For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs.
Ile, Runar (2021)
Deformation theory of Cohen-Macaulay approximation
In [22] we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel we study the induced maps of deformation functors and deduce properties like smoothness and injectivity under general, mainly cohomological conditions on the module.
Gustavsen, Trond Stølen & Ile, Runar (2018)
Deformations of rational surface singularities and reflexive modules with an application to flops
Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.
Deforming syzygies of liftable modules and generalised Knörrer functors
Collectanea Mathematica, 58(3), s. 255- 277.
Ile, Runar (2004)
Change of rings in deformation theory of modules
Transactions of the American Mathematical Society, 356(12), s. 4873- 4896.
Ile, Runar (2004)
Deformation Theory of Rank 1 maximal Cohen-Macaulay Modules on Hypersurface Singularities and the Scandinavian Complex
Compositio Mathematica, 140(2), s. 435- 446.
Gustavsen, Trond Stølen & Ile, Runar (2004)
The versal deformation space of a reflexive module on a rational cone
Journal of Algebra, 279(2), s. 613- 637.
Gustavsen, TS & Ile, Runar (2004)
The versal deformation space of a reflexive module on a rational cone
Journal of Algebra, 279, s. 613- 637.
Ile, Runar (2004)
Change of rings in deformation theory of modules
Transactions of the American Mathematical Society, 356, s. 4873- 4896.
Ile, Runar (2004)
Deformation theory of rank one maximal Cohen-Macaulay modules on hypersurface singularities and the Scandinavian complex
Compositio Mathematica, 140, s. 435- 446.
Ile, Runar (2004)
Deformation theory of reflexive modules on rational surface singularities II
[Academic lecture]. utenTitteltekst.
Ile, Runar (2004)
Nøkkellemma
[Academic lecture]. utenTitteltekst.
Ile, Runar (2004)
Introduction to the Weil conjectures
[Academic lecture]. utenTitteltekst.
Ile, Runar (2004)
Deforming syzygies of liftable modules and generalised Knörrer functors
[Report]. Dept. of Math.., University of Oslo.
Gustavsen, Trond Stølen & Ile, Runar (2004)
The versal deformation space of a maximal Cohen-Macaulay module on a simple singularity
[Report]. Dept. of Math.., University of Oslo.
Ile, Runar (2003)
Change of rings in deformation theory of modules
[Report]. Dept. of Math.., University of Oslo.
Gustavsen, Trond Stølen & Ile, Runar (2003)
The versal base space of a reflexive module on a rational cone
[Report]. Universitetet i Oslo. Matematisk institutt.
By an approach based on results of A. Ishii, we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m. To each component a resolution is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, we obtain the components by projection. In particular we give equations for the minimal stratum in the Chern class filtration of the versal deformation space. We obtain a combinatorial description of the local deformation relation and a classification of the components. In particular we give a formula for the number of components.
Ile, Runar (2002)
Deformation theory of rank 1 maximal Cohen-Macaulay modules on hypersurface singularities and the Scandinavian complex
[Report]. Dept. of Math.., University of Oslo.
Academic Degrees
Year
Academic Department
Degree
2001
University of Oslo
Ph.D.
1990
University of Oslo
Candidatus scientiarum
Work Experience
Year
Employer
Job Title
2024 - Present
BI Norwegian Business School
Professor
2017 - Present
University of Oslo
External advisor to the bachelor program «Mathematics with informatics» at the Department of Mathematics
2017 - 2024
BI Norwegian Business School, Department of Economics
Associate professor
2017 - 2022
University of Bergen, Department of Mathematics
Associate professor (adjunct position)
2005 - 2016
University of Bergen, Department of Mathematics
Associate professor
2011 - 2012
KTH Royal Institute of Technology, Department of Mathematics
Visiting Associate professor (with support from the Kurt and Alice Wallenberg foundation)
1999 - 2005
University College of Hedmark
Associate professor
2003 - 2004
University of Oslo
Guest researcher under the SUPREMA program
2002 - 2003
Norwegian University of Science and Technology, Department of Mathematical Sciences
