Derived scheme

In algebraic geometry, a derived scheme is a pair $(X,{\mathcal {O}})$ consisting of a topological space X and a sheaf ${\mathcal {O}}$ of commutative ring spectra ^[1] 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 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.^[2] It was introduced by Maxim Kontsevich^[3] "as the first approach to derived algebraic geometry."^[4] 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 $f_{1},\ldots ,f_{k}\in \mathbb {C} [x_{1},\ldots ,x_{n}]=R$ , then we can get a derived scheme $(X,{\mathcal {O}}_{\bullet })=\mathbf {RSpec} (R/(f_{1})\otimes _{R}^{\mathbf {L} }\cdots \otimes _{R}^{\mathbf {L} }R/(f_{k}))$ where

{\textbf {RSpec}}:({\textbf {dga}}_{\mathbb {C} })^{op}\to {\textbf {DerSch}}

is the étale spectrum. Since we can construct a resolution

{\begin{matrix}0\to &R&{\xrightarrow {\cdot f_{i}}}&R&\to 0\\&\downarrow &&\downarrow &\\0\to &0&\to &R/(f_{i})&\to 0\end{matrix}}

the derived ring $R/(f_{1})\otimes _{R}^{\mathbf {L} }\cdots \otimes _{R}^{\mathbf {L} }R/(f_{k})$ is the koszul complex $K_{R}(f_{1},\ldots ,f_{k})$ .

Cotangent Complex

Examples

The cotangent complex of a hypersurface $X=\mathbb {V} (f)\subset \mathbb {A} _{\mathbb {C} }^{n}$ can easily be computed: since we have the dga $K_{R}(f)$ representing the derived enhancement of $X$ , we can compute the cotangent complex as

0\to R\cdot ds{\xrightarrow {\Phi }}\oplus _{i}R\cdot dx_{i}\to 0

where $\Phi (gds)=g\cdot df$ and $d$ is the usual universal derivation. If we take a complete intersection, then the koszul complex

R^{\bullet }={\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{(f_{1})}}\otimes _{\mathbb {C} [x_{1},\ldots ,x_{n}]}^{\mathbf {L} }\cdots \otimes _{\mathbb {C} [x_{1},\ldots ,x_{n}]}^{\mathbf {L} }{\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{(f_{k})}}

is quasi-isomorphic to the complex

{\frac {\mathbb {C} [x_{1},\ldots ,x_{n}]}{(f_{1},\ldots ,f_{k})}}[+0]

This implies we can construct the cotangent complex of the derived ring $R^{\bullet }$ as the tensor product of the cotangent complex above for each $f_{i}$ .

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 $f$ 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

Tangent Complexes

Polynomial Functions

Given a polynomial function $f:\mathbb {A} ^{n}\to \mathbb {A} ^{m}$ , then consider the (homotopy) pullback diagram

{\begin{matrix}Z&\to &\mathbb {A} ^{n}\\\downarrow &&\downarrow f\\\{pt\}&{\xrightarrow {0}}&\mathbb {A} ^{m}\end{matrix}}

where the bottom arrow is the inclusion of a point at the origin. Then, the derived scheme $Z$ has tangent complex at $x\in Z$ is given by the morphism

\mathbf {T} _{x}=T_{x}\mathbb {A} ^{n}{\xrightarrow {df_{x}}}T_{0}\mathbb {A} ^{m}

where the complex is of amplitude $[-1,0]$ . Notice that the tangent space can be recovered using $H^{0}$ and the $H^{{-1}}$ measures how far away $x\in Z$ is from being a smooth point.

Stack Quotients

Given a stack $[X/G]$ there is a nice description for the tangent complex:

\mathbf {T} _{x}={\mathfrak {g}}_{x}\to T_{x}X

If the morphism is not injective, the $H^{{-1}}$ 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 $G$ -bundles, then the tangent complex is just ${\mathfrak {g}}[+1]$ .

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 $M\subset \mathbb {A} ^{n}$ . If we take a regular function $f:M\to \mathbb {C}$ and consider the section of $\Omega_M$

\Gamma _{df}:M\to \Omega _{M}

sending

x\mapsto (x,df(x))

Then, we can take the derived pullback diagram

{\begin{matrix}X&\to &M\\\downarrow &&\downarrow 0\\M&{\xrightarrow {\Gamma _{df}}}&\Omega _{M}\end{matrix}}

where $0$ is the zero section, constructing a derived critical locus of the regular function $f$ .

Example

Consider the affine variety

M={\text{Spec}}\left(\mathbb {C} [x,y]\right)

and the regular function given by $f(x,y)=x^{2}+y^{3}$ . Then,

\Gamma _{df}(a,b)=(a,b,2a,3b^{2})

where we treat the last two coordinates as $dx,dy$ . The derived critical locus is then the derived scheme

{\textbf {RSpec}}\left({\frac {\mathbb {C} [x,y,dx,dy]}{(dx,dy)}}\otimes _{\mathbb {C} [x,y,dx,dy]}^{\mathbf {L} }{\frac {\mathbb {C} [x,y,dx,dy]}{(2x-dx,3y^{2}-dy)}}\right)

Note that since the left term in the derived intersection is a complete intersection, we can compute a complex representing the derived ring as

K_{dx,dy}^{\bullet }(\mathbb {C} [x,y,dx,dy])\otimes _{\mathbb {C} [x,y,dx,dy]}{\frac {\mathbb {C} [x,y,dx,dy]}{(2-dx,3y^{2}-dy)}}

where $K_{dx,dy}^{\bullet }(\mathbb {C} [x,y,dx,dy])$ is the koszul complex.

Notes

↑ also often called $E_\infty$ -ring spectra
↑ Behrend, Kai (2002-12-16). "Differential Graded Schemes I: Perfect Resolving Algebras". arXiv:math/0212225 .
↑ Kontsevich, M. (1994-05-05). "Enumeration of rational curves via torus actions". arXiv:hep-th/9405035 .
↑ http://ncatlab.org/nlab/show/dg-scheme

References

Reaching Derived Algebraic Geometry - Mathoverflow
M. Anel, The Geometry of Ambiguity
K. Behrand, On the Virtual Fundamental Class
P. Goerss, Topological Modular Forms [after Hopkins, Miller, and Lurie]
B. Toën, Introduction to derived algebraic geometry
M. Manetti, The cotangent complex in characteristic 0

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.