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\section*{formal scheme}

\begin{quote}%
under construction


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

\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{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{formal_spectra_of_noetherian_rings}{Formal spectra of Noetherian rings}\dotfill \pageref*{formal_spectra_of_noetherian_rings} \linebreak
\noindent\hyperlink{more_general_indschemes}{More general ind-schemes}\dotfill \pageref*{more_general_indschemes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FormalPowerSeries}{Formal power series and their formal spectra}\dotfill \pageref*{FormalPowerSeries} \linebreak
\noindent\hyperlink{formal_neighbourhood_of_the_diagonal}{Formal neighbourhood of the diagonal}\dotfill \pageref*{formal_neighbourhood_of_the_diagonal} \linebreak
\noindent\hyperlink{formal_groups}{Formal groups}\dotfill \pageref*{formal_groups} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{formal scheme} is a [[formal neighbourhood]] of a [[scheme]], a scheme with infinitesimal thickening.

Generally this may be formalized via [[ind-objects]] of schemes. Hence regarded as a [[sheaf]] on [[affine schemes]], a formal scheme is a [[filtered colimit]] of ordinary schemes (e.g. \hyperlink{Strickland00}{Strickland 00, section 4}).

Regarded as a [[locally ringed space]] a formal scheme has the underlying [[topological space]] of an ordinary [[scheme]], but its [[structure sheaf]] contains extra nilpotent elements.

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

Formal [[power series]] [[ring]]s $k[\![x_1,\ldots,x_n]\!]$ are limits of their truncations (e.g. in one variable $k[\![x]\!]/(x^n)$); they can be viewed as completions and they get equipped with a natural filtration and [[adic topology]].

They do not converge as a series (and make sense) in an open set or in any of the standard topologies (e.g. Zariski and complex topology over $\mathbb{Z}$), but they are rather ``localized'' in an \emph{[[infinitesimal neighborhood|infinitesimal neighborhood]]} of the origin. One would like to be able to talk about functions supported only infinitesimally (in the transverse direction) to a [[closed subscheme]]. The formal schemes of [[Grothendieck]] are [[ringed spaces]] containing the information on all infinitesimal neighborhoods. Zariski's theorem on formal functions and establishing the theory of [[formal group]]s were some of the concrete motivations.

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

There are roughly four equivalent definitions of a $k$-formal scheme for a field $k$ (check this):

Let $Mf_k$ denote the category of finite dimensional $k$-rings (=$k$-algebras which are rings).

\begin{enumerate}%
\item A $k$-scheme is called a $k$\emph{-formal scheme} if it is is equivalent to a [[filtered colimit|codirected colimit]] of finite (affine) $k$-schemes.


\item A $k$-scheme is a $k$-formal scheme if it is presented by a [[profinite group|profinite]] $k$-ring; i.e a $k$-ring which is the limit of topologically discrete quotients which are finite $k$-rings. If $A$ is such a topological $k$-ring $Spf_k(A)(R)$ denotes the set of continous morphisms from $A$ to the topologically discrete ring $R$. We have $Spf_k$ is a (contravariant) equivalence between the category of profinite $k$-rings and the category $fSch_k$ of formal $k$-schemes.


\item Instead of defining the category $fSch_k$ of formal $k$-schemes as the [[opposite category|opposite]] of $Mf_k$ define it instead covariantly on the category of finite dimensional $k$-[[coring|corings]].


\item A formal $k$-scheme is precisely a left exact (commuting with finite limits) functor $X:Mf_k\to Set$.



\end{enumerate}
(\hyperlink{Demp}{Demazure, p-divisible groups, chapter I})

The inclusion $Mf_k\hookrightarrow M_k$ induces a functor

\begin{displaymath}
{}^\hat\; :Sch_k\to fSch_k
\end{displaymath}
called \emph{completion functor}.

\hypertarget{formal_spectra_of_noetherian_rings}{}\subsubsection*{{Formal spectra of Noetherian rings}}\label{formal_spectra_of_noetherian_rings}

If $X$ is a [[scheme]], a [[closed subscheme]] $Y\subset X$ is given by an embedding of topological spaces with the [[comorphism]] $\mathcal{O}_X\to f_*\mathcal{O}_Y$ which is a surjection; but alternatively $\mathcal{O}_Y$ can be recovered from $X$ and the defining sheaf of ideals $\mathcal{I}\subset \mathcal{O}_X$. Then one defines the [[structure sheaf]] of the completion $\mathcal{O}_{\hat{X}}$ as $lim_n \mathcal{O}_X/\mathcal{I}^n$ restricted to $Y$ and the \textbf{completion} $\hat{X} := (Y,\mathcal{O}_{\hat{X}})$. If $X = Spec\,R$ where $R$ is a noetherian $I$-adic ring, and $Y=Spec\,R/I$ then the completion is called the \textbf{[[formal spectrum]]} of $R$ denoted

\begin{displaymath}
Spf\,R = \hat{X}
\end{displaymath}
(where $R$ is viewed as a topological ring). The formal spectrum is an [[ind-object]] in the [[category]] of [[algebraic schemes]], viewed as a formal colimit $colim_n Spec (R/I^n)$.

A (locally) noetherian formal scheme is a formal completion of a (locally) noetherian scheme along a closed subscheme. Equivalently, a locally noetherian scheme is a locally ringed space which is locally isomorphic to the formal spectrum of a complete separated adic noetherian ring.

\hypertarget{more_general_indschemes}{}\subsubsection*{{More general ind-schemes}}\label{more_general_indschemes}

The formal spectrum can be extended to a somewhat bigger class of topological rings than the noetherian ones; Grothendieck developed the theory in the generality of [[pseudocompact ring|pseudocompact]] topological rings. However, some important rings, e.g. the ring of [[integers]] $\mathbb{Z}$, do not have a pseudocompact topology. Thus one could try to consider a more general subcategory of [[ind-schemes]] (with at least the requirement that the [[ind-object]] be represented by a diagram where the connecting morphisms are [[closed subscheme|closed immersions of schemes]]); one such approach is outlined in some detail in (\hyperlink{BeilinsonDrinfeld}{Beilinson-Drinfeld}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{FormalPowerSeries}{}\subsubsection*{{Formal power series and their formal spectra}}\label{FormalPowerSeries}

The [[completion of a ring]] of the [[polynomial ring]] $\mathbb{Z}[x]$ (the [[formal dual]] to the [[affine line]] $\mathbb{A}^1$) at the ideal $(x)$ is the [[limit]] -- formed in the [[category]] [[CRing]] --

\begin{displaymath}
\mathbb{Z}[x]^\wedge_{(x)} = \underset{\longleftarrow}{lim}_n (\mathbb{Z}[x])/(x^{n+1})
\end{displaymath}
of the [[quotient rings]] of $\mathbb{Z}[x]$ by the ideals generated by $x^{n+1}$ (the [[ring of dual numbers]] and its higher analogs). This yields the [[formal power series]] ring

\begin{displaymath}
\mathbb{Z}[x]^\wedge_{(x)}
  \simeq
   \mathbb{Z}[ [x] ]
  \,.
\end{displaymath}
Notice that the formal power series ring contains \emph{no} nilpotent elements except for zero -- even though each filtering stage $(\mathbb{Z}[x])/(x^n)$ does.

In contrast, the [[filtered colimit]] of the [[sheaves]] on [[affine varieties]] $Spec(R)$ that are [[representable functor|represented]] by this system of nilpotent rings is the formal scheme

\begin{displaymath}
\widehat{\mathbb{A}^1}
  =
  \underset{\longrightarrow}{lim}_n Spec(\mathbb{Z}[x]/(x^{n+1}))
  \;\;\;
  \in 
  Sh(Aff) = Sh(CRing^{op})
\end{displaymath}
given by the sheaf which sends each ring to its [[nilradical]]

\begin{displaymath}
\widehat{\mathbb{A}^1} \;\colon\;  Spec(R) \mapsto Nil(R)
  \,.
\end{displaymath}
(see e.g. \hyperlink{Strickland00}{Strickland 00, example 4.2, example 4.18}).

\hypertarget{formal_neighbourhood_of_the_diagonal}{}\subsubsection*{{Formal neighbourhood of the diagonal}}\label{formal_neighbourhood_of_the_diagonal}

See at \emph{[[formal neighbourhood of the diagonal]]}.

\hypertarget{formal_groups}{}\subsubsection*{{Formal groups}}\label{formal_groups}

The [[group objects]] in formal schemes are the [[formal groups]]. See there for more.

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

Classical discussion includes

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{G\'e{}om\'e{}trie formelle et g\'e{}om\'e{}trie alg\'e{}brique}, FGA 2 (S\'e{}minaire Bourbaki, t. 11, 1958/59, no. 182)


\item [[Luc Illusie]], \emph{Grothendieck existence theorem in formal geometry}, chapter 8 in Fantechi, Gottsche, Illusie, Kleiman, Nitsure, Vistoli, \emph{Fundamental algebraic geometry, Grothendieck's [[FGA explained]], Math. Surveys and Monographs} \textbf{123}, AMS 2005 (draft version \href{http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes}{pdf})


\item A. Grothendieck (avec J. Dieudonne), [[EGA]] I.10


\item A. Grothendieck et al. [[SGA]] III.2 (Exp. 7a, P. Gabriel, \'E{}tude infinit\'e{}simale des sch\'e{}mas en groupe et groupes formels; Exp. 7b, P. Gabriel, Groupes formels)


\item R. Hartshorne, \emph{Algebraic geometry}, II.9


\item M. Demazure, P. Gabriel, \emph{Groupes algebriques}, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970


\item Michel Demazure, lectures on p-divisible groups \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}


\item [[Daniel Murfet]], notes \href{http://therisingsea.org/notes/Section2.9-FormalSchemes.pdf}{Section2.9-FormalSchemes.pdf}


\item Leovigildo Alonso, Ana Jeremias, Marta Perez, \emph{Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes} (\href{http://arxiv.org/abs/math/0604241}{arXiv:math/0604241})



\end{itemize}
More general discussion in terms of ind-schemes includes

\begin{itemize}%
\item A. Beilinson, V. Drinfel'd, \emph{Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system}, preliminary version (\href{http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf}{pdf})

\end{itemize}
Another approach using a certain topological extension of the [[Yoneda lemma]] on $k Alg^{op}$ has been proposed in

\begin{itemize}%
\item B. Pareigis, R. A. Morris, \emph{Formal groups and Hopf algebras over discrete rings}, Trans. Amer. Math. Soc. \textbf{197} (1974), 113--129 (\href{http://dx.doi.org/10.2307/1996930}{doi}, [[Morris-Pareigis formal scheme|nlab entry]]).

\end{itemize}
[[Nikolai Durov]] has considered a flexible bigger category (which inludes the usual schemes) of covariant functors from the category of pairs $(R,I)$ where $R$ is a commutative ring and $I$ a [[nilpotent ideal]]; the correspondence between formal groups and [[Lie algebras]] based on Hausdorff series is neatly developed and used in that language; see chapters 7--9 of

\begin{itemize}%
\item N. Durov, S. Meljanac, A. Samsarov, Z. \v{S}koda, \emph{A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra}, Journal of Algebra 309, n. 1, 318--359 (2007) (\href{http://dx.doi.org/10.1016/j.jalgebra.2006.08.025}{doi:jalgebra}) (\href{http://front.math.ucdavis.edu/math.RT/0604096}{math.RT/0604096}).

\end{itemize}
For another generalization of formal schemes see

\begin{itemize}%
\item T. Yasuda, \emph{Non-adic formal schemes}, Int. Math. Research Notices 2009: 2417--2475, (\href{http://dx.doi.org/10.1093/imrn/rnp021}{doi:imrn}, \href{http://arxiv.org/abs/0711.0434}{arxiv})

\end{itemize}
In a fundamental article in [[noncommutative algebraic geometry]],

\begin{itemize}%
\item M. Kapranov, \emph{Noncommutative geometry based on commutator expansions}, \href{http://arxiv.org/abs/math.AG/9802041}{math.AG/9802041},

\end{itemize}
Kapranov introduced objects which should be interpreted as the infinitesimal neighborhoods of those commutative schemes with a closed immersion into a noncommutative scheme which is locally isomorphic to the spectrum of a free associative algebra.

And a decidedly functorial perspective is in (section 4 of)

\begin{itemize}%
\item [[Neil Strickland]], \emph{Formal schemes and formal groups}, \href{http://arxiv.org/abs/math.AT/0011121}{math.AT/0011121}

\end{itemize}
For the scheme geometric picture behind the infinitesimal neighborhoods and [[D-modules]] see also

\begin{itemize}%
\item A. Belinson, J. Bernstein, J., \emph{A proof of Jantzen conjectures}, in I. M. Gelfand Seminar, 1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993 (MR1237825 (95a:22022))

\end{itemize}
Some aspects of formal completions from the point of view of the derived categories are in

\begin{itemize}%
\item [[Dmitri Orlov|D. Orlov]], \emph{Formal completions and idempotent completions of triangulated categories of singularities}, \href{http://arxiv.org/abs/0901.1859}{arxiv/0901.1859}


\item Alexander I. Efimov, \emph{Formal completion of a category along a subcategory}, \href{http://arxiv.org/abs/1006.4721}{arxiv/1006.4721}



\end{itemize}
[[!redirects formal schemes]]



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