×

Corestriction of Galois algebras. (English) Zbl 0737.16010

The author studies the corestriction map of C. Riehm (introduced in Invent. Math. 11, 73–98 (1970; Zbl 0199.34904)]). This map \(\hbox{Cor}_{L/K}: \text{Br}(L)\to \text{Br}(K)\), which maps the Brauer group of \(L\) to that of \(K\) for \(L/K\) finite separable, is given in a purely algebra- theoretic manner by assigning to every \(L\)-algebra \(A\), a \(K\)-algebra \(\text{Cor}_{L/K}(A)\) in a functorial manner. The author studies Riehm’s mapping for algebras carrying a Galois structure with respect to a finite dimensional Hopf algebra (a so-called Galois algebra introduced by S. U. Chase and M. E. Sweedler for commutative algebras [Hopf algebras and Galois theory (Lect. Notes Math. 97, 1969; Zbl 0197.01403)]). First the author deals with the behavior of Galois algebras under change of the Hopf algebra, then studies the corestriction of such algebras. This work constitutes part of the author’s thesis at Universität München (1989; Zbl 0731.16028).

MSC:

16K20 Finite-dimensional division rings
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
14F22 Brauer groups of schemes
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chase, S. U., On a variant of the Witt and Brauer groups, (Brauer Groups, Evanston, 1975. Brauer Groups, Evanston, 1975, Lecture Notes in Mathematics, Vol. 549 (1976), Springer-Verlag: Springer-Verlag New York/Berlin), 148-187
[2] Chase, S. U.; Sweedler, M. E., Hopf algebras and Galois theory, (Lecture notes in Mathematics, Vol. 97 (1966), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0197.01403
[3] Childs, L. N., Cyclic Stickelberger cohomology and descent of Kummer extensions, (Proc. Amer. Math. Soc., 90 (1984)), 505-510 · Zbl 0538.12013
[4] Draxl, P. K., Skew fields, (London Math. Soc. Note Series, Vol. 81 (1983), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0498.16015
[5] Early, T. E.; Kreimer, H. F., Galois algebras and Harrison cohomology, J. Algebra, 58, 136-147 (1979) · Zbl 0411.16027
[6] Garfinkel, G.; Orzech, M., Galois extensions as modules over the group ring, Canad. J. Math., 22, 242-248 (1968) · Zbl 0197.03401
[7] Greither, C., Cyclic Galois Extensions and Normal Bases, (Habilitationsschrift (1988), Universität München) · Zbl 0743.11060
[8] Greither, C.; Pareigis, B., Hopf Galois theory for separable field extensions, J. Algebra, 106, 239-258 (1987) · Zbl 0615.12026
[9] Harrison, D. K., Abelian extensions of commutative rings, Mem. Amer. Math. Soc., 52, 1-14 (1965) · Zbl 0143.06003
[10] Hasse, H., Die Multiplikationsgruppe der abelschen Körper mit fester Galois-Gruppe, Hamb. Abh., 16, 29-40 (1949) · Zbl 0039.26801
[11] Merkurjev, A. S., On the structure of the Brauer group of fields, Math. USSR Izv., 27, No. 1, 141-157 (1986) · Zbl 0598.12022
[12] Pareigis, B., Non-additive ring and module theory. IV. The Brauer group of a symmetric monoidal category, (Brauer Groups, Evanston, 1975. Brauer Groups, Evanston, 1975, Lecture Notes in Mathematics, Vol. 549 (1976), Springer-Verlag: Springer-Verlag New York/Berlin), 112-133
[13] Riehm, C., The corestriction of algebraic structures, Invent. Math., 11, 73-98 (1970) · Zbl 0199.34904
[14] Serre, J. P., Cohomologie Galoisienne, (Lecture Notes in Mathematics, Vol. 5 (1973), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0812.12002
[15] Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin New York · Zbl 0194.32901
[16] Tignol, J.-P, On the corestriction of central simple algebras, Math. Z., 194, 267-274 (1987) · Zbl 0595.16012
[17] Waterhouse, W. C., Introduction to affine group schemes, (Graduate Text in Mathematics, Vol. 66 (1979), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0212.25602
[18] Wenninger, C. H., Galois-Algebren zu Hopf-Algebren und verallgemeinerte Quaternionen, (Dissertation (1989), Universität München) · Zbl 0731.16028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.