Let $\Lambda$ be a finite dimensional algebra over an algebraically closedfield. Criteria are given which characterize existence of a fine or coarsemoduli space classifying, up to isomorphism, the representations of $\Lambda$with fixed dimension $d$ and fixed squarefree top $T$. Next to providing acomplete theoretical picture, some of these equivalent conditions are readilycheckable from quiver and relations. In case of existence of a moduli space --unexpectedly frequent in light of the stringency of fine classification -- thisspace is always projective and, in fact, arises as a closed subvariety${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduliproblem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen tohave distinctive properties recommending it as a substitute for a moduli space.As an application, a characterization of the algebras having only finitely manyrepresentations with fixed simple top is obtained; in this case of `finitelocal 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 themodules with top $T$. This relies on the following general fact obtained as abyproduct: Proper degenerations of a local module $M$ never have the sameradical layering as $M$.
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机译：令$ \ Lambda $是代数封闭域上的有限维代数。给出了对存在精细或粗糙模空间进行分类的标准，直到同构为止，具有固定维数$ d $和固定无平方顶部$ T $的$ \ Lambda $表示形式。除了提供完整的理论图，还可以从颤动和关系中容易地检查其中的一些等效条件。在存在模空间的情况下-鉴于精细分类的严格性而出乎意料地频繁-该空间始终是射影的，并且实际上是作为经典的闭合子变量$ {\ mathfrak {Grass}} ^ T_d $出现的格拉斯曼即使完整的模幂问题不能解决，变体$ {\ mathfrak {Grass}} ^ T_d $仍具有独特的性能，建议将其替换为模空间。作为一种应用，代数的特征仅有限获得具有固定简单顶部的许多表示；在这种情况下，“在给定的简单$ T $处为有限局部表示类型”中，将部首分层$ \ bigl（J ^ lM / J ^ {l + 1} M \ bigr）_ {l \ ge 0} $显示为是具有最高$ T $的模块的分类不变式。这取决于作为副产品获得的以下一般事实：本地模块$ M $的适当退化永远不会与$ M $具有相同的自由基分层。
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