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 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. 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.

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

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.

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

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

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

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

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

References

• 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