In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra.

From the functor of points point-of-view, a derived scheme is a sheaf X on the category of simplicial commutative rings which admits an open affine covering .

From the locally ringed space point-of-view, 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[1] on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module.

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

Differential graded scheme

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Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme.[2] By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology.[3] It was introduced by Maxim Kontsevich[4] "as the first approach to derived algebraic geometry."[5] and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine.

Connection with differential graded rings and examples

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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

 

where

 

is the étale spectrum.[citation needed] Since we can construct a resolution

 

the derived ring  , a derived tensor product, 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  

Cotangent complex

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Construction

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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:

  1. The underlying graded algebra   is a polynomial algebra over  , meaning it is isomorphic to  
  2. 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.

Examples

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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  .

Remarks

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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.

Tangent complexes

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Polynomial functions

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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.

Stack quotients

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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

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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  .

Example

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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

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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  .

Example

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For example, if

 

we have the complex

 

representing the derived enhancement of  .

Notes

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  1. ^ also often called  -ring spectra
  2. ^ section 1.2 of Eugster, J.; Pridham, J.P. (2021-10-25). "An introduction to derived (algebraic) geometry". arXiv:2109.14594 [math.AG].
  3. ^ Behrend, Kai (2002-12-16). "Differential Graded Schemes I: Perfect Resolving Algebras". arXiv:math/0212225.
  4. ^ Kontsevich, M. (1994-05-05). "Enumeration of rational curves via torus actions". arXiv:hep-th/9405035.
  5. ^ "Dg-scheme".

References

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