Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

TL;DR

This paper explores derived noncommutative schemes, their geometric realizations, and connections to finite dimensional algebras, introducing new phenomena and methods for constructing and understanding these complex structures.

Contribution

It introduces the concept of geometric realization for derived noncommutative schemes and develops new methods for their construction, especially relating to finite dimensional algebras.

Findings

Existence of geometric realizations for certain noncommutative schemes.

Construction of noncommutative schemes from finite dimensional algebras.

Introduction of well-formed quasi-hereditary algebras with specific geometric properties.

Abstract

The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a special type. We review and discuss different properties of both noncommutative schemes and morphisms between them. In addition, the concept of geometric realization for derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. We also study the construction of gluing noncommutative schemes via morphisms and consider some new phenomena, such as phantoms, quasi-phantoms, and Krull-Schmidt partners, arising in the world of noncommutative schemes and allowing us to find new noncommutative schemes. In the last sections we consider noncommutative schemes that are related to basic finite dimensional algebras. It is proved that such noncommutative schemes have special geometric realizations under which the algebra goes to a vector bundle on a smooth projective scheme. Such realizations are constructed in two steps, one of which is the well-known construction of Auslander, while the second step is connected with a new concept of a well-formed quasi-hereditary algebra for which there are very particular geometric realizations sending standard modules to line bundles.