Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T00:35:29.218Z Has data issue: false hasContentIssue false
  • English
  • Français

On Inflation and Torsion of Amitsur Cohomology

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
Rutgers University New Brunswick, New Jersey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Studies of torsion [2] and inflation [11] of Amitsur cohomology have primarily been concerned with module-finite faithful projective algebras. In this paper, our goal is to consider these topics for more general algebras. The fundamental tool, in case R is a domain with quotient field K, is the functor UK/U (defined in § 1), together with the monomorphic connecting map of Amitsur cohomology H1(S/R, UK/U) → H2(S/R, U) arising from a (commutative) flat R-algebra S and the unit functor U. This map was first considered in [10, Chapter IV, Theorem 1.6], where it was proved to be an isomorphism for certain étale faithfully flat algebras S in case R is an algebraic number ring with trivial Brauer group. In Corollary 1.5 it is shown, in the case of a module-finite faithful projective S over a regular domain R, that the connecting map is the kernel of the canonical homomorphism [8] from H2(S/R, U) to the split Brauer group B(S/R). As the latter map is often injective [8, Corollary 7.7], one might expect instances of vanishing of H1(S/R, UK/U) without assuming both R Noetherian and S module-finite R-projective. Much of § 1 (viz. (1.7)-(1.9)) is devoted to such examples.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 239 - 260
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Amitsur, S. A., Homology groups and double complexes for arbitrary fields, J. Math. Soc. Japan 14 (1962), 1–25.Google Scholar
2. Amitsur, S. A., Complexes of rings, Israel J. Math. 2 (1964), 143–154.Google Scholar
3. Artin, M., Grothendieck topologies, mimeographed notes (Harvard University, Cambridge, Mass., 1962).Google Scholar
4. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.Google Scholar
5. Bourbaki, N., Algèbre commutative, chapitres 1-2 (Act. scient, et ind. 1290) (Hermann, Paris, 1962).Google Scholar
6. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956).Google Scholar
7. Chase, S. U., Harrison, D. K., and Rosenberg, A., Galois theory and Galois cohomology of commutative rings, Memoirs Amer. Math. Soc, No. 52 (Amer. Math. Soc, Providence, R.I., 1965).Google Scholar
8. Chase, S. U. and Rosenberg, A., Amitsur cohomology and the Brauer group, Memoirs Amer. Math. Soc, No. 52 (Amer. Math. Soc, Providence, R.I., 1965).Google Scholar
9. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, New York, 1962).Google Scholar
10. Dobbs, D. E., Cech cohomological dimensions for commutative rings (Springer-Verlag, Berlin, 1970).Google Scholar
11. Dobbs, D. E., Amitsur cohomology of algebraic number rings, Pacific J. Math, (to appear).Google Scholar
12. Garfinkel, G., Amitsur cohomology and an exact sequence involving Pic and the Brauer group,. Ph.D. thesis, Cornell University, Ithaca, 1968.Google Scholar
13. Goldman, O., Determinants in projective modules, Nagoya Math. J. 18 (1961), 27–36.Google Scholar
14. Morris, R. A., The reciprocity isomorphisms of class field theory for separable field extensions,, Ph.D. thesis, Cornell University, Ithaca, 1970.Google Scholar
15. Rosenberg, A. and Zelinsky, D., Amitsur)s complex for inseparable fields, Osaka Math. J. 14 (1962), 219–240.Google Scholar
16. Serre, J.-P., Corps locaux (Act. scient, et ind. 1296) (Hermann, Paris, 1962).Google Scholar
17. Silver, L., Tame orders, tame ramification and Galois cohomology, Illinois J. Math. 12 (1968), 7–34.Google Scholar
18. Zariski, O. and Samuel, P., Commutative algebra, vol. I (Van Nostrand, Princeton, 1958).Google Scholar