Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

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 special type. Different properties of both noncommutative schemes and morphisms between them are reviewed and discussed. In addition, the concept of a geometric realization for a derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. Also studied are the construction of gluings of noncommutative schemes via morphisms, along with certain new phenomena such as phantoms, quasi-phantoms, and Krull-Schmidt partners which arise in the world of noncommutative schemes and which enable us to find new noncommutative schemes. The last sections consider noncommutative schemes connected with 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, the first of which is the well-known construction of Auslander, while the second step is connected with the new concept of a well-formed quasi-hereditary algebra, for which there are very special geometric realizations sending standard modules to line bundles. Bibliography: 50 titles.