arXiv 1407.26592014-07-10math.RTmath.RA

有限维代数分次表示的模空间

Moduli spaces of graded representations of finite dimensional algebras

E. Babson, B. Huisgen-Zimmermann, R. Thomas

发表信息：: in Algebra and its Applications (Athens, Ohio 2005), (D.V. Huynh, et al., Eds.), Contemp. Math. 419 (2006) 7-27

摘要：

设$Λ$是代数闭域上的基本有限维代数，表示为路代数模关系，进一步假设$Λ$是按路长分次的.研究了分次$Λ$ -模类的可分类性，通过模空间，证明了分次$Λ$ -模类具有固定维数$d$和固定顶$T$ .它表明，这样的模空间存在的频率远远超过他们做的ungraded模。在本地情况下（即，当$T$是简单的时），具有顶$T$的分次$d$ -维$Λ$ -模总是具有一个细模空间，该模空间将这些模分类到分次同构;而且，这个模空间是一个具有特殊仿射覆盖的投射簇，它可以由$Λ$的模和关系构造。当$T$不简单时，具有顶$T$的分次$d$ -维$Λ$ -模的粗模空间的存在迫使这些模成为局部模的直和;在后一种条件下，所讨论的模的同构不变量的有限集合产生子类的划分，每个子类具有参数化相应的分次同构类的细模空间（也是投射的）。

英文摘要：

Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $Λ$ is graded by lengths of paths. The paper addresses the classifiability, via moduli spaces, of classes of graded $Λ$-modules with fixed dimension $d$ and fixed top $T$. It is shown that such moduli spaces exist far more frequently than they do for ungraded modules. In the local case (i.e., when $T$ is simple), the graded $d$-dimensional $Λ$-modules with top $T$ always possess a fine moduli space which classifies these modules up to graded-isomorphism; moreover, this moduli space is a projective variety with a distinguished affine cover that can be constructed from quiver and relations of $Λ$. When $T$ is not simple, existence of a coarse moduli space for the graded $d$-dimensional $Λ$-modules with top $T$ forces these modules to be direct sums of local modules; under the latter condition, a finite collection of isomorphism invariants of the modules in question yields a partition into subclasses, each of which has a fine moduli space (again projective) parametrizing the corresponding graded-isomorphism classes.

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