Finite generation of class groups of rings of invariants
Finite generation of class groups of rings of invariants
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不变量环类群的有限生成
DOI：
10.1090/s0002-9939-1976-0427306-0
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发表时间：
1976
期刊：
影响因子：
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通讯作者：
A. Magid
中科院分区：
文献类型：
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作者：
A. Magid
设R为代数闭域k上的正规仿射域，设G为作用于R上的连通代数群，证明了RG的除数类群是R的类群的子群被G的字符群的子商扩展的同态象，特别地，如果R有有限生成的类群，RG也有有限生成的类群。本文的目的是建立如下定理：设R是代数闭域k上的正规仿射域，设G是作用于R上的连通代数群，则如果R有有限生成的除数类群，则RG也有有限生成的除数类群。（如果K是R的商域，则RG是rn KG，因此RG是Krull定义域，因此有一个除数类群。）采用以下约定：k为固定代数闭基域。对于可交换k代数a， U(a)表示a的单位群，且Uk(a) = U(a)/k*。我们从一些关于群体行动和单位的观察开始。命题1。让R积分域与商磁场k-algebra K suvh英国(R)是一个有限生成集团,让G是一个连接代数组作为k-algebra同构的R,这样每一个单能性的子群G的理性行为在R:(一)每个f U (R)是G的累积量(b)如果在K, G f (f) / f e U (R)为所有G e G,然后为G f是一个semiinvariant证明。首先我们考虑G是单幂函数，R是仿射k型V的坐标环的情况。如果f是V上的一个非消失函数，V是V的一个元素，那么G -> f （gv）是G上的一个非消失函数，因此G是常数，因为G是单幂函数。因此f是一个不变量。一般来说R是这种坐标环的直接极限，因此R的每一个单位在g的每一个幂偶子群下都是不变的。现在我们可以建立(a)。我们需要知道G对Uk(R)的作用是平凡的，并且由上段可以处理G = Gm的情况。现在Uk(R)是一个有限生成的自由阿贝尔群，并且由编辑于1975年12月29日收到的作用。AMS （MOS）学科分类（1970年）。主要13 a05;二级20G 15。版权CD 1977，美国数学学会
Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. It is shown that the divisor class group of RG is a homomorphic image of an extension of a subgroup of the class group of R by a subquotient of the character group of G. In particular, if R has finitely generated class group, so does RG. The object of this note is to establish the following theorem: Let R be a normal affine domain over the algebraically closed field k, and let G be a connected algebraic group acting rationally on R. Then if R has a finitely generated divisor class group, then so does RG. (If K is the quotient field of R, then RG is R n KG, so RG is a Krull domain and hence has a divisor class group.) The following conventions are adopted: k is the fixed algebraically closed base field. For a commutative k-algebra A, U(A) denoLes the group of units of A and Uk(A) = U(A)/k*. We begin with some observations regarding group actions and units. PROPOSITION 1. Let R be an integral domain k-algebra with quotient field K suvh that Uk (R) is a finitely generated group, and let G be a connected algebraic group acting as k-algebra automorphisms of R, such that every unipotent subgroup of G acts rationally on R. Then: (a) Every f in U(R) is a semi-invariant for G. (b) If f is in K such that g(f )/f e U(R) for all g e G, then f is a semiinvariant for G. PROOF. First we consider the case where G is unipotent and R is the coordinate ring of the affine k-variety V. If f is a nonvanishing function on V and v an element of V, then g -> f (gv) is a nonvanishing function on G, hence constant since G is unipotent. Thus f is an invariant. In general R is a direct limit of such coordinate rings, and hence every unit of R is invariant under every unipotent subgroup of G. Now we can establish (a). We need to know that G acts trivially on Uk(R), and by the above paragraph it is enough to treat the case G = Gm. Now Uk (R) is a finitely generated free abelian group, and the action of Received by the editors December 29, 1975. AMS (MOS) subject classifications (1970). Primary 13A05; Secondary 20G 15. Copyright CD 1977, American Mathematical Society