Galois cohomology and forms of algebras over Laurent polynomial rings

P. GILLE AND A. PIANZOLA

MATHEMATISCHE ANNALEN

Año: 2007 vol. 338 p. 497 - 543

0025-5831

The main thrust of this work is the study of two seemingly unrelated questions: Non-abelian Galois cohomology of Laurent polynomial rings on the one hand, while on the other,  a class of infinite dimensional Lie algebras which, as rough approximations, can be thought off as higher nullity analogues of the affine Kac-Moody Lie algebras. Though the algebras in question are in general infinite dimensional over the given base field (say the complex numbers), they can be thought as being finite  provided that the base field is now replaced by  the centroid of the algebras, (which turns out to be a Laurent polynomial ring). This leads us to the theory of reductive group schemes as developed by M. Demazure and A. Grothendieck. Once this point of view is taken, Algebraic Principal Homogeneous Spaces (torsors) and their accompanying non-abelian \'etale cohomology, arise naturally.