用格拉斯曼方法对表示进行分类
Classifying representations by way of Grassmannians
摘要：
设$Λ$是代数闭域上的有限维代数.给出了表征存在细或粗模空间分类的标准，直到同构，具有固定维数$d$和固定无平方顶$T$的$Λ$的表示。除了提供完整的理论图景之外，其中一些等价条件很容易通过颤动和关系进行检查。在存在模空间的情况下-出乎意料地频繁的光严格的精细分类-这个空间总是投射，事实上，出现作为一个封闭的子品种${\mathfrak{Grass}}^T_d$的经典格拉斯曼。即使当完全模问题无法求解时，品种${\mathfrak{Grass}}^T_d$被认为具有独特的属性，推荐它作为模空间的替代品。作为一个应用，得到了具有固定单顶的有限局部表示类型的代数的一个特征，在这种情况下，对于给定单$T$的有限局部表示类型，根分层$\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$是顶为$T$的模的分类不变量.这依赖于作为副产品获得的以下一般事实：局部模块$M$的适当退化永远不会具有与$M$相同的激进分层。
英文摘要：
Let $Λ$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $Λ$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations. In case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety ${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $T$', the radical layering $\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: Proper degenerations of a local module $M$ never have the same radical layering as $M$.
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