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

2017 - Present

BI Norwegian Business School, Department of Economics

Associate professor

2017 - Present

University of Bergen, Department of Mathematics

Associate professor (adjunct position)

2017 - Present

University of Oslo

External advisor to the bachelor program «Mathematics with informatics» at the Department of Mathematics

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