Derived scheme

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 ${\displaystyle f_{1},\ldots ,f_{k}\in \mathbb {C} [x_{1},\ldots ,x_{n}]=R}$, then we can get a derived scheme

${\displaystyle (X,{\mathcal {O}}_{\bullet })=\mathbf {RSpec} \left(R/(f_{1})\otimes _{R}^{\mathbf {L} }\cdots \otimes _{R}^{\mathbf {L} }R/(f_{k})\right)}$

where

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

is the étale spectrum. Since we can construct a resolution

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

the derived ring ${\displaystyle R/(f_{1})\otimes _{R}^{\mathbf {L} }\cdots \otimes _{R}^{\mathbf {L} }R/(f_{k})}$ is the koszul complex ${\displaystyle K_{R}(f_{1},\ldots ,f_{k})}$. The truncation of this derived scheme to amplitude ${\displaystyle [-1,0]}$ provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme

${\displaystyle \operatorname {Proj} \left({\frac {\mathbb {Z} [x_{0},\ldots ,x_{n}]}{(f_{1},\ldots ,f_{k})}}\right)}$

where ${\displaystyle \deg(f_{i})=d_{i}}$ we can construct the derived scheme ${\displaystyle (\mathbb {P} ^{n},{\mathcal {E}}^{\bullet },(f_{1},\ldots ,f_{k}))}$ where

${\displaystyle {\mathcal {E}}^{\bullet }=[{\mathcal {O}}(-d_{1})\oplus \cdots \oplus {\mathcal {O}}(-d_{k}){\xrightarrow {(\cdot f_{1},\ldots ,\cdot f_{k})}}{\mathcal {O}}]}$

with amplitude ${\displaystyle [-1,0]}$

Let ${\displaystyle (A_{\bullet },d)}$ be a fixed differential graded algebra defined over a field of characteristic ${\displaystyle 0}$. Then a ${\displaystyle A_{\bullet }}$-differential graded algebra ${\displaystyle (R_{\bullet },d_{R})}$ is called semi-free if the following conditions hold:

1. The underlying graded algebra ${\displaystyle R_{\bullet }}$ is a polynomial algebra over ${\displaystyle A_{\bullet }}$, meaning it is isomorphic to ${\displaystyle A_{\bullet }[\{x_{i}\}_{i\in I}]}$
2. There exists a filtration ${\displaystyle \varnothing =I_{0}\subseteq I_{1}\subseteq \cdots }$ on the indexing set ${\displaystyle I}$ where ${\displaystyle \cup _{n\in \mathbb {N} }I_{n}=I}$ and ${\displaystyle s(x_{i})\in A_{\bullet }[\{x_{j}\}_{j\in I_{n}}]}$ for any ${\displaystyle x_{i}\in I_{n+1}}$.

It turns out that every ${\displaystyle A_{\bullet }}$ differential graded algebra admits a surjective quasi-isomorphism from a semi-free ${\displaystyle (A_{\bullet },d)}$ 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 ${\displaystyle (A_{\bullet },d)}$-differential graded algebra ${\displaystyle (B_{\bullet },d_{B})}$ can be constructed using a semi-free resolution ${\displaystyle (R_{\bullet },d_{R})\to (B_{\bullet },d_{B})}$: it is defined as

${\displaystyle \mathbb {L} _{B_{\bullet }/A_{\bullet }}:=\Omega _{R_{\bullet }/A_{\bullet }}\otimes _{R_{\bullet }}B_{\bullet }}$

Many examples can be constructed by taking the algebra ${\displaystyle B}$ representing a variety over a field of characteristic 0, finding a presentation of ${\displaystyle R}$ 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 ${\displaystyle (B_{\bullet },0)}$ where ${\displaystyle B_{\bullet }}$ is the graded algebra with the non-trivial graded piece in degree 0.

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

${\displaystyle 0\to R\cdot ds{\xrightarrow {\Phi }}\bigoplus _{i}R\cdot dx_{i}\to 0}$

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

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

is quasi-isomorphic to the complex

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

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

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 ${\displaystyle f}$ 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 length.

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

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

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

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

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

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

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

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

Consider the affine variety

${\displaystyle M=\operatorname {Spec} (\mathbb {C} [x,y])}$

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

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

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

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

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

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

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

For example, if

${\displaystyle {\begin{cases}f:\mathbb {C} ^{2}\to \mathbb {C} \\f(x,y)=x^{2}+y^{3}\end{cases}}}$

we have the complex

${\displaystyle R\cdot \partial x\wedge \partial y{\xrightarrow {2xdx+3y^{2}dy}}R\cdot \partial x\oplus R\cdot \partial y{\xrightarrow {2xdx+3y^{2}dy}}R}$

representing the derived enhancement of ${\displaystyle \operatorname {Crit} (f)}$.