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\section*{cotangent complex}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{ModCat}{Model category presentation}\dotfill \pageref*{ModCat} \linebreak
\noindent\hyperlink{LeftDerivedFunctorOfKaehlerDifferentials}{Quillen's definition: left derived functor of K\"a{}hler differentials}\dotfill \pageref*{LeftDerivedFunctorOfKaehlerDifferentials} \linebreak
\noindent\hyperlink{explicit_resolutions}{Explicit resolutions}\dotfill \pageref*{explicit_resolutions} \linebreak
\noindent\hyperlink{InfCat}{$(\infty,1)$-categorical description}\dotfill \pageref*{InfCat} \linebreak
\noindent\hyperlink{further_properties_and_applications}{Further properties and applications}\dotfill \pageref*{further_properties_and_applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \textbf{cotangent complex} is a \emph{derived} or [[(∞,1)-category|(∞,1)-categorical]] refinement of the notion of [[Kähler differential]]s.

Traditionally this has been conceived in terms of [[model category]] presentations. This we discuss in the section

\begin{itemize}%
\item \hyperlink{ModCat}{Model category presentation}.

\end{itemize}
From the [[nPOV]], the notion of cotangent complex has a more intrinsic description as being the [[left adjoint]] to the [[tangent (∞,1)-category]] projection. This we discuss in the section

\begin{itemize}%
\item \hyperlink{InfCat}{(∞,1)-Categorical description}

\end{itemize}
\hypertarget{motivation}{}\subsubsection*{{Motivation}}\label{motivation}

The construction of [[moduli space]]s depends strongly on local properties like smoothness and [[transversal map|transversality]] of intersections when trying to [[representable functor|represent]] the [[functor]] of assigning families of objects to the varying base of family. When passing to the classes of equivalent objects, one faces the problem of having nontrivial [[automorphism]]s.

At the [[infinitesimal object|infinitesimal]] level [[automorphism]]s correspond to the [[derivation]]s. Taking derivations is represented by the [[module]] of relative [[Kähler differential]]s which sufficies in good cases. Its correct [[derived algebraic geometry|derived]] replacement is the \textbf{cotangent complex} of Grothendieck-Illusie. One can typically split the information about a map of higher rings into its ``discrete part'' and infinitesimal obstruction theory governed by the cotangent complex.

\hypertarget{ModCat}{}\subsection*{{Model category presentation}}\label{ModCat}

\hypertarget{LeftDerivedFunctorOfKaehlerDifferentials}{}\subsubsection*{{Quillen's definition: left derived functor of K\"a{}hler differentials}}\label{LeftDerivedFunctorOfKaehlerDifferentials}

The \textbf{cotangent complex} functor is effectively the left [[derived functor]] of the [[Kähler differential]]s assignment.

To talk about the nonabelian derived functors, Quillen introduced a [[simplicial ring|model structure on the category of simplicial commutative rings]].

Given a morphism $f: A\to B$ in [[CRing]], which makes $B$ an $A$-algebra, the fiber of the [[tangent category]] $AbGr(A Alg/B)$ of abelian group objects in the slice category $A Alg/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules.

In particular we can consider the forgetful functor $AbGr(A Alg/B)\to A Alg/B$ which has a [[left adjoint]] $\Omega_{A/B} Ab_{B/A} : A Alg/B\to AbGr(A Alg/B)\cong {}_B Mod$. This is the [[Kähler differential]]s functor.

All said is true for [[simplicial ring|simplicial commutative rings]] as well.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{relative cotangent complex} functor is the left [[derived functor]]

\begin{displaymath}
\mathbb{L} \Omega_{B/A} : 
   s A Alg/B\to s AbGr(A Alg/B)\cong s B Mod
\end{displaymath}
Its value on $B$ is the relative cotangent complex $L_{B/A}$

\end{defn}
The [[Andre-Quillen cohomology]] of $R$ is the [[cohomology]] of $\mathbb{L}\Omega(R)$.

\hypertarget{explicit_resolutions}{}\subsubsection*{{Explicit resolutions}}\label{explicit_resolutions}

Here is one way to compute the required cofibrant [[resolution]] for the construction of the left [[derived functor]] for the case that $A = k$ is a [[field]].

Let $P : CAlg \to CAlg$ be the [[comonad]] induced by the [[adjunction]]

\begin{displaymath}
U : CAlg \to Set : k[-]
\end{displaymath}
that sends a commutative $k$-algebra $R$ to the polynomial algebra on its underlying set.

Let $P_\bullet R$ be the corresponding [[bar construction]] [[simplicial ring|simplicial algebra]]. The canonical morphism $P_\bullet R \to R$ with $R$ on the right regarded as a constant simplicial object is a resolution of $R$.

Forming degreewise the module of [[Kähler differential]]s on this yields the simplicial object $\Omega_{k}(P_\bullet R)$, which is a $P_\bullet R$-module.

\begin{prop}
\label{}\hypertarget{}{}
The cotangent complex of $R$ is equivalent to

\begin{displaymath}
(\mathbb{L} \Omega_{/k}) (R) \simeq
  R \otimes_{P_\bullet R} \Omega_k(P_\bullet R)
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Notice that the universal property of the [[Kähler differential]]s is that for $R$ a ring and $N$ an $R$-module, we have

\begin{displaymath}
Hom(\Omega(R), N) \simeq Der(R,N)
  \,.
\end{displaymath}
Accordingly, it follows that the [[Andre-Quillen cohomology]] of $R$ with values in $N$, which is the cohomology of the cosimplicial object

\begin{displaymath}
Hom((\mathbb{L}\Omega)(R), N)
\end{displaymath}
is equivalently the cohomology of the object

\begin{displaymath}
\begin{aligned} 
    \cdots & \simeq
    Hom(R \otimes_{P_\bullet R} \Omega_k(P_\bullet R), N)
    \\
    & \simeq 
    Hom_{P_\bullet R}( \Omega_k(P_\bullet R), N)
    \\
    & \simeq
    Der(P_\bullet R, N)
  \end{aligned}
  \,.
\end{displaymath}
In particular we have the the degree-0 cohomology of this complex is the module of ordinary derivations

\begin{displaymath}
H^0(Der(P_\bullet R, N)) \simeq Der(R,N)
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
If in the above $k$ is [[field]] of characteristic 0, then [[Andre-Quillen cohomology]] of the $k$-algebra $R$ with coefficients in a [[module]] $N$ is a [[direct sum]]mand of the corresponding [[Hochschild cohomology]]:

\begin{displaymath}
H^q(Hom(\mathbb{L} \Omega (R)), N)
  \simeq
  HH^{q+1}_{(1)}(R,N)
  \,,
\end{displaymath}
where the subscript refers to [[Hodge decomposition]] of [[Hochschild cohomology]].

\end{prop}
This is in section 8.8 of

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}

\end{itemize}
\hypertarget{InfCat}{}\subsection*{{$(\infty,1)$-categorical description}}\label{InfCat}

The \emph{cotangent complex} is a generalization to [[higher category theory]] and [[higher algebra]] of the notion of [[cotangent bundle]] in the sense of [[Kähler differential]]s.

Recall from above that for $C =$ [[CRing]] the ordinary category of commutative rings, the cotangent complex functor is the [[section]]

\begin{displaymath}
\Omega_K : Ring \to Mod
\end{displaymath}
of the canonical [[bifibration]] $Mod \to Ring$ of [[module]]s over [[ring]]s that is on objects given by forming the module of [[Kähler differential]]s.

This generalizes to the case where [[CRing]] is replaced by any [[(∞,1)-category]] $C$: the cotangent complex functor for $C$ is here the [[left adjoint]] [[section]]

\begin{displaymath}
\Omega : C \to T_C
\end{displaymath}
of the [[tangent (∞,1)-category]] projection $dom : T_C \to C$.

In particular, when $C = ...$, then the cotangent complex assigns \ldots{} .

\hypertarget{further_properties_and_applications}{}\subsection*{{Further properties and applications}}\label{further_properties_and_applications}

For more background see [[deformation theory]].

Apart from simplicial rings we can consider $E_\infty$-rings. A map of connective $E_\infty$-rings is an equivalence, if it induces an isomorphism at the level of $\pi_0$ plus a condition on the relative cotangent complex. Similarly, one can express the descent properties of higher stacks via the usual gluing at the bottom level plus the obstruction theory for relative cotangent complex. Study of an appropriate version of the [[Postnikov tower]] is a systematic way to do this.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[deformation theory]], [[derived deformation theory]]


\item [[tangent complex]], [[André-Quillen cohomology]], [[Hochschild cohomology]]


\item \textbf{cotangent complex}, [[André-Quillen homology]], [[Hochschild homology]]


\item [[topological André-Quillen homology]], [[topological Hochschild homology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

See also [[deformation theory]] and references therein.

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{Cat\'e{}gories cofibr\'e{}es additives et complexe cotangent relatif}, Lec. Notes in Math. \textbf{79}


\item [[Luc Illusie]], \emph{Complexe cotangent et d\'e{}formations I}, Lec. Notes Math. \textbf{239}, Springer 1971, xv+355 pp.; \emph{II}, LNM \textbf{283}, Springer 1972. vii+304 xv+355 pp.


\item [[Kai Behrend]], [[Barbara Fantechi]], \emph{The intrinsic normal cone}, Invent. Math. \textbf{128} (1997), no. 1, 45--88, MR1437495 (98e:14022) \href{http://arxiv.org/abs/alg-geom/9601010}{arXiv:alg-geom/9601010}


\item [[Barbara Fantechi]], M. Manetti, \emph{Obstruction calculus for functors of Artin rings I}, J. Algebra \textbf{202} (1998), no. 2, 541--576, MR1617687 (99f:14004).


\item [[Jacob Lurie]], [[Deformation Theory]], (\href{http://arxiv.org/abs/0709.3091}{arXiv:0709.3091})


\item [[Stefan Schwede]], \emph{Spectra in model categories and applications to the algebraic cotangent complex}, J. Pure Appl. Alg. \textbf{120}, 77--104 (1997) ()


\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]} section 8.8.


\item [[Gabriele Vezzosi]], \emph{A note on the cotangent complex in derived algebraic geometry}, \href{http://arxiv.org/abs/1008.0601}{arxiv/1008.0601}


\item [[The Stacks Project]], \emph{The cotangent complex} (\href{http://stacks.math.columbia.edu/download/cotangent.pdf}{pdf})



\end{itemize}
A short exposition (from the point of view of [[formal schemes]]) is in

\begin{itemize}%
\item chapter 5 (5.29-5.31) in Luc Illusie, \emph{Grothendieck's existence theorem in formal geometry}, in [[FGA explained]] (179--233) MR2223409; (draft version \href{http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes}{pdf})

\end{itemize}
The cotangent complex for a general [[algebra over an operad]] in [[chain complexes]] is discussed in section 7 of

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras} Communications in algebra, 25(10). 3291-3323 (1997)(\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453}))

\end{itemize}
In terms of [[model category]] presentations for [[tangent (infinity,1)-categories]]:

\begin{itemize}%
\item [[Yonatan Harpaz]], [[Joost Nuiten]], [[Matan Prasma]], \emph{Tangent categories of algebras over operads} (\href{https://arxiv.org/abs/1612.02607}{arXiv:1612.02607})


\item \emph{The abstract cotangent complex and Quillen cohomology of enriched categories} (\href{https://arxiv.org/abs/1612.02608}{arXiv:1612.02608})



\end{itemize}
[[!redirects cotangent complexes]]



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