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 (2020)
The algebra of observables in noncommutative deformation theory
Proceedings of the Estonian Academy of Sciences : Physics, Mathematics, 59(4), s. 294- 300. Doi: 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.
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.
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
[Academic lecture]. Principles of Computer Security.
Eriksen, Eivind (2007)
Examples in noncommutative deformation theory
[Academic lecture]. Algebra, Geometry, and Mathematical Physics, Baltic-Nordic Workshop.
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Computing obstructions for existence of connections on modules
[Report]. ArXiv.
Eriksen, Eivind (2006)
Computing noncommutative global deformations of D-modules
[Report]. ArXiv.
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Connections on modules over singularities of finite CM representation type
[Report]. ArXiv.
Eriksen, Eivind (2006)
Connections on modules over simple curve singularities
[Report]. ArXiv.
Eriksen, Eivind & Gustavsen, Trond S. (2006)
Connections on modules over singularities of finite CM representation type
[Academic lecture]. Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop.
Eriksen, Eivind (2006)
Noncommutative deformations of differential structures
[Academic lecture]. Algebra-seminaret ved HiBu.
Eriksen, Eivind (2006)
Non-commutative deformations of differential structures
[Academic lecture]. Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop.
Eriksen, Eivind (2006)
Konneksjoner på moduler over singulariteter
[Academic lecture]. Algebra-seminaret.
Eriksen, Eivind (2004)
A_{\infty} algebras and A_{\infty} modules
[Academic lecture]. utenTitteltekst.
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
[Academic lecture]. 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.
Eriksen, Eivind (2004)
Noncommutative deformations of sheaves of modules
[Academic lecture]. 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.
Eriksen, Eivind (2004)
Iterated extensions in module categories
[Report]. 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.
Eriksen, Eivind (2004)
Noncommutative deformations of sheaves of modules
[Report]. 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.