Derived scheme

Cotangent complex

Construction

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 ${\displaystyle \bigcup _{n\in \mathbb{N}}{I}_n=I}$ and ${\displaystyle d_R(x_i)\in A_{\bullet }[\{x_j{\}}_{j\in I_n}]}$ 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

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

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.

Examples

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

${\displaystyle 0\to R\cdot ds\stackrel{\mathrm{\Phi }}{\to }\underset{i}{⨁}R\cdot d{x}_i\to 0}$

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

${\displaystyle R^{\bullet }=\frac{\mathbb{C}[x_1,\dots ,x_n]}{(f_1)}{\otimes }_{\mathbb{C}[x_1,\dots ,x_n]}^{\mathbf{L}}\cdots {\otimes }_{\mathbb{C}[x_1,\dots ,x_n]}^{\mathbf{L}}\frac{\mathbb{C}[x_1,\dots ,x_n]}{(f_k)}}$

is quasi-isomorphic to the complex

${\displaystyle \frac{\mathbb{C}[x_1,\dots ,x_n]}{(f_1,\dots ,f_k)}[+0].}$

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 ${\displaystyle f_i}$.

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 ${\displaystyle f}$ 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.

Tangent complexes

Polynomial functions

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

${\displaystyle \begin{array}{ccc}Z& \to & {\mathbb{A}}^n\\ \downarrow & & \downarrow f\\ \{pt\}& \stackrel{0}{\to }& {\mathbb{A}}^m\end{array}}$

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

${\displaystyle {\mathbf{T}}_x=T_x{\mathbb{A}}^n\stackrel{d{f}_x}{\to }{T}_0{\mathbb{A}}^m}$

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.

Stack quotients

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 ${\displaystyle \mathfrak{g}[+1]}$.

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

${\displaystyle \{\begin{array}{l}{\mathrm{\Gamma }}_{df}:M\to {\mathrm{\Omega }}_M\\ x\mapsto (x,df(x))\end{array}}$

Then, we can take the derived pullback diagram

${\displaystyle \begin{array}{ccc}X& \to & M\\ \downarrow & & \downarrow 0\\ M& \stackrel{{\mathrm{\Gamma }}_{df}}{\to }& {\mathrm{\Omega }}_M\end{array}}$

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

Example

Consider the affine variety

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

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

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

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

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

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.

Derived critical locus

Consider a smooth function ${\displaystyle f:M\to \mathbb{C}}$ where ${\displaystyle M}$ is smooth. The derived enhancement of ${\displaystyle \mathrm{Crit}(f)}$, the derived critical locus, is given by the differential graded scheme ${\displaystyle (M,{\mathcal{A}}^{\bullet },Q)}$ where the underlying graded ring are the polyvector fields

${\displaystyle {\mathcal{A}}^{-i}={\wedge }^i{T}_M}$

and the differential ${\displaystyle Q}$ is defined by contraction by ${\displaystyle df}$.

Example

For example, if

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

we have the complex

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

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