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Hopf algebra forms of the multiplicative group and other groups. (English) Zbl 0604.16005
Let k be a commutative ring and G a finitely generated group with finite group of automorphisms F. In this paper, the authors determine Hopf(kG), the set of Hopf algebra forms of kG, i.e. the set of bialgebras H over k which, after faithfully flat ring extension L, become isomorphic to \(kG\otimes_ kL\cong LG\). By using the descent theory of A. Grothendieck [Sém. Bourbaki 12, 1959/60, Exp. 190 (1960; Zbl 0229.14007)] or R. Haggenmüller [”Über Invarianten separabler Galoiserweiterungen kommutativer Ringe” (Dissertation, Univ. München 1979)] and the fact that \(Hopf\)-Aut\({}_ L(LG)\) is isomorphic to V(LF), the set of group-like elements of LF, and thus also to \(Gal\)-Aut\({}_ k(E^ F_ k)\) where \(E^ F_ k\) is the trivial F-Galois extension, the authors show that there is a bijection between Hopf(kG) and the set of F-Galois extensions of k. Explicitly, if K is an F-Galois extension of k, then the corresponding Hopf algebra form of kG is \[ H=\{\sum c_ gg\in KG| \quad \sum f(c_ g)f(g)=\sum c_ gg\text{ for all } f\in F\}. \] This correspondence together with known descriptions of quadratic extensions then yields an explicit description in terms of generators and relations of the Hopf algebra forms of kG for \(G={\mathbb{Z}}\), \(C_ 3\), \(C_ 4\) or \(C_ 6\) if \(Pic_ 2(k)=0\) and 2 is not a zero divisor in k.
Note that pages 125 and 126 in this paper should be reversed.
Reviewer: M.Beattie

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S34 Group rings
16W20 Automorphisms and endomorphisms
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References:
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