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# Contents

## 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 $k$:

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

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.