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Nicole Lemire, Ján Mináč, and John Swallow
When is Galois cohomology free or trivial?
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| Published: |
June 23, 2005
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| Keywords: |
Galois cohomology, Milnor K-theory, Bloch-Kato conjecture, pro-p-group, free product |
| Subject: |
12G05 (primary), 19D45 (secondary) |
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Abstract
Let p be a prime, F a field containing a primitive pth root of
unity, and E/F a cyclic extension of degree p. Using the
Bloch-Kato Conjecture we determine precise conditions for the
cohomology group Hn(E):=Hn(GE,Fp) to be free or trivial as an
Fp[\Gal(E/F)]-module, and we examine when these properties for
Hn(E) are inherited by Hk(E), k>n. By analogy with
cohomological dimension, we introduce notions of cohomological
freeness and cohomological triviality, and we give examples of
Hn(E) free or trivial for each n∈ N with prescribed
cohomological dimension.
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| Acknowledgements
The first author was supported in part by NSERC grant R3276A01.
The second author was supported in part by NSERC grant R0370A01, by the Institute for Advanced Study, Princeton, by a Distinguished Professorship during 2004-2005 at the University of Western Ontario, and by the Mathematical Sciences Research Institute, Berkeley.
The third author was supported in part by NSA grant MDA904-02-1-0061.
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| Author information
Nicole Lemire:
Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7, CANADA
nlemire@uwo.ca
Ján Mináč:
Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7, CANADA
minac@uwo.ca
John Swallow:
Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046, USA
joswallow@davidson.edu
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