Publications (3)0 Total impact
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ABSTRACT: We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings. We use an explicit classification of all reflexive rank one $R$-modules, which is given in terms of the class group of $R$, or in terms of $R$-semi-invariants. This result is implicitly contained in a paper of Nakajima (\cite{Nakajima:rel_inv}). Comment: 16 pages
11/2010;
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ABSTRACT: Let $G$ be a finite $p$-group and $k$ a field of characteristic $p>0$. We show that $G$ has a \emph{non-linear} faithful action on a polynomial ring $U$ of dimension $n=\mathrm{log}_p(|G|)$ such that the invariant ring $U^G$ is also polynomial. This contrasts with the case of \emph{linear and graded} group actions with polynomial rings of invariants, where the classical theorem of Chevalley-Shephard-Todd and Serre requires $G$ to be generated by pseudo-reflections. Our result is part of a general theory of "trace surjective $G$-algebras", which, in the case of $p$-groups, coincide with the Galois ring-extensions in the sense of \cite{chr}. We consider the \emph{dehomogenized symmetric algebra} $D_k$, a polynomial ring with non-linear $G$-action, containing $U$ as a retract and we show that $D_k^G$ is a polynomial ring. Thus $U$ turns out to be \emph{universal} in the sense that every trace surjective $G$-algebra can be constructed from $U$ by "forming quotients and extending invariants". As a consequence we obtain a general structure theorem for Galois-extensions with given $p$-group as Galois group and any prescribed commutative $k$-algebra $R$ as invariant ring. This is a generalization of the Artin-Schreier-Witt theory of modular Galois field extensions of degree $p^s$. Comment: 20 pages
11/2010;
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ABSTRACT: Let G be a finite group acting on a polynomial ring A over the field K and let AG denote the corresponding ring of invariants. Let B be the subalgebra of AG generated by all homogeneous elements of degree less than or equal to the group order |G|. Then in general B is not equal to AG if the characteristic of K divides |G|. However we prove that the field of fractions Quot(B) coincides with the field of invariants Quot(AG)=QuotG(A). We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for AG. We prove that there is always a nonzero transfer c∈AG of degree <|G|, such that the localization c(AG) can be generated by fractions of homogeneous invariants of degrees less than 2⋅|G|−1. If A=Sym(V⊕FG) with finite-dimensional FG-module V, then c can be chosen in degree one and 2⋅|G|−1 can be replaced by |G|. Let N denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that N contains the transfer ideal and thus can be used to calculate generators for AG by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for AG.
Journal of Algebra.
