On Projective Modules and Automorphisms of Central Separable Algebras

Childs, L. N.

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 44-53

Résumé

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

Détail

DOI : 10.4153/CJM-1969-005-6

Bibliographie

[1] 1. Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24. Google Scholar

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

[3] 3. Bass, H., The Morita theorems (mimeographed notes, University of Oregon, 1964). Google Scholar

[4] 4. Bass, H., K-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 5–60. Google Scholar

[5] 5. Childs, L. N. and DeMeyer, F. R., On automorphisms of separable algebras, Pacific J. Math. 23 (1967), 25–34. Google Scholar

[6] 6. DeMeyer, F. R., Projective modules over central separable algebras, Can. J. Math. 21 (1969) 39–43. Google Scholar

[7] 7. Deuring, M., Algebren (Ergebn. Math. Vol. 4, No. 1, Berlin, 1937). Google Scholar

[8] 8. Eichler, M., Bestimmung der Idealklassenzahl in gewissen normalen einfachen Algebren, J. Reine Angew. Math. 176 (1937), 192–202. Google Scholar

[9] 9. Eichler, M., Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z. J+$ (1937), 102–109. Google Scholar

[10] 10. Eichler, M., Über die Idealklassenzahl hyperkomplexer Système, Math. Z. 43 (1937), 481–494. Google Scholar

[11] 11. Harada, M., Hereditary orders, Trans. Amer. Math. Soc. 107 (1963), 273–290. Google Scholar

[12] 12. Hasse, H., Über P-adische Schiefkörper und ihre Bedeutungfilr die Arithmetik hyperkomplexer Zahlsysteme, Math. Ann. 104 (1931), 495–534. Google Scholar

[13] 13. Hoobler, R. T., A generalization of the Brauer group and Amitsur cohomology (Thesis, University of California at Berkeley, 1966). Google Scholar

[14] 14. O'Meara, O. T., Introduction to quadratic forms (Academic Press, New York, 1965). Google Scholar

[15] 15. Riley, J., Reflexive ideals in maximal orders, J. Algebra 2 (1965), 451–465. Google Scholar

[16] 16. Rosenberg, A. and Zelinsky, D., Automorphisms of separable algebras, Pacific J. Math. 11 (1961), 1109–1118. Google Scholar

[17] 17. Schilling, O., Arithmetic in a special class of algebras, Ann. of Math. (2) 38 (1937), 116–119. Google Scholar

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

[19] 19. Swan, R. G., Projective modules over group rings and maximal orders, Ann. of Math. (2) 76 (1962), 55–61. Google Scholar