In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf of commutative ring spectra  on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.
A derived stack is a stacky generalization of a derived scheme.
Differential graded scheme
Over a field of characteristic zero, the theory is equivalent 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.
Connection with differential graded rings and examples
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 , then we can get a derived scheme
is the étale spectrum. Since we can construct a resolution
the derived ring is the koszul complex . The truncation of this derived scheme to amplitude provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme
where we can construct the derived scheme where
Let be a fixed differential graded algebra defined over a field of characteristic . Then a -differential graded algebra is called semi-free if the following conditions hold:
- The underlying graded algebra is a polynomial algebra over , meaning it is isomorphic to
- There exists a filtration on the indexing set where and for any .
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. The (relative) cotangent complex of an -differential graded algebra can be constructed using a semi-free resolution : 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 where is the graded algebra with the non-trivial graded piece in degree 0.
The cotangent complex of a hypersurface can easily be computed: since we have the dga representing the derived enhancement of , we can compute the cotangent complex as
where and is the usual universal derivation. If we take a complete intersection, then the koszul complex
is quasi-isomorphic to the complex
This implies we can construct the cotangent complex of the derived ring as the tensor product of the cotangent complex above for each .
Please note that the cotangent complex in the context of derived geometry differs from the cotangent complex of classical schemes. Namely, if there was a singularity in the hypersurface defined by then the cotangent complex would have infinite amplitude. These observations provide motivation for the hidden smoothness philosophy of derived geometry since we are now working with a complex of finite length.
Given a polynomial function then consider the (homotopy) pullback diagram
where the bottom arrow is the inclusion of a point at the origin. Then, the derived scheme has tangent complex at is given by the morphism
where the complex is of amplitude . Notice that the tangent space can be recovered using and the measures how far away is from being a smooth point.
Given a stack there is a nice description for the tangent complex:
If the morphism is not injective, the measures again how singular the space is. In addition, the euler characteristic of this complex yields the correct (virtual) dimension of the quotient stack.
In particular, if we look at the moduli stack of principal -bundles, then the tangent complex is just .
Derived Schemes in Complex Morse Theory
Derived schemes can be used for analyzing topological properties of affine varieties. For example, consider a smooth affine variety . If we take a regular function and consider the section of
Then, we can take the derived pullback diagram
where is the zero section, constructing a derived critical locus of the regular function .
Consider the affine variety
and the regular function given by . Then,
where we treat the last two coordinates as . The derived critical locus is then the derived scheme
Note that since the left term in the derived intersection is a complete intersection, we can compute a complex representing the derived ring as
where is the koszul complex.
Derived Critical Locus
Consider a smooth function where is smooth. The derived enhancement of , the derived critical locus, is given by the differential graded scheme where the underlying graded ring are the polyvector fields
and the differential is defined by contraction by .
For example, if
we have the complex
representing the derived enhancement of .