nLab derived Deligne-Mumford stack

Idea

… derived algebraic geometry … higher algebra …generalized scheme…

…E-∞ scheme, locally representable structured (∞,1)-topos

Definition

Let $k$ be a commutative ring.

A derived Deligne-Mumford stack (over $k$) is a generalized scheme in the sense of locally affine $\mathcal{G}$-structured (infinity,1)-topos for $\mathcal{G} = \mathcal{G}_{et}(k)$ the étale geometry (for structured (infinity,1)-toposes).

Special cases

A 1-localic derived Deligne-Mumford stack is an ordinary Deligne-Mumford stack. See there for more details.

Properties

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 $(\infty,1)$-presheaves on $E_\infty$-rings

The (∞,1)-presheaves on E-∞ rings which are represented by spectral Deligne-Mumford stacks are described by the Artin-Lurie representability theorem.

Related concepts

Notice that for generalized schemes the étale geometry (for structured (infinity,1)-toposes) $\mathcal{G}_{et}(k)$ is not interchangeable with the Zariski geometry $\mathcal{G}_{et}(k)$. Instead $\mathcal{G}_{Zar}(k)$-generalized schemes are derived schemes.

References

• Jacob Lurie, section 4.3 Structured Spaces

In the context of E-infinity geometry (spectral Deligne-Mumford stacks):

• Jacob Lurie, section 1 of Quasi-Coherent Sheaves and Tannaka Duality Theorems

• Jacob Lurie, section 1 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem