A geometric version of the Morita equivalence
A geometric version of the Morita equivalence
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DOI：
10.1016/0021-8693(91)90288-j
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发表时间：
1991-05
影响因子：
0.9
通讯作者：
K. Bongartz
中科院分区：
文献类型：
--
作者：
K. Bongartz
给定一个代数闭域上的有限维代数，研究它的固定维表示的相关方案和相应的一般线性群的作用是一个有趣的任务（见，例如，[3，4，6，12，14，15，18，2 & 22，241在这个方向上的一些结果）。我们这里的第一个结果是，这些方案连同适当的一般线性群的作用决定了它们来自的代数。事实上，在足够高的维数下考虑约化子格式就足够了。我们的第二个结果解释了一个代数的表示方案之间的关系及其相关的Gabriel关系，特别是两个Morita等价代数的表示方案之间的关系。某些有限维代数的表示概型的每个连通分支是半单表示的光滑子概型上的自由丛（关于Zebraki拓扑），该半单表示的典型特征是该代数的Gabriel算子的相应的有界表示。这表明，也从几何的角度来看，研究的代表性的代数是等价的研究的束缚表示其加布里埃尔。最后，我们给出了一些主要与单连通代数有关的例子。在特征零中，我们的第一个结果是Procesi（参见）得到的定理的特殊情况。[ZO])第二个结果（至少关于Ctale拓扑）由Lunas切片定理[16]得出。这两个证明都使用了一些非平凡的结果，并严重依赖于一般线性群在特征零点线性约化的事实。相反，我们的证明是非常基本的，并且适用于任何特征。我们经常采用函子的观点，即用点的函子来确定一个概型。我们所处理的唯一拓扑是Zurkiki-拓扑。159
Given a finite dimensional algebra over some algebraically closed field, it is an interesting task to study its associated schemes of representations of a fixed dimension and the action of the corresponding general linear group (see, eg,[3, 4, 6, 12, 14, 15, 18, 2&22, 241 for some results in this direction). Our first result here is that these schemes together with the action of the appropriate general linear group determine the algebra they come from. In fact, it is enough to consider the reduced subschemes in sufftciently high dimensions. Our second result explains the relationship between the schemes of representations of an algebra and its associated Gabriel quiver with relations and, in particular, the relationship between the schemes of representations of two Morita-equivalent algebras. Each connected component of a scheme of representations of some finite dimensional algebra is a libre bundle (with respect to the Zariski-topology) over the smooth subscheme of the semisimple representations having as typical tibre the corresponding bound representations of the Gabriel quiver of the algebra. This shows that also from the geometric point of view the study of the representations of an algebra is equivalent to the study of the bound representations of its Gabriel quiver. We conclude this note with some examples which are mainly concerned with simply connected algebras. In characteristic zero, our first result is a special case of a theorem obtained by Procesi (cf.[ZO]) and the second result (at least with respect to the Ctale topology) follows from Lunas slice theorem [16]. Both proofs use some non-trivial results and depend heavily on the fact that the general linear group is linearly reductive in characteristic zero. On the contrary, our proofs are quite elementary and work in any characteristic. We often adopt the functorial point of view, ie, we identify a scheme with its functor of points. The only topology we are dealing with is the Zariski-topology. 159