Illinois Journal of Mathematics

Structure of the Brauer ring of a field extension

Hiroyuki Nakaoka

Full-text: Open access

Abstract

In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.

Article information

Source
Illinois J. Math., Volume 52, Number 1 (2008), 261-277.

Dates
First available in Project Euclid: 15 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1242414131

Digital Object Identifier
doi:10.1215/ijm/1242414131

Mathematical Reviews number (MathSciNet)
MR2507244

Zentralblatt MATH identifier
1170.18003

Subjects
Primary: 18A25: Functor categories, comma categories 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

Citation

Nakaoka, Hiroyuki. Structure of the Brauer ring of a field extension. Illinois J. Math. 52 (2008), no. 1, 261--277. doi:10.1215/ijm/1242414131. https://projecteuclid.org/euclid.ijm/1242414131


Export citation

References

  • R. Boltje, Mackey functors and related structures in representation theory and number theory, Habilitation-Thesis, Universität Augsburg, 1995, Available at http://math.ucsc.edu/boltje/.
  • S. Bouc, Green functors and $G$-sets, Lecture Notes in Mathematics, vol. 1671, Springer-Verlag, Berlin (1977).
  • C. Chen, The Brauer ring of a commutative ring, Portugal. Math. 50 (1993), 163–170.
  • E.T. Jacobson, The Brauer ring of a field, Illinois J. Math. 30 (1986), 479–510.