SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES

By Dmitri Orlov. 2014

141 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b73d7b915c87c31c1b8ebe781310828619e8c095

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. V. Lunts
2. Daniel Bergh
3. Olaf M. Schnürer

In Geometry and Topology

1. Olaf M. Schnürer
2. V. Lunts
3. Daniel Bergh
4. Gonçalo Tabuada
5. A. Neeman
6. Dmitri Orlov

In Statistical and Nonlinear Physics

1. V. Lunts
2. Daniel Bergh
3. Olaf M. Schnürer

Citations in the corpus

• Geometricity for derived categories of algebraic stacks (2016, background): “We then recall Orlov’s notion of a geometric dg category and state his main glueing result.”