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Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category.

This is motivated by the fact that homological invariants of an "ordinary" scheme and of its $\mathrm{DG}$-category of quasicoherent sheaves coincide, so we can study a scheme by studying it's $\mathrm{DG}$-category of quasicoherent sheaves. Then we generalize to general $\mathrm{DG}$-categories over $k$ and get a generalized, noncommutative space (over $k$).

Note, that another, but quite similar framework for derived noncommutative geometry is defining a space to be a linear $A_{\infty}$-category. And both approaches ($\mathrm{DG}$ and $A_{\infty}$) are really the models for linear stable $\infty$-categories.

But this noncommutative geometry was motivated by the desire to study noncommutative versions of schemes, wasn't it?

However, in Lurie's "Structured Spaces" (http://www.math.harvard.edu/~lurie/papers/DAG-V.pdf) there is an abstract definition of "generalized" schemes which seems to include derived and spectral schemes, derived and spectral stacks.

It really seems that (at least according to ncatlab: https://ncatlab.org/nlab/show/derived+algebraic+geometry#RelationToDerivedNoncommutativeGeometry) that such "generalized" schemes in the sense of Lurie are also can be represented by stable $\infty$-categories of quasicoherent sheaves, that is, a noncommutative "generalized" scheme over $k$ would also be $k$-linear $\mathrm{DG}$-category, thus a special case of Kontsevich's noncommutative algebraic geometry?

That said, is it true that in derived noncommutative geometry (such as studied by Kontsevich, Katzarkov, Kaledin, Orlov, Tabuada) one studies "noncommutative" versions not only of ordinary schemes but also of such geometric objects as:

  • algebraic stacks (Artin or Deligne-Mumford)
  • derived schemes (in the "simplicial commutative rings" sense)
  • spectral schemes (in the "$E_{\infty}$-rings" sense, see Lurie "Spectral Algebraic Geometry)
  • derived algebraic stacks (Artin or Deligne-Mumford)
  • spectral Deligne-Mumford stacks

If not, which objects are included in "derived noncommutative geometry" framework, and which are not?

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There are several functors (for instance, $QCoh$ or $Perf$) from derived or spectral stacks (geometric or not) to stable $\infty$-categories. It is thus possible to study objects of derived/spectral algebraic geometry à la Toën-Vezzosi-Lurie from the point of view of derived noncommutative geometry à la Kontsevich.

Though it is not clear to me at all how much information is loss by applying these functors (some information is, for sure). For instance, I don't think one can detect geometricity from the non-commutative view-point.

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    $\begingroup$ what do you mean by geometricity? $\endgroup$
    – Yosemite Sam
    Jul 5, 2017 at 23:08
  • $\begingroup$ In Toën-Vezzosi there is a notion of a geometric stack (roughly speaking, derived geometric stacks are obtained from derived schemes by taking iterated quotient by smooth groupoids). These stacks have very nice properties (such as existence of a cotangent complex). $\endgroup$
    – DamienC
    Jul 6, 2017 at 14:30

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