Runar Ile

Journal of Pure and Applied Algebra 229(1) Doi: https://doi.org/10.1016/j.jpaa.2024.107779

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.

Journal of Algebra 568 s. 437-466 Doi: https://doi.org/10.1016/j.jalgebra.2020.09.039

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.

Advances in Mathematics 340 s. 1108-1140 Doi: https://doi.org/10.1016/j.aim.2018.10.023

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.

Journal of Algebra 397 s. 141-167 Doi: https://doi.org/10.1016/j.jalgebra.2013.08.024

Journal of Algebra 367 s. 142-165 Doi: https://doi.org/10.1016/j.jalgebra.2012.06.006

utenTitteltekst

utenTitteltekst

utenTitteltekst

Universitetet i Oslo

Several maps of deformation functors of modules are given which generalise the maps induced by the Knörrer functors. These maps become isomorphisms after introducing linear equations in the target functor. In this manner the obstruction ideal for one module occurs as obstruction ideals for other modules over other rings. In particular we obtain a map to the deformation functor of the maximal Cohen-Macaulay approximation over a quotient ring defined by a regular sequence.

Universitetet i Oslo

We prove that the versal deformation space R of a (not necessarily indecomposable) maximal Cohen-Macaulay module M on a simple singularity X of even dimension is irreducible (in contrast to the odd dimensional case). Assuming dim X even, we show that Sing Rred has one or two components, and we give the codimension. We give further properties of the local deformation relation, in particular we completely describe how any indecomposable M (locally) deforms. In dimension two the proofs proceed by investigating a partial order defined by the intersection form, applying the McKay correspondence and an existence result of A. Ishii. The general case follows by applying the functor introduced by H. Knörrer.