# Derived scheme

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In [algebraic geometry](/source/algebraic_geometry), a '''derived scheme''' is a [homotopy](/source/homotopy)-theoretic generalization of a [scheme](/source/scheme_(mathematics)) in which classical [commutative rings](/source/commutative_ring) are replaced with derived versions such as [differential graded algebra](/source/differential_graded_algebra)s, commutative [simplicial ring](/source/simplicial_ring)s, or [commutative ring spectra](/source/commutative_ring_spectrum).

From the [functor of points](/source/functor_of_points) point-of-view, a derived scheme is a [sheaf](/source/Grothendieck_topology) ''X'' on the category of simplicial commutative rings which admits an open affine covering <math>\{Spec(A_i) \to X\}</math>.

From the locally [ringed space](/source/ringed_space) point-of-view, a derived scheme is a pair <math>(X, \mathcal{O})</math> consisting of a [topological space](/source/topological_space) ''X'' and a [sheaf](/source/sheaf_of_spectra) <math>\mathcal{O}</math> either of simplicial commutative rings or of [commutative ring spectra](/source/commutative_ring_spectrum)<ref>also often called <math>E_\infty</math>-ring spectra</ref> on ''X'' such that (1) the pair <math>(X, \pi_0 \mathcal{O})</math> is a [scheme](/source/scheme_(mathematics)) and (2) <math>\pi_k \mathcal{O}</math> is a [quasi-coherent](/source/quasi-coherent_sheaf) <math>\pi_0 \mathcal{O}</math>-[module](/source/module_(mathematics)). 

A [derived stack](/source/derived_stack) is a stacky generalization of a derived scheme.

==Differential graded scheme==
Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme.<ref>section 1.2 of {{Cite arXiv|title = An introduction to derived (algebraic) geometry |eprint = 2109.14594|date = 2021-10-25|first1 = J. | last1 = Eugster | first2= J.P. | last2 = Pridham | class= math.AG }}</ref>  By definition, a '''differential graded scheme''' is obtained by gluing affine differential graded schemes, with respect to [étale topology](/source/%C3%A9tale_topology).<ref>{{Cite arXiv|title = Differential Graded Schemes I: Perfect Resolving Algebras|eprint = math/0212225|date = 2002-12-16|first = Kai|last = Behrend }}</ref> It was introduced by [Maxim Kontsevich](/source/Maxim_Kontsevich)<ref>{{Cite arXiv |title = Enumeration of rational curves via torus actions |eprint = hep-th/9405035|date = 1994-05-05|first = M.|last = Kontsevich }}</ref> "as the first approach to derived algebraic geometry"<ref>{{Cite web|url=http://ncatlab.org/nlab/show/dg-scheme|title = Dg-scheme}}</ref> and was developed further by [Mikhail Kapranov](/source/Mikhail_Kapranov) and Ionut Ciocan-Fontanine.

===Connection with differential graded rings and examples===
Just as [affine](/source/Affine_space) algebraic geometry is equivalent (in [categorical sense](/source/Category_theory)) to the theory of [commutative ring](/source/commutative_ring)s (commonly called [commutative algebra](/source/commutative_algebra)), affine [derived algebraic geometry](/source/derived_algebraic_geometry) over characteristic zero is equivalent to the theory of [commutative differential graded rings](/source/differential_graded_algebra). One of the main example of derived schemes comes from the derived intersection of subschemes of a scheme, giving the [Koszul complex](/source/Koszul_complex). For example, let <math>f_1, \ldots, f_k \in \Complex[x_1,\ldots, x_n] = R</math>, then we can get a derived scheme

:<math>(X,\mathcal{O}_\bullet) = \mathbf{RSpec} \left (R/(f_1) \otimes_R^\mathbf{L} \cdots \otimes_R^\mathbf{L} R/(f_k) \right )</math>

where

:<math>\textbf{RSpec}:(\textbf{dga}_\Complex)^{op} \to \textbf{DerSch}</math>

is the [étale spectrum](/source/%C3%A9tale_spectrum).{{citation needed|date=July 2017}} Since we can construct a resolution

:<math>\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}</math>

the derived ring <math>R/(f_1) \otimes_R^\mathbf{L} \cdots \otimes_R^\mathbf{L} R/(f_k)</math>, a [derived tensor product](/source/derived_tensor_product), is the Koszul complex <math>K_R(f_1,\ldots, f_k)</math>. The truncation of this derived scheme to amplitude <math>[-1,0]</math> provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme

:<math>\operatorname{Proj}\left( \frac{\Z[x_0,\ldots,x_n]}{(f_1,\ldots, f_k)} \right)</math>

where <math>\deg(f_i) = d_i</math> we can construct the derived scheme <math>(\mathbb{P}^n, \mathcal{E}^\bullet,(f_1,\ldots, f_k))</math> where

:<math>\mathcal{E}^\bullet = [\mathcal{O}(-d_1)\oplus\cdots\oplus\mathcal{O}(-d_k) \xrightarrow{(\cdot f_1,\ldots,\cdot f_k)} \mathcal{O}]</math>

with amplitude <math>[-1,0]</math>.

==Cotangent complex==
<!-- Cite manetti for this section, also, are there any universal properties the cotangent complex satisfies? -->

=== Construction ===
Let <math>(A_\bullet,d)</math> be a fixed differential graded algebra defined over a field of characteristic <math>0</math>. Then a <math>A_\bullet</math>-differential graded algebra <math>(R_\bullet,d_R)</math> is called '''semi-free''' if the following conditions hold:
# The underlying graded algebra <math>R_\bullet</math> is a polynomial algebra over <math>A_\bullet</math>, meaning it is isomorphic to <math>A_\bullet[\{x_i \}_{i \in I}]</math>
# There exists a filtration <math>\varnothing = I_0 \subseteq I_1 \subseteq \cdots</math> on the indexing set <math>I</math> where <math>\textstyle\bigcup_{n \in \N} I_n = I</math> and <math>d_R(x_i) \in A_\bullet[\{x_j\}_{j \in I_n}]</math> for any <math>x_i \in I_{n+1}</math>.
It turns out that every <math>A_\bullet</math>-differential graded algebra admits a surjective quasi-isomorphism from a semi-free <math>(A_\bullet,d)</math> differential graded algebra, called a semi-free resolution. These are unique up to [homotopy equivalence](/source/Chain_homotopy_equivalence) in a suitable [model category](/source/model_category). The (relative) '''cotangent complex''' of an <math>(A_\bullet,d)</math>-differential graded algebra <math>(B_\bullet, d_B)</math> can be constructed using a semi-free resolution <math>(R_\bullet,d_R) \to (B_\bullet, d_B)</math>: it is defined as

:<math>\mathbb{L}_{B_\bullet/A_\bullet} := \Omega_{R_\bullet/A_\bullet}\otimes_{R_\bullet} B_\bullet.</math>

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

===Examples===
The [cotangent complex](/source/cotangent_complex) of a [hypersurface](/source/hypersurface) <math>X = \mathbb{V}(f) \subset \mathbb{A}^n_\Complex</math> can easily be computed: since we have the dga <math>K_R(f)</math> representing the '''derived enhancement''' of <math>X</math>, we can compute the cotangent complex as

:<math>0 \to R\cdot ds \xrightarrow{\Phi} \bigoplus_i R \cdot dx_i \to 0</math>

where <math>\Phi(gds) = g\cdot df</math> and <math>d</math> is the usual universal derivation. If we take a complete intersection, then the Koszul complex

:<math>R^\bullet = \frac{\Complex[x_1,\ldots,x_n]}{(f_1)} \otimes^\mathbf{L}_{\Complex[x_1,\ldots,x_n]} \cdots \otimes^\mathbf{L}_{\Complex[x_1, \ldots, x_n]} \frac{\Complex[x_1,\ldots,x_n]}{(f_k)}</math>

is quasi-isomorphic to the complex

:<math>\frac{\Complex[x_1,\ldots,x_n]}{(f_1,\ldots,f_k)}[+0].</math>

This implies we can construct the cotangent complex of the derived ring <math>R^\bullet</math> as the tensor product of the cotangent complex above for each <math>f_i</math>.

===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 <math>f</math> 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 <math>f:\mathbb{A}^n \to \mathbb{A}^m,</math> then consider the (homotopy) pullback diagram

:<math>\begin{matrix}
Z & \to & \mathbb{A}^n \\
\downarrow & & \downarrow f \\
\{ pt \} & \xrightarrow{0} & \mathbb{A}^m
\end{matrix}</math>

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

:<math>\mathbf{T}_x = T_x\mathbb{A}^n \xrightarrow{df_x} T_0\mathbb{A}^m</math>

where the complex is of amplitude <math>[-1,0]</math>. Notice that the [tangent space](/source/tangent_space) can be recovered using <math>H^0</math> and the <math>H^{-1}</math> measures how far away <math>x \in Z</math> is from being a smooth point.
<!-- should discuss the L^\infty structure with explicit examples!. look at the top of page 9 -->

===Stack quotients===
Given a stack <math>[X/G]</math> there is a nice description for the tangent complex:

:<math> \mathbf{T}_x = \mathfrak{g}_x \to T_xX .</math>

If the morphism is not injective, the <math>H^{-1}</math> measures again how singular the space is. In addition, the [Euler characteristic](/source/Euler_characteristic) of this complex yields the correct (virtual) dimension of the [quotient stack](/source/quotient_stack).
<!-- should discuss the L^\infty structure... -->
In particular, if we look at the [moduli stack of principal <math>G</math>-bundles](/source/Moduli_stack_of_principal_bundles), then the tangent complex is just <math>\mathfrak{g}[+1]</math>.

==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 <math>M \subset \mathbb{A}^n</math>. If we take a regular function <math>f:M \to \Complex</math> and consider the section of <math>\Omega_M</math>

:<math>\begin{cases} \Gamma_{df}: M \to \Omega_M \\ x \mapsto (x,df(x)) \end{cases}</math>

Then, we can take the derived pullback diagram

:<math>\begin{matrix}
X & \to & M \\
\downarrow & & \downarrow 0 \\
M & \xrightarrow{\Gamma_{df}} & \Omega_M
\end{matrix}</math>

where <math>0</math> is the zero section, constructing a '''derived critical locus''' of the regular function <math>f</math>.

===Example===
Consider the affine variety

:<math>M = \operatorname{Spec} (\Complex[x,y])</math>

and the regular function given by <math>f(x,y) = x^2 + y^3</math>. Then,

:<math>\Gamma_{df}(a,b) = (a,b,2a,3b^2)</math>

where we treat the last two coordinates as <math>dx, dy</math>. The derived critical locus is then the derived scheme

:<math>\textbf{RSpec}\left( \frac{\Complex[x,y,dx,dy]}{(dx,dy)} \otimes_{\Complex [x,y,dx,dy]}^{\mathbf{L}} \frac{\Complex [x,y,dx,dy]}{(2x - dx, 3y^2 - dy)} \right).</math>

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

:<math>K_{dx,dy}^\bullet(\Complex [x,y,dx,dy])\otimes_{\Complex [x,y,dx,dy]} \frac{\Complex [x,y,dx,dy]}{(2 - dx, 3y^2 - dy)}</math>

where <math>K_{dx,dy}^\bullet(\Complex [x,y,dx,dy])</math> is the Koszul complex.
<!-- The previous computation gave the derived intersection of the cotangent bundle of the hypersurface with the zero section of the embedded space... -->

==Derived critical locus==
Consider a [smooth function](/source/Smoothness) <math>f:M \to \Complex</math> where <math>M</math> is smooth. The derived enhancement of <math>\operatorname{Crit}(f)</math>, the '''derived critical locus''', is given by the differential graded scheme <math>(M,\mathcal{A}^\bullet, Q)</math> where the underlying graded ring are the polyvector fields

:<math>\mathcal{A}^{-i} = \wedge^i T_M</math>

and the differential <math>Q</math> is defined by contraction by <math>df</math>.

===Example===
For example, if

:<math>\begin{cases} f:\Complex^2 \to \Complex \\ f(x,y) = x^2 + y^3 \end{cases}</math>

we have the complex

:<math> R\cdot \partial x\wedge \partial y \xrightarrow{2xdx + 3y^2dy} R \cdot \partial x \oplus R \cdot \partial y \xrightarrow{2xdx + 3y^2dy} R</math>

representing the derived enhancement of <math>\operatorname{Crit}(f)</math>.

==Notes==
{{reflist}}

==References==
* [https://mathoverflow.net/q/217792 Reaching Derived Algebraic Geometry - Mathoverflow]
* M. Anel, [https://web.archive.org/web/20171124204930/http://mathieu.anel.free.fr/mat/doc/Anel%20-%20DerivedGeometry.pdf The Geometry of Ambiguity]
* [K. Behrend](/source/Kai_Behrend), [https://www.math.ubc.ca/~behrend/talks/seoul14.pdf On the Virtual Fundamental Class]
* P. Goerss, [https://web.archive.org/web/20171117065616/http://www.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf Topological Modular Forms <nowiki>[after Hopkins, Miller, and Lurie]</nowiki>]
* B. Toën, [https://web.archive.org/web/20160310212544/https://math.berkeley.edu/~aaron/gaelxx/DAG.pdf Introduction to derived algebraic geometry]
* M. Manetti, [https://web.archive.org/web/20171124205016/http://www1.mat.uniroma1.it/people/manetti/DT2011/marco2.pdf The cotangent complex in characteristic 0]
* G. Vezzosi, [https://arxiv.org/abs/1109.5213 The derived critical locus I - basics]

Category:Algebraic geometry
Category:Homotopy theory

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Adapted from the Wikipedia article [Derived scheme](https://en.wikipedia.org/wiki/Derived_scheme) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Derived_scheme?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.