Some Homological Properties in Geometric Invariant Theory

复数域C上的约化代数群G的表示(V，G)称为余正则的，如果V//G是非奇异的。对于半单G，P.Littelemann确定了不可约余正则表示，而对于单G，所有的余正则表示由V.L.Popov、G.W.Schwarz、O.M.Adomovich和E.O.Golovina分类。在这项研究中，我们确定了单半单部分具有闭环轨道的非半单还原群G的余正则表示。这是基于代数环面在正规簇上的作用被分解为余维1的非爆破作用和余维2的爆破作用。Chow群在余维1的作用下保持商态射。此外，在此基础上，我们还研究了非半单还原群作用的相对等维性和相对稳定性，得到了一些对余正则或等维Rep…分类有用的结果本文将有限Galois群的经典分支理论的一部分推广到仿射作用下的商态射，并给出了一个类似于有限覆盖情形下的结果在仿射群情形下成立的判据，这与有限群的半不变量的一些结果在中心双连通情形下的推广有关。为了研究局部环上的经典群的不变理论，给出了定义在局部环上(由于Ishibashi)的渐近群Sp(V)的生成系(由于Ishibashi)的辛向移集的一个很好的判据。关于有限群的表示理论：我们确定了有限交换2-群(由于Ogawa)的mod_2-上同调环的本质理想和初等分解，并得到了一些保持诱导特征标不可逆性的2-群扩张的部分结果(由于Sekiguchi)。这些结果似乎对研究有限群不变理论中的函子性质是有用的。较少

A representation (V,G) of a reductive algebraic group G over the complex number field C is said to be coregular, if V//G is non-singular. For a semisimple G, irreducible coregular representations are determined by P.Littelemann, and for a simple G, all coregular representations are classified by V.L. Popov, G.W. Schwarz, O.M. Adomovich and E.O. Golovina. In this research, we have determined coregular representations of non-semisimple reductive groups G with simple semisimple parts having enouch closed orbits. This is based on the decomposition of actions of algebraic tori on normal varieties into no-blowing-up actions of codimension one and blowing-up actions of codimension 2. The Chow groups preserve under quotient morphisms in the latter actions. Moreover, in the relaion with this, we have studied relative equidimensionalities and relative stabilities of actions of non-semisimple reductive groups and obtain some results which are useful in classifying coregular or equidimensional rep … More resentaions.We generalize a part of the classical ramification theory of finite Galois groups to one of quotient morphisms under affine group actions and give a criterion the result similar to in finite covering cases to hold in affine groups case, which is related to an extension of some results on semi-invariants of finite groups to in the case of centric diconnected tori.In order to study on invariant theory of classical groups over local rings, we give a nice criterion for a set of symplectic trasvections to be a genrating system of the sympectic group Sp(V) defined over local rings (due to Ishibashi).On representaion theory of finite groups: We determine essential ideals and primary decompositions of mod 2-cohomology ring of finite abelian 2-groups (due to Ogawa) and obtain partial results on extensions of some 2-groups which preserve the irresucibilities of induced characters (due to Sekiguchi). These results seem to be useful in studying functor properties in invariant theory of finite groups. Less