Algebra & Number Theory

On the algebra of some group schemes

Daniel Ferrand

Full-text: Open access


The algebra of a finite group over a field k of characteristic zero is known to be a projective separable k-algebra; but these separable algebras are of a very special type, characterized by Brauer and Witt.

In contrast with that, we prove that any projective separable k-algebra is a quotient of the group algebra of a suitable group scheme, finite étale over k. In particular, any finite separable field extension KL, even a noncyclotomic one, may be generated by a finite étale K-group scheme.

Article information

Algebra Number Theory, Volume 2, Number 4 (2008), 435-466.

Received: 12 December 2007
Revised: 31 March 2008
Accepted: 6 May 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C05: Group rings of finite groups and their modules [See also 16S34]
Secondary: 14L15: Group schemes 16S34: Group rings [See also 20C05, 20C07], Laurent polynomial rings 16S35: Twisted and skew group rings, crossed products 16W30

group algebra finite étale group scheme Weil restriction separable algebra


Ferrand, Daniel. On the algebra of some group schemes. Algebra Number Theory 2 (2008), no. 4, 435--466. doi:10.2140/ant.2008.2.435.

Export citation


  • M. Artin, “Grothendieck topologies”, mimeographed notes, Harvard University, 1962.
  • M. Auslander and O. Goldman, “The Brauer group of a commutative ring”, Trans. Amer. Math. Soc. 97 (1960), 367–409.
  • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik $(3)$ 21, Springer, Berlin, 1990.
  • N. Bourbaki, Éléments de mathématique, Masson, Paris, 1981. Algèbre, chapitres IV à VII.
  • M. Demazure and P. Gabriel, Groupes algébriques, I: Géométrie algébrique, généralités, groupes commutatifs, Masson, Paris, 1970.
  • D. Ferrand, “Un foncteur norme”, Bull. Soc. Math. France 126:1 (1998), 1–49.
  • J.-M. Fontaine, “Sur la décomposition des algèbres de groupes”, Ann. Sci. École Norm. Sup. $(4)$ 4 (1971), 121–180.
  • M.-A. Knus and M. Ojanguren, Théorie de la descente et algèbres d'Azumaya, Lecture Notes in Math. 389, Springer, Berlin, 1974.
  • M. Orzech and C. Small, The Brauer group of commutative rings, Lecture Notes Pure Appl. Math. 11, Dekker, New York, 1975.
  • J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer, New York, 1977.
  • W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Math. 66, Springer, New York, 1979.
  • T. Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Math. 397, Springer, Berlin, 1974.