# Functorial point of view for formal schemes

Giving a scheme is the same as giving the corresponding functor from the category of rings to the category of set, and there are characterization of what functors arise in this way. This is explained in the book by Demazure and Gabriel. This "functorial point of view" is sometimes very useful, so I was wondering whether there is something similar for formal schemes rather that algebraic schemes. I searched for this but I wasn't able to find anything, it seems that Demazure and Gabriel don't speak about formal schemes at all.

Ricky

• Hint: rings correspond to affine schemes. But more usefully, observe that the proof allows one to often restrict attention to a much small class of objects than "all" rings (e.g., only need finite type $k$-algebras when working with schemes locally of finite type over a ring $k$). It is important that in various functorial arguments, we can restrict attention to a more limited class of rings which have more useful properties (e.g., noetherian, complete local noetherian with alg. closed residue field, etc.) Jul 23, 2010 at 10:15

You can try having a look at this paper:

http://arxiv.org/abs/math.AT/0011121

It's the most functorial-minded paper on algebraic geometry I'ver ever seen. It's written by an algebraic topologist. He cares mostly about affine and formal schemes.

The definition you're looking for is in section 4 of the paper. The functorial point of view for a formal scheme is a small filtered colimit of schemes, the colimit taken in the functor category.

• This reference defines the question out of existence by changing the def'n of formal schemes into certain functors on rings instead of as EGA defines them. It is akin to redefining schemes as certain functors on rings, which has the effect of bypassing the part of the argument with geometric content. So this shifts the work to proving a presence link between "formal scheme" as defined in this link and as in EGA. (Actually, the definition in the link doesn't quite capture all formal schemes as in EGA; it corresponds to a slightly different concept. ) Jul 23, 2010 at 10:32
• How big is the difference between these formal schemes and EGA's? Would you say they're really essentially different kinds of objects, or just two formalisms for roughly the same idea that have some technical differences in corner cases? And if the latter, what are the pros and cons of each? Speaking as a category theorist, this functorial definition seems much clearer to me than the EGA definition, so I'd hope/expect that it should work better in practice; but IANAAG, so that's a pretty uninformed opinion and I'd love to know what the experts think! Jul 23, 2010 at 11:02
• Dear Peter: they're closely related, but to prove deep theorems about formal schemes (algebraization thm, formal GAGA...) the geometric definition is crucial. The situation is analogous to schemes: one can make a categorical definition with functors, but to do geometry (involving concepts like irreducibility, connectedness, dimension, etc.) the ringed-space definition is important. So both viewpoints are needed. A good analogy is completion of a module with respect to an ideal and not only having the inverse system; flatness of completion (in noetherian case) is painful to express otherwise. Jul 23, 2010 at 11:40
• I've note read that paper yet, but maybe the situation is simpler in the case of adic formal schemes? Jul 23, 2010 at 11:40
• Ricky, it's true that the link in the answer is more closely related to adic formal schemes, but this is still sweeping the issue under the rug by changing the definitions. I think you'll learn more by following my hint above to adapt the argument for usual schemes directly to the formal scheme case (and then focus on formal spectrum of complete local noetherian rings by proving they're determined by points valued in artin local rings...that will also show you how to come back to the viewpoint in the link given in the above answer). Jul 23, 2010 at 12:48

There are many nonequivalent generalities in which one can define a formal scheme, for example the definitions in Hartshorne and in EGA are not quite the same. (I use in this answer some parts of my own editing in nlab's entry where more references can be found). In my understanding, whatever the definition is, the category of formal schemes is a realization of certain subcategory of Ind-schemes. Typically one requires at least that the ind-object in the subcategory may be represented by a diagram whose connecting morphisms are closed immersions of schemes. A pretty modern treatment is in

• A. Beilinson, V. Drinfel'd, Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system, preliminary version (pdf)

Some subcategories of Ind-objects in many algebraic categories can be described by putting the topology on algebraic objects. Thus the complete local rings, or more general the pseudocompact case, in the Grothendieck's approach to local schemes. One can use a topological version of Yoneda on rings to get a nice theory of formal schemes, over an arbitrary ring:

• B. Pareigis, R. A. Morris, Formal groups and Hopf algebras over discrete rings, Trans. Amer. Math. Soc. 197 (1974), 113--129 (doi:10.2307/1996930).

Nikolai Durov suggests to use directly the Gabriel-Demazure approach but not over Aff but over the opposite to the category of pairs (commutative ring, nilpotent ideal). Formal schemes should be an appropriate subcategory of that category of presheaves. That larger category (but without singling out there the smaller subcategory which would correspond more precisely to Grothendieck's formal schemes) is sketched in ch. 7-9 of

• N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, 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) (doi:10.1016/j.jalgebra.2006.08.025) (math.RT/0604096).

Although this doesn't cover every example, formal completions $$\hat{X}_Y$$ of a scheme $$X$$ along a closed subscheme $$Y$$ have a particularly nice functorial presentation. We have $$\hat{X}_Y \simeq Y_{dR} \times_{X_{dR}} X.$$ Here the subscript $$dR$$ means we are considering the de Rham stack of a space $$X$$; i.e. the stack whose $$S$$-points are $$\operatorname{Hom}(S^{red},X)$$.

In general, a formal scheme is a ind-scheme $$X$$ that has a presentation $$X = \operatorname{colim}_{i \in I} X_i,$$ where each $$X_i$$ is a finite scheme and the transition maps $$X_i \to X_j$$ are closed embeddings. By definition, this colimit takes place in some category of sheaves (e.g. fppf) on the category of affine schemes, and therefore automatically has a functor of points interpretation.

• How did I not notice this was posted 10 years ago... Oct 5, 2021 at 3:22
• Do you know a presentation for $\mathop{\mathrm{Spec}}(A)_{\mathrm{dR}}$ as an affine groupoid? Oct 5, 2021 at 9:45
• The fact that I asked this question more than 10 years ago makes me feel old :) Oct 5, 2021 at 14:48
• @LeoAlonso I believe you can think of it as the coequalizer of the two projections from the formal completion of the diagonal to the space itself Oct 5, 2021 at 20:57