Noncommutative algebraic geometry
Noncommutative algebraic geometry
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非交换代数几何
DOI：
10.4171/rmi/360
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发表时间：
2003
影响因子：
1.2
通讯作者：
O. A. Laudal
中科院分区：
文献类型：
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作者：
O. A. Laudal
在经典不变模理论中，对非交换代数几何的需要是显而易见的。一般来说，不可能找到可交换参数来参数化作用在一个方案上的李群的所有轨道。当一个轨道包含在另一个轨道的闭包中时，轨道空间不能以自然的方式给定方案结构。在本文中，我们将证明可以通过引入非交换代数几何来克服这些困难，其中仿射“方案”是在结合代数上建模的。这种仿射格式的点是代数的简单模，并且该格式在有限族点处的局部结构用作者在[10]中提出的非交换变形理论表示。更一般地说，理论中的几何是由一个群来表示的，即在给定的k-线性阿贝尔范畴（k a域）中满足一些合理条件的物体（如果愿意，也可以是箭头）的图（有限或无限）。上面提到的非交换变形理论，允许构造一个关联k代数的预集，局部参数化图。结果表明，该理论自然地推广了经典方案理论。此外，它还为处理不变量理论问题和模问题提供了一个有前途的框架。特别是证明了经典代数几何中的许多模空间是包含附加信息的非交换格式的交换化。2000数学学科分类：14A22、16E、16D90、16G、13D。
The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine “schemes” are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in [10]. More generally, the geometry in the theory is represented by a swarm, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given k-linear Abelian category (k a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative k-algebras, locally parametrizing the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information. 2000 Mathematics Subject Classification: 14A22, 16E, 16D90, 16G, 13D.