higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A formal scheme is a completion of an algebraic scheme. It is a ringed space being an enlarged version of a scheme including an infinitesimal neighborhood, which is expressed in terms of structure sheaves which are closer to sheaves of completions like formal power series rings rather than the polynomial rings.
Formal power series rings $k[\![x_1,\ldots,x_n]\!]$ 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 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 groups were some of the concrete motivations.
There are roughly four equivalent definitions of a $k$-formal scheme for a field $k$:
Let $Mf_k$ denote the category of finite dimensional $k$-rings (=$k$-algebras which are rings).
A $k$-scheme is called a $k$-formal scheme if it is is equivalent to a codirected colimit of finite (affine) $k$-schemes.
A $k$-scheme is a $k$-formal scheme if it is presented by a 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.
Instead of defining the category $fSch_k$ of formal $k$-schemes as the opposite of $Mf_k$ define it instead covariantly on the category of finite dimensional $k$-corings.
A formal $k$-scheme is precisely a left exact (commuting with finite limits) functor $X:Mf_k\to Set$.
(Demazure, p-divisible groups, chapter I)
The inclusion $Mf_k\hookrightarrow M_k$ induces a functor
called completion functor.
Given any ring $R$ and an ideal $I$ of $R$, there is a natural homomorphisms of rings $R/I^{n+1}\to R/I^n$ for all $n\geq 0$. The inverse limit $\hat{R}:=lim_n R/I^n$ is called the completion of $R$ at the ideal $I$ or the $I$-adic completion of $R$. If $R$ is noetherian, the completion is noetherian as well.
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 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 formal spectrum of $R$ denoted
(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.
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? 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 immersions of schemes); one such approach is outlined in some detail in
Another approach using a certain topological extension of the Yoneda lemma on $k Alg^{op}$ has been proposed in
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
For another generalization of formal schemes see
In a fundamental article in noncommutative algebraic geometry,
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.
A topologists viewpoint:
A. Grothendieck, Géométrie formelle et géométrie algébrique, FGA 2 (Séminaire Bourbaki, t. 11, 1958/59, no. 182)
Luc Illusie, Grothendieck existence theorem in formal geometry, chapter 8 in Fantechi, Gottsche, Illusie, Kleiman, Nitsure, Vistoli, Fundamental algebraic geometry, Grothendieck’s FGA explained, Math. Surveys and Monographs 123, AMS 2005 (draft version pdf)
A. Grothendieck (avec J. Dieudonne), EGA I.10
A. Grothendieck et al. SGA III.2 (Exp. 7a, P. Gabriel, Étude infinitésimale des schémas en groupe et groupes formels; Exp. 7b, P. Gabriel, Groupes formels)
R. Hartshorne, Algebraic geometry, II.9
M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970
Michel Demazure, lectures on p-divisible groups web
notes of Daniel Murfet, Section2.9-FormalSchemes.pdf
For the scheme geometric picture behind the infinitesimal neighborhoods and D-modules see also
Some aspects of formal completions from the point of view of the derived categories are in
D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, arxiv/0901.1859
Alexander I. Efimov, Formal completion of a category along a subcategory, arxiv/1006.4721