Séminaire de Géométrie et Quantification
(Université de Nantes)
On the first cohomology of Lie-Rinehart algebras
Résumé : This is joint work in progress with Bas Janssens (Utrecht). Motivation to our work is the absence of general computational methods to tackle the Lie algebra cohomology of the Lie algebra of algebraic vector fields on algebraic varieties. In pioneering work from 2004, Serge Skryabin computes the first cohomology of subalgebras W of derivations of a commutative ring R with values in modules from some carefully constructed category under very general hypotheses (the R-module W should be finitely generated projective, differentials are generated by dR, 2 and 3 are invertible). One key step is to show that 1-cocycles are differential operators of degree at most 3, in order to have the order filtration on the cohomology space. In our work with Bas Janssens, we generalize Skyabin's methods to Lie-Rinehart algebras and manage (for the moment - the work is in progress) to show under mild conditions on the anchor that 1-cocycles are differential operators of order at most 3.