Shephard-todd-chevalley Theorem for Skew Polynomial Rings
نویسنده
چکیده
We prove the following generalization of the classical ShephardTodd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := kpij [x1, · · · , xn]. Then the fixed subring A has finite global dimension if and only if G is generated by quasireflections. In this case the fixed subring A is isomorphic a skew polynomial ring with possibly different pij ’s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.
منابع مشابه
The Chevalley-shephard-todd Theorem for Finite Linearly Reductive Group Schemes
We generalize the classical Chevalley-Shephard-Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme X which is étale locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli) whose stacky structure is supported o...
متن کاملOn constant products of elements in skew polynomial rings
Let $R$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $R$ and $f(X)=a_0+a_1X+cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+cdots+b_mX^m$ in $R[X;alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such tha...
متن کاملN ov 2 00 8 EXTENDING THE COINVARIANT THEOREMS OF CHEVALLEY , SHEPHARD – TODD , MITCHELL , AND SPRINGER
We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.
متن کاملOn a generalization of central Armendariz rings
In this paper, some properties of $alpha$-skew Armendariz and central Armendariz rings have been studied by variety of others. We generalize the notions to central $alpha$-skew Armendariz rings and investigate their properties. Also, we show that if $alpha(e)=e$ for each idempotent $e^{2}=e in R$ and $R$ is $alpha$-skew Armendariz, then $R$ is abelian. Moreover, if $R$ is central $alpha$-skew A...
متن کاملm at h . A C ] 2 4 M ay 2 00 8 EXTENDING THE COINVARIANT THEOREMS OF CHEVALLEY , SHEPHARD – TODD , MITCHELL , AND SPRINGER
We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.
متن کامل