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On the homology groups of the Brauer complex for a triquadratic field extension.
- Source :
-
Mathematische Nachrichten . Feb2018, Vol. 291 Issue 2/3, p518-538. 21p. - Publication Year :
- 2018
-
Abstract
- Abstract: The homology groups h 1 ( l / k ), h 2 ( l / k ), and h 3 ( l / k ) of the Brauer complex for a triquadratic field extension l = k ( a , b , c ) are studied. In particular, given D ∈ 2 Br ( k ( a , b , c ) / k ), we find equivalent conditions for the image of <italic>D</italic> in h 2 ( l / k ) to be zero. We consider as well the second divided power operation γ 2 : 2 Br ( l / k ) → H 4 ( k , Z / 2 Z ), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h 2 ⊗ h 1 → H 3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F / k such that the cohomology group h 1 ( F ( a , b , c ) / F ) of the Brauer complex contains the images of prescribed elements of k ∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L / F with nontrivial h 3 ( L / F ) are given. As a consequence we show that the homology group h 3 ( L / F ) can be arbitrarily big. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0025584X
- Volume :
- 291
- Issue :
- 2/3
- Database :
- Academic Search Index
- Journal :
- Mathematische Nachrichten
- Publication Type :
- Academic Journal
- Accession number :
- 127968342
- Full Text :
- https://doi.org/10.1002/mana.201600022