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derived Deligne-Mumford stack

Contents

Idea

derived algebraic geometryhigher algebrageneralized 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

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

Revised on May 22, 2014 09:38:20 by Urs Schreiber (88.128.80.84)