Girish Mahajan (Editor)

Derived scheme

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

Contents

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 is (roughly in homotopical sense) 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 f 1 , … , f k ∈ C [ x 1 , … , x n ] = R , then we can get a derived scheme ( X , O ∙ ) = R S p e c ( R / ( f 1 ) ⊗ L ⋯ ⊗ L R / ( f k ) )

Generalizations

A derived stack is a stacky generalization of a derived scheme.

References

Derived scheme Wikipedia
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