## Séminaire de Géométrie et Quantification |

Friedrich
Wagemann

(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.