nLab
derived noncommutative geometry

Idea

In noncommutative algebraic geometry one represents a scheme by an abelian category of quasicoherent sheaves on the scheme. This loses a bit of information but sometimes the information is sufficient.

In derived noncommutative (algebraic) geometry one instead considers the derived category of quasicoherent sheaves, or more precisely its dg-enhancement or A-infinity-enhancement; dg-enhancements for the derived categories of quasiprojective smooth varieties are essentially unique.

In general one represents complex noncommutative spaces by pretriangulated dg-categories. This is well into homotopy theory area. Quillen model category structures and homotopy limits in this context were studied by a number of people (including the impressive thesis by Tabuada). On the other hand, over a mixed characteristics, the meaning of such representations is less well understood.

Derived noncommutative geometry has been introduced by Kapranov-Bondal and later Orlov around 1990; contemporary main works belong also to Kontsevich, Lunts, van den Bergh, Katzarkov, Kuznetsov and Kaledin. Some of the works of Toen, Vaquie, Keller are properly in this area as well.

Definitions

In

• L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry), arxiv:0806.0107.

the following definition is given.

Definition (graded complex noncommutative space)

A graded complex nc-space is a $ℂ$-linear differential graded category $C$ which is homotopy complete and cocomplete (has all homotopy limits and colimits).

The derived categories of quasicoherent sheaves on a scheme over $\mathrm{Spec}(ℂ)$ is one of the examples; another example is the category of dg-modules over a fixed dg-algebra $A$, which are such that $A$ admits an exhaustive filtration such that the associated graded is a sum of shifts of $A$. Call that category $A$-$\mathrm{Mod}$.

Kontsevich calls a complex differential $\mathbb{Z}/2$-graded algebra

• smooth if $A$ is a perfect object in the category of $A$-$A$-dg-bimodules (perfect object here means that $\mathrm{Hom}(A,-)$ preserves small homotopy colimits);

• compact if the total complex dimension of its cohomology $H^{•}(A,d_A)$ is finite

The category $A$-$\mathrm{Mod}$ is a smooth (resp. compact) nc space if the underlying dg-algebras $A$ is; this notion depends on the category and not on the underlying dg-algebra.

The above definition implies that a category $C$ which represents a nc-space in the sense above is triangulated and Karoubi closed. Sometimes this are requirements in another variant of the definition.

Definition (noncommutative space (M. Kontsevich))

A noncommutative space $X$ is a small triangulated category $C_X$ which is Karoubi closed (=every idempotent is a split idempotent) and appropriately enriched over either

• spectra ${\mathrm{Hom}}_{C_X}(E,F[i])={\pi }_{-i}{\mathrm{Hom}}_{C_X}(E,F)$

• complexes of $k$-vector spaces (i.e. is a dg-category): $\mathrm{Hom}(E,F[i])=H^i({\mathrm{Hom}}_{C_X}(E,F))$. $C_X$ is $k$-linear over a field $k$ and one writes $X/k$. If instead $k$ is replaced by a ring $R$ then one enriches over complexes of $R$-modules which are cofibrant.