Derived scheme

In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra.

From the functor of points point-of-view, a derived scheme is a sheaf X on the category of simplicial commutative rings which admits an open affine covering ${\displaystyle \{Spec(A_{i})\to X\}}$.

From the locally ringed space point-of-view, a derived scheme is a pair ${\displaystyle (X,{\mathcal {O}})}$ consisting of a topological space X and a sheaf ${\displaystyle {\mathcal {O}}}$ either of simplicial commutative rings or of commutative ring spectra^[1] on X such that (1) the pair ${\displaystyle (X,\pi _{0}{\mathcal {O}})}$ is a scheme and (2) ${\displaystyle \pi _{k}{\mathcal {O}}}$ is a quasi-coherent ${\displaystyle \pi _{0}{\mathcal {O}}}$-module.

A derived stack is a stacky generalization of a derived scheme.