Derived scheme

In algebraic geometry, a derived scheme is a pair formula_1 consisting of a topological space "X" and a sheaf formula_2 of commutative ring spectra on "X" such that (1) the pair formula_3 is a scheme and (2) formula_4 is a quasi-coherent formula_5-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 formula_6, then we can get a derived scheme 
where
is the étale spectrum. Since we can construct a resolution
the derived ring formula_10 is the koszul complex formula_11. The truncation of this derived scheme to amplitude formula_12 provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme
where formula_14 we can construct the derived scheme formula_15 where
with amplitude formula_12
Cotangent Complex.
Construction.
Let formula_18 be a fixed differential graded algebra defined over a field of characteristic formula_19. Then a formula_20-differential graded algebra formula_21 is called semi-free if the following conditions hold:
It turns out that every formula_20 differential graded algebra admits a surjective quasi-isomorphism from a semi-free formula_18 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 formula_18-differential graded algebra formula_33 can be constructed using a semi-free resolution formula_34: it is defined as
Many examples can be constructed by taking the algebra formula_36 representing a variety over a field of characteristic 0, finding a presentation of formula_37 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 formula_38 where formula_39 is the graded algebra with the non-trivial graded piece in degree 0.
Examples.
The cotangent complex of a hypersurface formula_40 can easily be computed: since we have the dga formula_41 representing the derived enhancement of formula_42, we can compute the cotangent complex as
where formula_44 and formula_45 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 formula_48 as the tensor product of the cotangent complex above for each formula_49.
Remarks.
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 formula_50 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.
Tangent Complexes.
Polynomial Functions.
Given a polynomial function formula_51 then consider the (homotopy) pullback diagram
where the bottom arrow is the inclusion of a point at the origin. Then, the derived scheme formula_53 has tangent complex at formula_54 is given by the morphism
where the complex is of amplitude formula_12. Notice that the tangent space can be recovered using formula_57 and the formula_58 measures how far away formula_59 is from being a smooth point.
Stack Quotients.
Given a stack formula_60 there is a nice description for the tangent complex:
If the morphism is not injective, the formula_58 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 formula_63-bundles, then the tangent complex is just formula_64.
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 formula_65. If we take a regular function formula_66 and consider the section of formula_67
Then, we can take the derived pullback diagram
where formula_19 is the zero section, constructing a derived critical locus of the regular function formula_50.
Example.
Consider the affine variety
and the regular function given by formula_73. Then,
where we treat the last two coordinates as formula_75. 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 formula_78 is the koszul complex.
Derived Critical Locus.
Consider a smooth function formula_66 where formula_80 is smooth. The derived enhancement of formula_81, the derived critical locus, is given by the differential graded scheme formula_82 where the underlying graded ring are the polyvector fields
and the differential formula_84 is defined by contraction by formula_85.
Example.
For example, if 
we have the complex
representing the derived enhancement of formula_81.