derived Deligne-Mumford stack
… derived algebraic geometry … higher algebra …generalized scheme…
…E-∞ scheme, locally representable structured (∞,1)-topos
Let be a commutative ring.
A derived Deligne-Mumford stack (over ) is a generalized scheme in the sense of locally affine -structured (infinity,1)-topos for the étale geometry (for structured (infinity,1)-toposes).
A 1-localic derived Deligne-Mumford stack is an ordinary Deligne-Mumford stack. See there for more details.
Relation to derived algebraic spaces
(LurieProp, theorem 1.3.8):
spectral Deligne-Mumford stack is quasi-compact, quasi-separated E-∞ algebraic space? precisely if it admits a scallop decomposition.
Characterization as -presheaves on -rings
The (∞,1)-presheaves on E-∞ rings which are represented by spectral Deligne-Mumford stacks are described by the Artin-Lurie representability theorem.
Notice that for generalized schemes the étale geometry (for structured (infinity,1)-toposes) is not interchangeable with the Zariski geometry . Instead -generalized schemes are derived schemes.
In the context of E-infinity geometry (spectral Deligne-Mumford stacks):