In this post we discuss the notion of an ‘infinitesimal neighborhood’ of a point of a scheme , and how this relates to the ring .
For the sake of unencumbering ourselves of needless technicalities, we shall assume that is a scheme which is ‘sufficiently nice’. The vast majority of the time, it will suffice to assume that is locally Noetherian.
What is an ‘Infinitesimal Neighborhood’?
While there are many nebulous concepts and phrases in algebraic geometry. Some of the most mysterious, at least to me, were those of the form ‘take an infinitesimal neighborhood’ or ‘it’s true formally close’. These were always used to mean something like ‘really close’ or ‘zoomed in to an extreme degree’. They were supposed to capture information about a locality of a point , but a locality which is much finer than ‘Zariski close’.
This confusion got much worse as I learned more algebraic geometry. Namely, I started learning other notions of ‘extremely close’, in a sense much finer than the Zariski topology. In particular, for every Grothendieck topology on , there is a notion of a ‘-neighborhood’ of a point . For example, we can talk about something happening ‘etale locally’ at to mean that there is an etale neighborhood of (i.e. an etale map whose image contains ) for which that thing happens (slightly more precisely, that the ‘pullback’ of that thing satisfies whatever property).
This completely shoots the idea of just thinking about an infinitesimal neighborhood as being ‘really, really close’ or ‘much closer than the Zariski topology’. Namely, the things that holds ‘etale close’ and the things that hold ‘infinitesimally close’ aren’t the same. So, the question becomes, what type of closeness is ‘infinitesimally close’, and what does it ‘mean’?
Let us give a possible formal definition of an ‘infinitesimal neighborhood’, and then discuss why it makes sense. Without any more ado, let us define an infinitesimal neighborhood of a point to be a mapping , where is an Artin local ring with residue field (the residue field of at ) such that the following diagram commutes:
where and correspond to the natural maps.
Remark: For whatever reason, the term ‘Artin local ring’ seems to be preferred in this context to the more common ‘Artinian local ring’, although they mean the same thing.
As an example of an infinitesimal neighborhood, note that for , and the -point , the map given on -algebras by is an infinitesimal neighborhood of .
The rest of this post will be dedicated to understanding what type of information infinitesimal neighborhoods of contain.
What Information is Contained in the Completion of a Point?
To begin to understand what information is contained in the infinitesimal neighborhoods of a point, we need to go back and first study what information is contained in . We don’t mean ‘information’ in a very technical sense, but more in a holistic, intuitive sort of way.
We start by recalling precisely how we think of the local ring . Since is thought of as being ‘functions on ‘, it makes intuitive sense to consider (where the colimits is over Zariski neighborhoods of ) to be ‘germs’ of functions at . Namely, we should think about elements as being some function on a neighborhood of , but for which we can’t distinguish and for any neighborhood . So, it only captures the properties of a function which are true in any Zariski neighborhood . That said, since the Zariski topology is so coarse, this information about a function is still quite strong.
Now, is a local ring, with maximal ideal which we think of as being the ‘ideal of functions which vanish at ‘. The quotient map (where, by definition, ) is then thought of as being the ‘evaluation at map’ which sends a germ to ‘the value ‘.
This line of though is justified by more ostensibly geometric analogues. For example, let be a complex manifold, and denote the sheaf of holomorphic functions on . Then, is indeed just germs of holomorphic functions , for a neighborhood of . The ideal is indeed the germs which vanish at , and the map sending does indeed have as its kernel .
How then should we interpret the ring ? Well, let’s examine our more geometric counterpart to get an idea. Namely, what is ? Well, given any germ we certainly have that
(where is an indterminate), and in fact, one sees that
if and only if and . Thus, we may identify as keeping track not only of the value of a function at , but also of its derivative–it keeps track of ‘first order’ data of a function at . Thus, this is how we should think of , for a scheme, and –it keeps track of first order data of a function at .
In fact, generalizing the above techniques, one can show that on a complex manifold :
so that keeps track of the ‘-order data’ of functions at –its value at that point, and the values of its first -derivatives. This is also how we intuit the character of –it keeps track of the -order data of a germ at .
From this intuition, we can quite well understand the true nature of the ring . Formally, is , as varies over the naturla numbers. But, an explicit model of can be given as:
(this is just an explicit model for the inverse limit).
Thus, we see that can be thought of as “the set of all consistent differential data”. In other words, elements of are just tuples where is -order data, and such that if you take , which is -order data, and forget the highest order term, the -order data you obtain is just .
The natural map can then be interpreted as taking a germ to the tuple which, in -coordinate, has the -differential data (in manifold land, this would send to its Taylor polynomial).
Now, the map is injective (this is essentially the Krull intersection theorem), but far from surjective in general. For example, if we take , and , then , and . The map is far, far from surjective. For example, has a solution to for all primes , whereas as no such solution.
In essence, one thinks about the elements of as being ‘power series’. The non-surjectivity of can then be interpreted as saying that not every power series defines a germ of a function at . The reason for this, intuitively, is that the Zariski topology, more operatively germs in the Zariski topology, are defined on too large of open sets–even though we can shrink arbitrarily down, the sets are always very big, too big to support power series. Thus, is, in some sense, capturing the behavior of functions on much smaller ‘opens’ than the conventional Zariski open sets.
Let us give another some reason as to why this surjectivity should not, in general hold. One should think about ‘badness’ of a ring, in some sense, as being an indication of how much information it holds. The ‘badder’ it is, the more information it contains. For example, rings of functions of functions on larger open sets contain more information, because they obtain ‘obstructions’ contributed by each point. Thus, this gives another indication that, perhaps, we can think about as being functions on a ‘smaller than Zariski’ neighborhood of .
Regardless of whether these intuitions are clear to you, one thing is certainly true. The ring is a ring whose obstructions to any given problem (e.g. the existence of an inverse, or more generally the solution to an integral equation) are dictated entirely by differential obstructions (i.e. obstructions in ). Explicitly, things like equations should be solvable, if and only if one can produce consistent solutions in each of the -order data rings.
To sum up what we just said, we think of as being a ring which captures the differential data of germs at . A fact should be true about if and only if there is no obstruction to checking that this fact is consistently true in all of the -order. This allows us to think about to be capturing information about germs of functions, but germs in a ‘topology’ finer than the Zariski topology. This topology is one which contains open sets so small that the only thing a germ of function in this topology tells you about the function is its value at the point, and where it goes in an infinitesimal (using this term loosely from calculus!) neighborhood, as dictated by knowing all of this functions derivatives.
How They Relate
Now that we have reminded ourselves of the intuition of what information contains, we can definitively answer “what information is contained in an ‘infinitesimal neighborhood’ of a point”. In fact, we claim that it contains precisely the same information that contains–the differential data of at . Of course, this statement is not rigorous as stated, and in fact will require a bit of set up to make it so.
To begin, for a field , let us define to be the category of local Artin rings with residue field , for which a map is ring map such that
commutes, where the maps and are the projection maps.
In a similar vein, let us define to be the category of schemes equipped with a map from , and where a map is an morphism such that . In some circles, this would be called ‘the coslice category of over the object ‘, but this is not one of those circles.
Note then that has a full-subcategory consisting of objects of the form , where is a local Artin ring with residue field , and is induced from the quotient map . Then, this subcategory is anti-equivalent to .
Now, let’s suppose that is a scheme, and is such that . Then, we obtain an object of by letting be the natural map (where we abusively write for both the point of , and the associated map). With this, we consider the contravariant Hom functor .
Now, Yoneda’s lemma tells us precisely what information is contained inside the functor , namely the isomorphism type of (the whole shebang, one might say). But, the premise of this post was to understand what information was contained in all infinitesimal neighborhoods of . And, note that, by definition, an infinitesimal neighborhood of is an element of for some . Thus, a slightly more rigorous formulation of ‘what information is contained in infinitesimal neighborhoods’ is ‘what information is contained in ?”
To put this on more easy to understand (read ‘ring theoretic’) footing, let us first see what an element of looks like. The first thing to notice, is that from basic algebraic geometry, since is local, that any element of factors uniquely as
where and are the obvious maps. This then defines a natural isomorphism . Thus we shall only concern ourselves with .
But, now we can turn into a purely algebraic functor. Namely, under the anti-equivalence , the functor is carried to the covariant functor which takes to the set of maps such that
More generally, one can show that for any affine object , one obtains a functor defined similarly to the above. For reasons that may be clear to some of the more knowledgable readers, we denote by .
Thus, with this set-up, we can begin to state why the information contained in infinitesimal neighborhoods is ‘the same’ as that contained in . The first part of this assertion is the following.
Theorem 1: The map , induced by the natural map
is an isomorphism.
The second part is the real ‘oomph’ of this post:
Theorem 2: Let and be objects of . Then, if and only if as rings, with compatible projections to (i.e. as objects of ).
Let us begin by proving Theorem 1:
Proof(Theorem 1): Let’s turn this entirely into a theorem about algebra. Namely, the map being an isomorphism, essentially means that if is an Artin local ring with residue field , and we have a map , for which the projections to the residue field commute, then there exists a unique map , commuting with projections again, such that is .
To see firstly that such a map exists, let us note that since is Artinian, that is nilpotent, and so must contain in its kernel for some . This gives rise to a factorization
We then define by
It’s clear then that is the same as
since is the same as .
For unicity, let’s suppose that we have a factorization of as . By the same argument as above, must factor as for some . But, since
is supposed to , this forces to be . Thus, the only possibility for non-unicity of factorizations, is if perhaps and are both in the kernel, for different and , and the resulting factorizations (as described above) are different. But, the fact that these induce the same map is elementary.
So, let us now prove Theorem 2. Before we do, we need to prove a lemma which essentially says that all isomorphisms are induced by a map .
Lemma 3: Let be an isomorphism of functors. Then, there exists a morphism such that is the map induced by .
Proof: Since Theorem 1 For each , we have a bijection from to . Let us define to be the unique element of which maps, by , to the projection maps , which is an element of .
We claim that the maps glue together to give a map , commuting with projections to , such that . To see this, we need only show that for , , where is the natural map. To do this, we merely appeal to the naturality of the isomorphism . Namely, since is a morphism in , we have by assumption that
commutes. But, taking , an element of the set on the top left, and following it along the top, and then the right arrow, gives , and then . But, following it along the left, and then bottom arrow, gives . But, by very definition , and so this second map is just . But, then since, as we have just shown, , and was the unique preimage of under , we may conclude that as desired. Thus, we do obtain the desired map .
We now claim that is given by the map given by sending to . To see this, let in be arbitrary. We must show that . To see this, note that, as we have remarked multiple times before, must factor as
for some , an arrow in . But, by naturality of , we see that
But, note that
from where the conclusion follows.
Let us now prove Theorem 2:
Proof(Theorem 2): If we have isomorphisms , then by Theorem 1, we obtain isomorphisms . By Lemma 3, this isomorphism is induced by some map , commuting with projections to . Playing the exact same game with the inverse isomorphism , we obtain a morphism . It suffices to show that is an isomomorphism. But, this corresponds to a map giving an isomorphism . And, so it suffices to show that if is a ring map, commuting with projections to for which induces a bijection
where ranges over elements of , and the morphisms to are projections.
But, since is Noetherian, any ring endomorphism, which is surjective, is actually a ring isomorphism. So, it suffices to show that our morphism is surjective. But, by using the associated property with the projection maps , this easily follows.
Conversely, if we have isomorphisms , then (by Theorem 1) we obviously obtain isomorphisms as desired.
So, from this, we can honestly say what information is contained in the ‘infinitesimal neighborhoods’ of a point. It is precisely the information contained inside (made formal by Theorem 2). But, by the previous discussion, this is precisely the ‘differential data’ of at . So, infinitesimal neighborhoods are the ones which are close enough so that functions are determined by their derivatives, and so studying properties of infinitesimal neighborhoods can be thought of as studying the purely differential properties of a scheme at a point.