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\section*{formally smooth morphism}

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

\hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms}

[[!include etale morphisms - contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{GeneralAbstractNotion}{General abstract notion}\dotfill \pageref*{GeneralAbstractNotion} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ConcreteNotion}{Concrete notion}\dotfill \pageref*{ConcreteNotion} \linebreak
\noindent\hyperlink{OverCommutativeRings}{Over commutative rings}\dotfill \pageref*{OverCommutativeRings} \linebreak
\noindent\hyperlink{smoothness_versus_formal_smoothness}{Smoothness versus formal smoothness}\dotfill \pageref*{smoothness_versus_formal_smoothness} \linebreak
\noindent\hyperlink{formally_smooth_scheme}{Formally smooth scheme}\dotfill \pageref*{formally_smooth_scheme} \linebreak
\noindent\hyperlink{over_noncommutative_algebras}{Over noncommutative algebras}\dotfill \pageref*{over_noncommutative_algebras} \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}

A [[space]] $X$ is called \textbf{formally smooth} if every [[morphism]]s $Y \to X$ into it has all possible [[infinitesimal object|infinitesimal]] extensions.

(If there is at most one extension per infinitesimal extension of $Y$ with no guarantee of existence it is called a [[formally unramified morphism]]. If the thickenings exist uniquely, it is called a [[formally etale morphism]]).

Traditionally this has considered in the context of [[geometry]] over formal duals of [[ring]]s and [[associative algebra]]s. This we discuss in the section (\hyperlink{ConcreteNotion}{Concrete notion}). But generally the notion makes sense in any context of . This we discuss in the section \hyperlink{GeneralAbstractNotion}{General abstract notion}.

\hypertarget{GeneralAbstractNotion}{}\subsection*{{General abstract notion}}\label{GeneralAbstractNotion}

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

Let

\begin{displaymath}
\mathbf{H}
   \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}}
  \mathbf{H}_{th}
\end{displaymath}
be an [[adjoint triple]] of [[functors]] with $u^*$ a [[full and faithful functor]] that preserves the [[terminal object]].

We may think of this as exhibiting  (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as [[cohesive topos|cohesive space]]s and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with [[infinitesimal object|infinitesimal extension]].

As a class of examples that is useful to keep in mind consider a [[Q-category]] $(cod \dashv \epsilon \dashv dom) : \bar A \to A$ of  and let

\begin{displaymath}
((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := 
  ([dom,Set] \dashv [\epsilon, Set] \dashv [codom,Set] : [\bar A, Set] \to [A,Set])
\end{displaymath}
be the corresponding .

For any such setup there is a canonical [[natural transformation]]

\begin{displaymath}
u^* \to u^!
  \,.
\end{displaymath}
Details of this are in the section  at [[cohesive topos]].

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

\begin{equation}
u^* X \to u^* Y \prod_{u^! Y} u^! X
  \,.
\label{MorphismIntoPullback}\end{equation}
\begin{defn}
\label{AbstractFormallySmoothMorphism}\hypertarget{AbstractFormallySmoothMorphism}{}
A morphism $f : X \to Y$ in $\mathbf{H}$ is called \textbf{formally smooth} if \eqref{MorphismIntoPullback} is an [[effective epimorphism]].

\end{defn}
This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, def. 5.1, prop. 5.3.1.1}).

The dual notion, where the above morphism is a [[monomorphism]] is that of [[formally unramified morphism]]. If both conditions hold, hence if the above morphism is an [[isomorphism]], one speaks of a [[formally étale morphism]].

\begin{defn}
\label{AbstractFormallySmoothObject}\hypertarget{AbstractFormallySmoothObject}{}
An object $X \in \mathbf{H}$ is called \textbf{formally smooth} if the morphism $X \to *$ to the [[terminal object]] is formally smooth.

\end{defn}
\begin{prop}
\label{AbstractFormallySmoothObjectDirect}\hypertarget{AbstractFormallySmoothObjectDirect}{}
The object $X$ is formally smooth precisely if

\begin{displaymath}
u^* X \to u^! X
\end{displaymath}
is an [[effective epimorphism]].

\end{prop}
This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, def. 5.3.2}).

\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
Formally smooth morphisms are closed under composition.

\end{prop}
This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenberg, prop. 5.4}).

\hypertarget{ConcreteNotion}{}\subsection*{{Concrete notion}}\label{ConcreteNotion}

\hypertarget{OverCommutativeRings}{}\subsubsection*{{Over commutative rings}}\label{OverCommutativeRings}

Let $k$ be a [[field]] and let $CAlk_k$ be the [[category]] of commutative [[associative algebra]]s over $k$. Write

\begin{displaymath}
\mathbf{H} = [CAlg_k, Set]
\end{displaymath}
for the [[presheaf topos]] over the [[opposite category]] $CAlg_k^{op}$. This is the context in which [[scheme]]s and [[algebraic space]]s over $k$ live.

\begin{defn}
\label{CFormalSmoothness}\hypertarget{CFormalSmoothness}{}
A morphism $f :X\to Y$ in $\mathbf{H} = [CAlg_k, Set]$ is \textbf{formally smooth} if it satisfies the \emph{infinitesimal lifting property}: for every algebra $A$ and nilpotent [[ideal]] $I\subset A$ and morphism $Spec(A)\to Y$ the induced map

\begin{displaymath}
Hom_Y(Spec(A), X)\to Hom_Y(Spec(A/I),X)
\end{displaymath}
is surjective.

\end{defn}
This is due to ([[EGAIV]]${}_4$ 17.1.1)

\begin{prop}
\label{AbstractAndConcreteCommutativeFormallySmoothCoincidesOnObjects}\hypertarget{AbstractAndConcreteCommutativeFormallySmoothCoincidesOnObjects}{}
An object $X \in [CAlg_k, Set]$ is formally smooth in the concrete sense of def. \ref{CFormalSmoothness} precisely if it is so in the abstract sense of def. \ref{AbstractFormallySmoothObjectDirect}.

\end{prop}
This appears as (\hyperlink{KontsevichRosenbergSpaces}{KontsevichRosenbergSpaces, 4.1}).

\hypertarget{smoothness_versus_formal_smoothness}{}\paragraph*{{Smoothness versus formal smoothness}}\label{smoothness_versus_formal_smoothness}

For a morphism $f:X\to Y$ of schemes, and $x$ a point of $X$, the following are equivalent

(i) $f$ is a [[smooth morphism of schemes]] at $x$

(ii) $f$ is locally of finite presentation at $x$ and there is an open neighborhood $U\subset X$ of $x$ such that $f|_U: U\to Y$ is formally smooth

(iii) $f$ is flat at $x$, locally of finite presentation at $x$ and the sheaf of [[Kähler differential]]s $\Omega_{X/Y}$ is locally free in a neighborhood of $x$

The relative dimension of $f$ at $x$ will equal the rank of the module of K\"a{}hler differentials.

This is ([[EGAIV]]${}_4$ 17.5.2 and 17.15.15)

\hypertarget{formally_smooth_scheme}{}\paragraph*{{Formally smooth scheme}}\label{formally_smooth_scheme}

A [[scheme]] $S$, i.e. a scheme over the ground ring $k$, is a [[formally smooth scheme]] if the corresponding morphism $S \to Spec(k)$ is a formally smooth morphism.

There is also an interpretation of formal smoothness via the formalism of [[Q-categories]].

\hypertarget{over_noncommutative_algebras}{}\subsubsection*{{Over noncommutative algebras}}\label{over_noncommutative_algebras}

Let $k$ be a [[field]] and let $Alg_k$ be the category of [[associative algebra]]s over $k$ (not necessarily commutative). Let

\begin{displaymath}
Alg_k^{inf} : \bar A \to Alg_k
\end{displaymath}
be the [[Q-category]] of  of $k$-algebras (whose objects are surjective $k$-algebra morphisms with nilpotent kernel). Notice that the [[presheaf topos]]

\begin{displaymath}
\mathbf{H} := [Alg_k, Set]
\end{displaymath}
is the context in which [[noncommutative scheme]]s live. Let $\mathbf{H}_{th} \to \mathbf{Q}$ be the  over $Alg_k^{inf}$.

\begin{prop}
\label{NCFormalSmoothness}\hypertarget{NCFormalSmoothness}{}
Let $f : R \to S$ be a morphism in $Alg_k$ such that $R$ is a [[separable algebra]]. Write $Spec f : Spec S \to Spec R$ for the corresponding morphism in $\mathbf{H} = [Alg_k, Set]$.

This $Spec f$ is formally smooth in the sense of def. \ref{AbstractFormallySmoothMorphism} precisely if the $S \otimes_k S^{op}$-[[module]]

\begin{displaymath}
\Omega^1_{S|R} := ker ( R \otimes_k R \stackrel{mult}{\to} R \stackrel{f}{\to} S)
\end{displaymath}
is a [[projective object]] in $S \otimes_k S^{op}$[[Mod]].

\end{prop}
In particular, setting $R = k$ we have that an object of the form $Spec S$ is formally smooth according to def. \ref{AbstractFormallySmoothObject} precisely if $\Omega^1(S|k)$ is projective. This is what in (\hyperlink{CuntzQuillen}{CuntzQuillen}) is called the condition for a [[quasi-free algebra]].

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

\textbf{formally smooth morphism} and [[formally unramified morphism]] $\Rightarrow$ [[formally étale morphism]].

[[!include cohesion - table]]

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

The definition over commutative rings is in

\begin{itemize}%
\item [[EGAIV]]${}_4$, Publ. IH\'E{}S \textbf{32} (1967), p. 5-361, \href{http://www.numdam.org/item?id=PMIHES_1967__32__5_0}{numdam}

\end{itemize}
The definition over noncommutative algebras is in

\begin{itemize}%
\item [[J. Cuntz]], [[D. Quillen]], \emph{Algebra extensions and nonsingularity}, J. Amer. Math. Soc. \textbf{8} (1995), 251--289.

\end{itemize}
The general abstract definition and its relation to the standard definitions is in

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2303}{dvi})

\end{itemize}
See also

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/22393/is-formal-smoothness-a-local-property}{MO:is-formal-smoothness-a-local-property}


\item A. Ardizzoni, \emph{Separable functors and [[formal smoothness]]}, J. K-Theory 1 (2008), no. 3, 535--582, \href{http://arxiv.org/abs/math.QA/0407095}{math.QA/0407095}, \href{http://dx.doi.org/10.1017/is007011015jkt012}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=2009k:16069}{MR2009k:16069}


\item T. Brzeziski, \emph{Notes on formal smoothness}, \emph{in}: Modules and Comodules (series \emph{Trends in Mathematics}). T Brzeziski, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkh\"a{}user, Basel, 2008, pp. 113-124 (\href{http://dx.doi.org/10.1007/978-3-7643-8742-6}{doi}, \href{http://arxiv.org/abs/0710.5527}{arXiv:0710.5527})


\item Guillermo Corti\~n{}as, \emph{The structure of smooth algebras in Kapranov's framework for noncommutative geometry}, J. of Algebra \textbf{281} (2004) 679-694, \href{http://arxiv.org/abs/math/0002177}{math.RA/0002177}



\end{itemize}
[[!redirects formal smoothness]] [[!redirects formally smooth]] [[!redirects formally smooth morphisms]]



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