derived Deligne-Mumford stack

… derived algebraic geometry … higher algebra …generalized scheme…

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

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).

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

spectral Deligne-Mumford stack is quasi-compact, quasi-separated E-∞ algebraic space? precisely if it admits a scallop decomposition.

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) $\mathcal{G}_{et}(k)$ is not interchangeable with the Zariski geometry $\mathcal{G}_{et}(k)$. Instead $\mathcal{G}_{Zar}(k)$-generalized schemes are derived schemes.

- Jacob Lurie, section 4.3
*Structured Spaces*

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

Last revised on May 22, 2014 at 09:38:20. See the history of this page for a list of all contributions to it.