On Inflation and Torsion of Amitsur Cohomology

Dobbs, David E.

Canadian journal of mathematics, Tome 24 (1972) no. 2, pp. 239-260

Résumé

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.

Détail

DOI : 10.4153/CJM-1972-019-2

Bibliographie

[1] 1. Amitsur, S. A., Homology groups and double complexes for arbitrary fields, J. Math. Soc. Japan 14 (1962), 1–25. Google Scholar

[2] 2. Amitsur, S. A., Complexes of rings, Israel J. Math. 2 (1964), 143–154. Google Scholar

[3] 3. Artin, M., Grothendieck topologies, mimeographed notes (Harvard University, Cambridge, Mass., 1962). Google Scholar

[4] 4. Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409. Google Scholar

[5] 5. Bourbaki, N., Algèbre commutative, chapitres 1-2 (Act. scient, et ind. 1290) (Hermann, Paris, 1962). Google Scholar

[6] 6. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956). Google Scholar

[7] 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] 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] 9. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, New York, 1962). Google Scholar

[10] 10. Dobbs, D. E., Cech cohomological dimensions for commutative rings (Springer-Verlag, Berlin, 1970). Google Scholar

[11] 11. Dobbs, D. E., Amitsur cohomology of algebraic number rings, Pacific J. Math, (to appear). Google Scholar

[12] 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] 13. Goldman, O., Determinants in projective modules, Nagoya Math. J. 18 (1961), 27–36. Google Scholar

[14] 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] 15. Rosenberg, A. and Zelinsky, D., Amitsur)s complex for inseparable fields, Osaka Math. J. 14 (1962), 219–240. Google Scholar

[16] 16. Serre, J.-P., Corps locaux (Act. scient, et ind. 1296) (Hermann, Paris, 1962). Google Scholar

[17] 17. Silver, L., Tame orders, tame ramification and Galois cohomology, Illinois J. Math. 12 (1968), 7–34. Google Scholar

[18] 18. Zariski, O. and Samuel, P., Commutative algebra, vol. I (Van Nostrand, Princeton, 1958). Google Scholar