønningsæter, Yvon Halbwachs, Per Øyvind Hvidsten, Johannes Kaasa, Kyrre Strøm, Geir Westgaard, Jan Sigurd Røtnes (2004)

A comprehensive platform for laparoscopic education

SMIT Conference

Report Eivind Eriksen (2004)

Iterated extensions in module categories

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.

Academic Degrees
Year	Academic Department	Degree
2000	University of Oslo	Ph.D Dr. Scient.
1994	University of Oslo	Master Cand. Scient.
1992	University of Oslo	Cand.Mag

Work Experience
Year	Employer	Job Title
2019 - Present	BI Norwegian Business School	Professor
2010 - 2019	BI Norwegian Business School	Associate Professor
2005 - 2010	Oslo University College	Associate Professor
2004 - 2005	Buskerud University College	Associate Professor
2003 - 2004	University of Oslo	NRC Postdoctoral Fellow
2001 - 2003	University of Warwick	Marie Curie Postdoctoral Fellow
1995 - 2000	University of Oslo	Research Fellow