Universal Galois algebras and cohomology of p-groups

Fleischmann, Peter and Woodcock, Chris F. (2013) Universal Galois algebras and cohomology of p-groups. Journal of Pure and Applied Algebra, 217 (3). pp. 530-545. ISSN 0022-4049. (The full text of this publication is not available from this repository)

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Let G be a finite p-group and k a field of characteristic p>0. A universal Galois algebra of G is a weakly initial object in the category Ts of trace-surjective (commutative) k−G algebras. The objects in Ts are precisely the k−G-algebras that are Galois ring extensions over the ring of G-invariants. They are also characterized as k−G algebras which are free kG-modules. One example is Dk, the dehomogenized symmetric algebra of the regular representation, which is also an s-projective object in Ts (see the definition in the paper). In the previous work we proved that the polynomial ring Dk (of dimension ∣G∣−1) contains a polynomial retract U∈Ts of dimension , such that the invariant rings UG and are again polynomial rings. The G-action on U will in general be highly non-linear, but in special cases it can be chosen to be “almost linear”. In this paper we investigate such almost linear universal algebras, generalizing the construction of the algebras Dk and U. It is known that for k=Fp the minimal dimension of a polynomial universal algebra is n. Among other things we prove that such an algebra can be realized in an “almost linear” way, if and only if G is “crossed isomorphic” to an Fp-vector space. This is equivalent to the existence of a kG-module V such that there is 0≠[ρ]∈H1(G,V) with ρ∈Z1(G,V) being a bijective cocycle.

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Pure Mathematics
Depositing User: Chris F Woodcock
Date Deposited: 09 Oct 2012 16:51
Last Modified: 29 Apr 2014 14:49
Resource URI: http://kar.kent.ac.uk/id/eprint/31465 (The current URI for this page, for reference purposes)
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