Reductive groups are geometrically reductive
Reductive groups are geometrically reductive
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还原群在几何上是还原的
DOI：
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
W. Haboush
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文献类型：
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作者：
W. Haboush
设G是代数闭域k上的半单代数群。设G在R生成的k-代数R上自同构有理作用。证明不变量环RG是非线性生成的问题起源于世纪的不变量理论家。当k = C，复数，和G GL（n，C）的问题是肯定的回答希尔伯特的“基本定理不变理论”。证明涉及到从R到RG构造一个G等变投影，然后用它来证明结果代数。当k的特征为0，G为任意半单群时，利用H. Weyl，G的每个有限维表示都是完全可约的。50年代D.芒福德和其他人（卡地亚，岩堀，永田）应用外尔定理来构建从R到RG的任何半单群的投影。这使得希尔伯特的证明可以推广到任意的半单群。某些几何应用，特别是理论的模，作出了推广，以集团的领域积极的特点非常可取的。在积极的特点，完全还原肯定失败。因此，有人试图用一个较弱的条件来代替完全归约，这个条件将立即适用于所有半单群，并使有限生成RG的证明成为可能。以下是陈述完全还原的最弱方法。如果V是一个有限维G-模，其中包含一个余维为1的G-稳定子空间VT，则存在一条G-稳定线LC V使得V0 E L = V。这是猜想，因为它是在前言中所述[16]：
Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of invariants, RG, is finitely generated originates with the invariant theorists of the nineteenth century. When k = C, the complex numbers, and G GL (n, C) the question is answered affirmatively by Hilbert's "fundamental theorem of invariant theory". The proof involved constructing a G equivariant projection from R to RG and then using it to prove the result algebraically. When k is of characteristic 0 and G is any semi-simple group, by a theorem of H. Weyl, every finite dimensional representation of G is completely reducible. In the 1950's D. Mumford and others (Cartier, Iwahori, Nagata) applied Weyl's theorem to construct a projection from R to RG for any semi-simple group. This made it possible to generalize Hilbert's proof to an arbitrary semi-simple group. Certain geometric applications, particularly to the theory of moduli, made a generalization to groups over fields of positive characteristic highly desirable. In positive characteristic, complete reducibility definitely fails. Hence attempts were made to replace complete reducibility with a weaker condition which would at once hold for all semi-simple groups and make a proof of finite generation of RG possible. The weakest way to state complete reducibility is the following. If V is a finite dimensional G-module containing a G-stable sub-space of co-dimension one, VT, then there is a G-stable line LC V such that V0 E L = V. Mumford conjectured a weaker version of this statement by seeking a complement only in a higher symmetric power of V, SI( V). This is the conjecture as it is stated in the preface to [16]: