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Concept
Derived scheme
Summary
In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or 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.
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 ) 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
where
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
with amplitude
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.
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