Derived scheme

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a derived scheme is a <a href="/facts/Homotopy/1nWrPxdj">homotopy</a>-theoretic generalization of a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> in which classical <a href="/facts/Commutative_ring/QS1r2ovR"> commutative rings</a> are replaced with derived versions such as <a href="/facts/Differential_graded_algebra/zySud4Kl">differential graded algebras</a>, commutative <a href="/facts/Simplicial_ring/YWY6cWoR">simplicial rings</a>, or <a href="/facts/Commutative_ring_spectrum/34WgWhWd">commutative ring spectra</a>.
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 
 
 
 
 {
 S
 p
 e
 c
 (
 
 A
 
 i
 
 
 )
 →
 X
 }
 
 
 {\displaystyle \{Spec(A_{i})\to X\}}
 
.
From the locally <a href="/facts/Ringed_space/JyeaTMtX">ringed space</a> point-of-view, a derived scheme is a pair 
 
 
 
 (
 X
 ,
 
 
 O
 
 
 )
 
 
 {\displaystyle (X,{\mathcal {O}})}
 
 consisting of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X and a <a href="/facts/Sheaf_of_spectra/zMYP4XUb">sheaf</a> 
 
 
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {O}}}
 
 either of simplicial commutative rings or of <a href="/facts/Commutative_ring_spectrum/34WgWhWd">commutative ring spectra</a> on X such that (1) the pair 
 
 
 
 (
 X
 ,
 
 π
 
 0
 
 
 
 
 O
 
 
 )
 
 
 {\displaystyle (X,\pi _{0}{\mathcal {O}})}
 
 is a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> and (2) 
 
 
 
 
 π
 
 k
 
 
 
 
 O
 
 
 
 
 {\displaystyle \pi _{k}{\mathcal {O}}}
 
 is a <a href="/facts/Quasi-coherent_sheaf/h2NoUN7e">quasi-coherent</a> 
 
 
 
 
 π
 
 0
 
 
 
 
 O
 
 
 
 
 {\displaystyle \pi _{0}{\mathcal {O}}}
 
-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>. 
A <a href="/facts/Derived_stack/UzP9BsnQ">derived stack</a> is a stacky generalization of a derived scheme.

Derived scheme open-in-new

Derived scheme