Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

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.

Finite-dimensional differential graded algebras and their geometric realizations

“…A derived noncommutative scheme X over a field k is a k -linear DG category of the form Perf -R, where R is a cohomologically bounded DG k -algebra. Under such a definition the triangulated category perf-R is called the category of perfect complexes on the scheme X , while the derived category DpRq will be called the derived category of quasi-coherent sheaves on it (see [O5,Def. 2.1]).…”

“…A geometric realization of a derived noncommutative scheme X " Perf -R consists of a usual commutative scheme Z and a localizing subcategory L Ď D Qcoh pZq, the natural enhancement L of which is quasi-equivalent to the DG category SF -R (see [O5,Definition 2.17]). In other words a geometric realization is a fully faithful functor DpRq Ñ D Qcoh pZq that preserves all direct sums and is defined on the level of DG categories.…”

“…Such geometric realizations will be called plain. In the case when the subcategory perf-R Ă perf-Z is admissible, the geometric realization will be called pure (see [O5,Sect. 2.4.]).…”

“…HH ˚pPerf -Rq " 0, and K 0 pPerf -Rq is a finite abelian group. It is called a phantom if, in addition, K 0 pPerf -Rq " 0 (see [GO], [O5,Def. 3.18]).…”

“…a DG B ˝b A -module. One can define a so called lower triangular DG category that is related to the data pA , B; Tq (see [Ta,KL,O3,O4,O5]).…”