derived Deligne-Mumford stack in nLab

Contents

Contents

Idea

… derived algebraic geometry … higher algebra …generalized scheme…

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

Definition

Let kk be a commutative ring.

A derived Deligne-Mumford stack (over kk) is a generalized scheme in the sense of locally affine 𝒢\mathcal{G}-structured (infinity,1)-topos for 𝒢=𝒢 et(k)\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 (∞,1)(\infty,1)-presheaves on E ∞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.

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