Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COHOMOLOGY GROUPS OF RADICAL EXTENSIONS

  • Published : 2007.01.31
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Abstract

If k is a subfield of $\mathbb{Q}(\varepsilon_m)$ then the cohomology group $H^2(k(\varepsilon_n)/k)$ is isomorphic to $H^2(k(\varepsilon_{n'})/k)$ with gcd(m, n') = 1. This enables us to reduce a cyclotomic k-algebra over $k(\varepsilon_n)$ to the one over $k(\varepsilon_{n'})$. A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.

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References

  1. E. Aljadeff and J. Sonn, Projective Schur algebras have abelian splitting fields, J. Algebra 175 (1995), no. 1, 179-187 https://doi.org/10.1006/jabr.1995.1181
  2. E. Aljadeff and J. Sonn, Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras, J. Algebra 220 (1999), no. 2, 401-414 https://doi.org/10.1006/jabr.1999.7860
  3. A. Babakhanian, Cohomological methods in group theory, Marcel Dekker Inc., New York, 1972
  4. E. Choi and H. Lee, Crossed product algebras over cyclotomic extension fields, Math. Japon. 50 (1999), no. 2, 207-210
  5. E. Choi and H. Lee, The crossed product theorem for projective Schur algebras, Glasg. Math. J. 43 (2001), no. 1, 135-143
  6. G. Janusz, The Schur group of cyclotomic fields, J. Number Theory 7 (1975), no. 3, 345-352 https://doi.org/10.1016/0022-314X(75)90025-6
  7. G. Karpilovsky, Projective representations of finite groups, Monographs and Textbooks in Pure and Applied Mathematics, 94, Marcel Dekker Inc., New York, 1985
  8. G. Karpilovsky, Field Theory, Monographs and Textbooks in Pure and Applied Mathematics, 120, Marcel Dekker Inc., New York, 1988
  9. F. Lorenz and H. Opolka, Einfache Algebren und projektive Darstellungen Äuber ZahlkÄorpern, Math. Z. 162 (1978), no. 2, 175-182 https://doi.org/10.1007/BF01215073
  10. I. Reiner, Maximal orders, London Math. Soc. Monogr. Ser. 5, London, Academic Press, New York, 1975
  11. J. P. Serre, Local Fields, Grad. Text. Math. 67, Springer-Verlag, NewYork, 1979
  12. E. Weiss, Cohomology of groups , Academic press, New York, 1969
  13. T. Yamada, The Schur subgroup of the Brauer group, Lect. Notes Math. 397, Springer- Verlag, NewYork, 1974

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