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derived scheme

Contents

Idea

derived algebraic geometryhigher algebrageneralized scheme

Definition

Let kk be a commutative ring.

A derived scheme (over kk) is a generalized scheme in the sense of locally affine 𝒢\mathcal{G}-structured (infinity,1)-topos for 𝒢=𝒢 Zar(k)\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 0 0(𝒢 Zar(k))Sch(𝒢 Zar(k))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)Sch(k) of schemes over kk:

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

For more comments on this see also

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

References

section 4.2 in

Revised on December 8, 2010 17:07:52 by Urs Schreiber (131.211.233.8)