Noncommutative descent and nonabelian cohomology.

Philippe Nuss (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We study the descent problem for modules in the case of a faithfully flat extension of noncommutative rings. We show that the modules which can be descended are exactly the modules endowed with a flat connection or equivalently, with what we call a symmetry operator. We apply this to bialgebra Galois extensions of rings and obtain a descent theorem. As an other application, we give a descent theorem for hermitian modules over an extension of noncommutative rings with involutions. In the second part of the paper, we study the nonabelian cohomology sets arising in noncommutative descent. We show that they classify as well descent data as twisted forms of certain modules. In the case of Galois extensions of noncommutative rings, we prove that these sets coincide with Galois nonabelian cohomology sets. We deduce a noncommutative version of the Hilbert's Theorem 90.

Mots clés: non commutative descent

Domaines: Anneaux et algèbres [math.RA]

Philippe Nuss. Noncommutative descent and nonabelian cohomology.. 1995. ⟨hal-00129718⟩