arXiv subjects

Runar Ile

Publications and source records attributed to Runar Ile.

5 recordsLinked to original sources

arXiv2019

Versal deformations of pairs and Cohen-Macaulay approximation

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 Andr\'e-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. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions imply properties like smoothness, isomorphism and linear section.

math.AC↗

arXiv2019

Deformation theory of Cohen-Macaulay approximation

In a previous article (J. Algebra 367 (2012), 142-165) 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.

math.RT↗

arXiv2016

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.

math.AG↗

arXiv2013

Stably reflexive modules and a lemma of Knudsen

In his fundamental work on the stack of stable n-pointed genus g curves, Finn F. Knudsen introduced the concept of a stably reflexive module in order to prove a key technical lemma. We propose an alternative definition and generalise the results in the appendix to his article. Then we give a `coordinate free' generalisation of his lemma, generalise a construction used in Knudsen's proof concerning versal families of pointed algebras, and show that Knudsen's stabilisation construction works for plane curve singularities. In addition we prove approximation theorems generalising Cohen-Macaulay approximation with stably reflexive modules in flat families. The generalisation is not covered (even in the closed fibres) by the Auslander-Buchweitz axioms.

math.AC↗

arXiv2011

Cohen-Macaulay approximation in fibred categories

We extend the Auslander-Buchweitz axioms and prove Cohen-Macaulay approximation results for fibred categories. Then we show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In particular such a pair admits an approximation with a flat family of maximal Cohen-Macaulay modules and a hull with a flat family of modules with finite injective dimension. The existence of minimal approximations and hulls in the local, flat case implies extension of upper semi-continuous invariants. As an example of MCM approximation we define a relative version of Auslander's fundamental module. In the second part we study the induced maps of deformation functors and deduce properties like injectivity, isomorphism and smoothness under general, mainly cohomological conditions on the module. We also provide deformation theory for pairs (algebra, module), e.g. a cohomology for such pairs, a long exact sequence linking this cohomology to the Andr\'e-Quillen cohomology of the algebra and the Ext cohomology of the module, Kodaira-Spencer classes and maps including a secondary Kodaira-Spencer class, and existence of a versal family for pairs with isolated singularity.

math.AC↗