Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes
Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes
复制标题
任意基方案上的等变分辨率、线性化和希尔伯特第十四个问题
DOI：
10.1016/0001-8708(87)90016-8
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发表时间：
1987
影响因子：
1.7
通讯作者：
R. Thomason
中科院分区：
文献类型：
--
作者：
R. Thomason
考虑一个Noether基概型S上的平坦代数群概型G作用在Noether概型X上。设S和X有充足的线丛族，例如，S和X是正则的或仿射的，或在一个仿射概型上是拟投射的。本文讨论了G作用的局部自由模X上的层是否都允许G作用的局部自由模的等变分解。在经典的情况下，S是一个域，这个问题的肯定解决方案是容易的，但有用的。在现代的情况下，S是一个一般的计划，这个问题要困难得多，一个好的一般解决方案仍然是未知的。本文将证明在许多有趣的情况下存在等变分解，例如，如果G是半单的或在正规基S上是约化的，或者如果G是在维数小于或等于2（2.18，2.5）的正则Noether基S上的闭光滑连通利夫雷斯.关于还原群的陈述是Seshadri的一个猜想，他在[Se]中证明了它意味着还原群作用的不变量的环的有限生成。这是Nagata对域上希尔伯特第14问题（式3.8）的解的广泛推广。分解结果还证明了任何基S上的半单群、维数至多为2的正则基S上的连通纤维光滑的afline群以及维数至多为1的正则基上的任何仿射群都是线性的，因为它们可以嵌入为Aut（-t^）的闭子群，其中Y是S上的向量丛，（3.2）。如果dim S= 0，这是众所周知的，如果dim S= 1，这是雷诺已知的，但在其他情况下是新的。类似地，分解结果允许我们将X上的G作用等变嵌入到向量空间丛或射影空间丛上的线性作用中，如果X分别是S上的有限型非等变后线或正规和拟射影（3.7，3.4）。这大大加强了Sumihiro的结果。
Consider a flat algebraic group scheme G over a noetherian base scheme S and acting on a noetherian scheme X. Suppose S and X have ample families of line bundles, eg, that S and X are regular or afline, or are quasiprojective over an aftine scheme. This paper addresses the problem of whether all sheaves on X of finitely generated modules with G action admit equivariant resolutions by finitely generated locally free modules with G-action. In the classical case where S is a field, the affirmative solution of this problem is easy but useful. In the modern case where S is a general scheme, the question is much harder, and a good general solution is still unknown. This paper will show that equivariant resolutions exist in many interesting cases, for example, if G is semisimple or is reductive over a normal base S, or if G is afline and smooth with connected libres over a regular noetherian base S of dimension less than or equal to 2 (2.18, 2.5). The statement about reductive groups is a conjecture of Seshadri, who showed in [Se] that it implies finite generation of rings of invariants of reductive group actions. This is a wide generalization of Nagata’s solution of Hilbert’s 14th Problem over a field,(3.8). The resolution results also yield proofs that semisimple groups over any base S, and afline groups smooth with connected fibres over a regular base S of dimension at most 2, and any affine groups over a regular base of dimension at most 1, are linear in that they can be embedded as closed subgroups of Aut (-t^) for Y a vector bundle over S,(3.2). This is well known if dim S= 0, and known to Raynaud if dim S= 1, but otherwise new. Similarly the resolution results allow one to equivariantly embed a G action on X into a linear action on a vector space bundle or projective space bundle if X is, respectively, nonequivariantly aftine of finite type or normal and quasiprojective over S (3.7, 3.4). This greatly strengthens results of Sumihiro.