Derived scheme

In algebraic geometry, a derived scheme, by definition, is a pair $(X,{\mathcal {O}})$ consisting of a topological space X and a sheaf ${\mathcal {O}}$ of commutative ring spectra on X such that (1) the pair $(X,\pi _{0}{\mathcal {O}})$ is a scheme and (2) $\pi _{k}{\mathcal {O}}$ is a quasi-coherent $\pi _{0}{\mathcal {O}}$-module. The notion gives a homotopy-theoretic generalization of a scheme.

A basic example, besides a scheme, is a stack. Loop spaces are also 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 is (roughly) equivalent to the theory of differential graded commutative rings.

References

• P. Goerss, Topological Modular Forms [after Hopkins, Miller, and Lurie]
• B. Toën, Introduction to derived algebraic geometry