Tags: #todo #todo/stub Refs: ?

derived scheme

Replace structure sheaf with a sheaf of simplicial commutative rings \((X, {\mathcal{O}}_X)\) where \((X, \pi_0 {\mathcal{O}}_X)\in {\mathsf{Sch}}\) and \(\pi_i{\mathcal{O}}_X \in {\mathsf{QCoh}}( {}_{\pi_0{\mathcal{O}}_X}{\mathsf{Mod}} )\).

Example: take a surface and a curve, the self-intersection of the curve will be a derived scheme.

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See cotangent complex

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Unsorted/simplicial commutative ring in Unsorted/derived scheme, -Unsorted/pregeometry
pregeometry

Idea: think of \(\mathsf{T}_{\mathrm{an}}(k){\hbox{-}}\)structure \({\mathcal{O}}\) as a sheaf of derived rings equipped with an analytic structure. Then \({\mathcal{O}}({\mathbf{A}}^1)\) is like a sheaf of simplicial commutative rings and \({\mathcal{O}}({\mathbb{D}}_1)\) is like a subsheaf of functions \(f\) with \({\left\lVert {f} \right\rVert}\leq 1\).
Gromov-Witten invariants

derived scheme