nLab derived scheme

Idea

… derived algebraic geometry … higher algebra …generalized scheme…

Definition

Let $k$ be a commutative ring.

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

Special cases

A 0-trucated and 0-localic derived scheme is precisely an ordinary scheme.

More precisely:

Proposition (StSp, 4.2.9)

Let $Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k)) \subset Sch(\mathcal{G}_{Zar}(k))$ be the full subcategory of all derived schemes on the 0-trucated and 0-localic ones. This is canonically equivalent to the ordinary category $Sch(k)$ of schemes over $k$:

$$Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k)) \simeq Sch(k) \,.$$

For more comments on this see also

• scheme as locally affine structured (infinity,1)-topos.

Related concepts

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

References

section 4.2 in

• Jacob Lurie, Structured Spaces