Author: lschmsc Format: TextGreetings!
(Derived) noncommutative geometry seems to be motivated by studying noncommutative versions of (ordinaty) schemes (and, possible, algebraic stacks), as the key observation was that homological invariants associated to both scheme and it's DG-category of quasicoherent sheaves coincide.
So, derived noncommutative space is defined to be k-linear DG-category (or, in modern language, k-linear stable ∞-category).
However, according to this
https://ncatlab.org/nlab/show/derived+algebraic+geometry#RelationToDerivedNoncommutativeGeometry
stable ∞-categories (or DG-categories) seem to also encode the same information about "derived schemes". As I understand what ncatlab really means for "derived scheme" is "generalized space" in the sence of Lurie (see "Structured Spaces"), that is, G-scheme for a geometry G. Such "generalized spaces" (derived schemes, spectral schemes, stacks) are also studied over a commutative ring k, as in Lurie's "Generalized spaces".
That said, am I right to think that derived noncommutative space, that is, k-linear DG/stable ∞-category also include noncommutative versions of higher-geometric structures (they are called "G-schemes", but they also include stacks, if I'm correct, again, I'm talking about "higher spaces" in the sense of Lurie)? That is, k-linear stable ∞-categories (or k-linear DG-categories) for a commutative ring k also include derived schemes, derived stacks (based on simplicial commutative rings), spectral schemes, spectral Deligne-Mumford stacks (based on commutative ring spectra) and their noncommutative versions?
It seems that all that time people like Kontsevich, Orlov, Tabuada have been thinking that they study noncommutative spaces as noncommutative versions of schemes, but what they really have been studying all along are noncommutative versions of everything from (higher) geometry.
Greetings!
(Derived) noncommutative geometry seems to be motivated by studying noncommutative versions of (ordinaty) schemes (and, possible, algebraic stacks), as the key observation was that homological invariants associated to both scheme and it's DG-category of quasicoherent sheaves coincide.
So, derived noncommutative space is defined to be k-linear DG-category (or, in modern language, k-linear stable ∞-category).
stable ∞-categories (or DG-categories) seem to also encode the same information about "derived schemes". As I understand what ncatlab really means for "derived scheme" is "generalized space" in the sence of Lurie (see "Structured Spaces"), that is, G-scheme for a geometry G. Such "generalized spaces" (derived schemes, spectral schemes, stacks) are also studied over a commutative ring k, as in Lurie's "Generalized spaces".
That said, am I right to think that derived noncommutative space, that is, k-linear DG/stable ∞-category also include noncommutative versions of higher-geometric structures (they are called "G-schemes", but they also include stacks, if I'm correct, again, I'm talking about "higher spaces" in the sense of Lurie)? That is, k-linear stable ∞-categories (or k-linear DG-categories) for a commutative ring k also include derived schemes, derived stacks (based on simplicial commutative rings), spectral schemes, spectral Deligne-Mumford stacks (based on commutative ring spectra) and their noncommutative versions?
It seems that all that time people like Kontsevich, Orlov, Tabuada have been thinking that they study noncommutative spaces as noncommutative versions of schemes, but what they really have been studying all along are noncommutative versions of everything from (higher) geometry.