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 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.

Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology. It was introduced by Maxim Kontsevich "as the first approach to derived algebraic geometry." and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine.

Just as affine algebraic geometry is equivalent (in categorical sense) to the theory of commutative rings (commonly called commutative algebra), affine derived algebraic geometry over characteristic zero is equivalent to the theory of commutative differential graded rings. One of the main example of derived schemes comes from the derived intersection of subschemes of a scheme, giving the Koszul complex. For example, let ${\displaystyle f_{1},\ldots ,f_{k}\in \mathbb {C} [x_{1},\ldots ,x_{n}]=R}$, then we can get a derived scheme

where

is the étale spectrum.^{[citation needed]} Since we can construct a resolution