各种各样的均匀表示IV。几何亲缘关系
Varieties of uniserial representations IV. Kinship to geometric quotients
摘要：
设$Λ$是代数闭域上的有限维代数，${\Bbb S}$是简单左$Λ$ -模的有限序列.在文献[6，9]中，引入了具有可达仿射开覆盖的拟投射代数簇，用于对具有连续合成因子序列${\Bbb S}$的$Λ$的unisolate表示进行分类.我们在这里的主要目标有三个方面：一是证明这些品种是“良好的近似”-在某种意义上是精确的-几何的经典品种$\operatorname{Mod-Uni}({\Bbb S})$ parametrizing有关unisolation表示，模通常的共轭行动的一般线性群。在某种程度上，这填补了由于经常不存在这种代理人而留下的信息空白。第二个目标是促进“主”品种之间的信息传递，所考虑的单一品种可以嵌入其中。这些工具，然后适用于对第三个目标，关于存在的几何concurents：我们证明，$\operatorname{Mod-Uni}({\Bbb S})$有一个几何商的$GL$ -行动时，正是unischant品种有一个几何商模一定的自然代数群行动，在这种情况下，两个concurents一致。我们的主要结果是利用在一个代表性理论的背景下：在其他后果，他们产生的几何特征的代数的有限unisbrands类型的补充现有的描述，但更干净，更容易检查。
英文摘要：
Let $Λ$ be a finite dimensional algebra over an algebraically closed field, and ${\Bbb S}$ a finite sequence of simple left $Λ$-modules. In [6, 9], quasiprojective algebraic varieties with accessible affine open covers were introduced, for use in classifying the uniserial representations of $Λ$ having sequence ${\Bbb S}$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations' -- in a sense to be made precise -- to geometric quotients of the classical varieties $\operatorname{Mod-Uni}({\Bbb S})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. To some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered uniserial varieties can be embedded. These tools are then applied towards the third objective, concerning the existence of geometric quotients: We prove that $\operatorname{Mod-Uni}({\Bbb S})$ has a geometric quotient by the $GL$-action precisely when the uniserial variety has a geometric quotient modulo a certain natural algebraic group action, in which case the two quotients coincide. Our main results are exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.
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