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
From the locally ringed space point-of-view, a derived scheme is a pair
Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme.
[2] By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology.
[3] It was introduced by Maxim Kontsevich[4] "as the first approach to derived algebraic geometry.
"[5] 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.
[citation needed] Since we can construct a resolution the derived ring
, a derived tensor product, is the koszul complex
provides a classical model motivating derived algebraic geometry.
be a fixed differential graded algebra defined over a field of characteristic
is called semi-free if the following conditions hold: It turns out that every
differential graded algebra admits a surjective quasi-isomorphism from a semi-free
differential graded algebra, called a semi-free resolution.
These are unique up to homotopy equivalence in a suitable model category.
: it is defined as Many examples can be constructed by taking the algebra
representing a variety over a field of characteristic 0, finding a presentation of
as a quotient of a polynomial algebra and taking the Koszul complex associated to this presentation.
The Koszul complex acts as a semi-free resolution of the differential graded algebra
These observations provide motivation for the hidden smoothness philosophy of derived geometry since we are now working with a complex of finite length.
then consider the (homotopy) pullback diagram where the bottom arrow is the inclusion of a point at the origin.
there is a nice description for the tangent complex: If the morphism is not injective, the
In addition, the Euler characteristic of this complex yields the correct (virtual) dimension of the quotient stack.
Derived schemes can be used for analyzing topological properties of affine varieties.
is the zero section, constructing a derived critical locus of the regular function
, the derived critical locus, is given by the differential graded scheme
where the underlying graded ring are the polyvector fields and the differential
For example, if we have the complex representing the derived enhancement of