SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES
SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES
复制标题
平稳且适当的非交换方案以及 DG 类别的粘合
DOI：
10.1016/j.aim.2016.07.014
复制
发表时间：
2014
影响因子：
1.7
通讯作者：
Dmitri Orlov
中科院分区：
文献类型：
--
作者：
Dmitri Orlov
本文讨论了域上非交换概型的不同性质。我们定义一个非交换概型为一个特殊类型的微分分次范畴。我们研究了非交换格式的正则性、光滑性和适当性。光滑射影概型上的完全复形范畴的容许子范畴提供了光滑和适当的非交换概型的自然例子，称为几何非交换概型。在本文中，我们证明了世界的所有几何非交换计划下的一个操作下的胶合微分分次范畴通过双模关闭。作为主要定理的一个结果，我们得到，对于任何有限维代数可分半单部分的范畴的完美复形，它是等价于一个完整的子范畴的范畴的完美复形的光滑投射计划。此外，如果代数具有有限的整体维数，则满子范畴是可容许的。我们还提供了一个光滑的投影计划，承认一个完整的例外收集，并包含作为一个子集合的例外收集预先给定的建设。作为主要定理的另一个应用，我们得到，在特征0，存在一个完整的嵌入的范畴的完美复形的任何正常计划的范畴的完美复形的一个光滑的投射计划。
In this paper we discuss different properties of noncommutative schemes over a field. We define a noncommutative scheme as a differential graded category of a special type. We study regularity, smoothness and properness for noncommutative schemes. Admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes. In this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules. As a consequence of the main theorem we obtain that for any finite-dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme. Moreover, if the algebra has finite global dimension, then the full subcategory is admissible. We also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance. As another application of the main theorem we obtain, in characteristic 0, an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme.