# What Information is Contained in an ‘Infinitesimal Neighborhood’ of a Point?

In this post we discuss the notion of an ‘infinitesimal neighborhood’ of a point of a scheme $X$, and how this relates to the ring $\widehat{\mathcal{O}_{X,x}}$.

For the sake of unencumbering ourselves of needless technicalities, we shall assume that $X$ is a scheme which is ‘sufficiently nice’. The vast majority of the time, it will suffice to assume that $X$ 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 $x\in X$, 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 $\mathcal{T}$ topology on $\text{Sch}/X$, there is a notion of a ‘$\mathcal{T}$-neighborhood’ of a point $x\in X$. For example, we can talk about something happening ‘etale locally’ at $x\in X$ to mean that there is an etale neighborhood $U$ of $X$ (i.e. an etale map $U\to X$ whose image contains $x$) 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 $x\in X$ to be a mapping $\text{Spec}(A)\to X$, where $A$ is an Artin local ring with residue field $k(x)$ (the residue field of $X$ at $x$) such that the following diagram commutes:

$\begin{matrix}\text{Spec}(A) & &\\ \uparrow & \searrow & \\ \text{Spec}(k(x)) & \to & X\end{matrix}$

where $\text{Spec}(k(x))\to\text{Spec}(A)$ and $\text{Spec}(k(x))\to X$ 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 $\mathbb{A}^1_k$, and the $k$-point $(x)$, the map $\text{Spec}(k[\varepsilon]/(\varepsilon^2))\to \mathbb{A}^1_k$ given on $k$-algebras by $k[x]\to k[\varepsilon]/(\varepsilon^2):x\mapsto \varepsilon$ is an infinitesimal neighborhood of $(x)\in\mathbb{A}^1_k$.

The rest of this post will be dedicated to understanding what type of information infinitesimal neighborhoods of $x\in X$ 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 $\widehat{\mathcal{O}_{X,x}}$. 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 $\mathcal{O}_{X,x}$. Since $\mathcal{O}_X(U)$ is thought of as being ‘functions on $U$‘, it makes intuitive sense to consider $\mathcal{O}_{X,x}=\varinjlim \mathcal{O}_{X}(U)$ (where the colimits is over Zariski neighborhoods of $U$) to be ‘germs’ of functions at $x$. Namely, we should think about elements $f\in\mathcal{O}_{X,x}$ as being some function $f$ on a neighborhood $U$ of $x$, but for which we can’t distinguish $f$ and $f\mid_V$ for any neighborhood $x\in V\subseteq U$. So, it only captures the properties of a function which are true in any Zariski neighborhood $x$. That said, since the Zariski topology is so coarse, this information about a function is still quite strong.

Now, $\mathcal{O}_{X,x}$ is a local ring, with maximal ideal $\mathfrak{m}_x$ which we think of as being the ‘ideal of functions which vanish at $x$‘. The quotient map $\mathcal{O}_{X,x}\to k(x)$ (where, by definition, $k(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$) is then thought of as being the ‘evaluation at $x$ map’ which sends a germ $f\in\mathcal{O}_{X,x}$ to ‘the value $f(x)$‘.

This line of though is justified by more ostensibly geometric analogues. For example, let $M$ be a complex manifold, and $\mathcal{O}_M$ denote the sheaf of holomorphic functions on $M$. Then, $\mathcal{O}_{M,x}$ is indeed just germs of holomorphic functions $f:U\to\mathbb{C}$, for $U$ a neighborhood of $x$. The ideal $\mathfrak{m}_x\subseteq\mathcal{O}_{M,x}$ is indeed the germs which vanish at $x$, and the map $\mathcal{O}_{M,x}\to \mathbb{C}$ sending $f\mapsto f(x)$ does indeed have as its kernel $\mathfrak{m}_x$.

How then should we interpret the ring $\mathcal{O}_{X,x}/\mathfrak{m}_x^2$? Well, let’s examine our more geometric counterpart to get an idea. Namely, what is $\mathcal{O}_{M,x}/\mathfrak{m}_x^2$? Well, given any germ $f\in\mathcal{O}_{M,x}$ we certainly have that

$f\equiv f(x)+f'(x)T\mod \mathfrak{m}_x^2$

(where $T$ is an indterminate), and in fact, one sees that

$a+bT\equiv c+dT\mod \mathfrak{m}_x^2$

if and only if $a=c$ and $b=d$. Thus, we may identify $\mathcal{O}_{M,x}$ as keeping track not only of the value of a function $f$ at $x$, but also of its derivative–it keeps track of ‘first order’ data of a function at $x$. Thus, this is how we should think of $\mathcal{O}_{X,x}/\mathfrak{m}_x^2$, for $X$ a scheme, and $x\in X$–it keeps track of first order data of a function at $x$.

In fact, generalizing the above techniques, one can show that on a complex manifold $M$:

$f\equiv f(x)+f'(x)T+f''(x)T^2+\cdots+f^{(n-1)}T^{n-1}\mod \mathfrak{m}_x^{n}$

so that $\mathcal{O}_{M,x}/\mathfrak{m}_x^n$ keeps track of the ‘$(n-1)^\text{st}$-order data’ of functions at $x$–its value at that point, and the values of its first $n-1$-derivatives. This is also how we intuit the character of $\mathcal{O}_{X,x}/\mathfrak{m}_x^n$–it keeps track of the $(n-1)^\text{st}$-order data of a germ at $x$.

From this intuition, we can quite well understand the true nature of the ring $\widehat{\mathcal{O}_{X,x}}$. Formally, $\widehat{\mathcal{O}_{X,x}}$ is $\varprojlim \mathcal{O}_{X,x}/\mathfrak{m}_x^n$, as $n$ varies over the naturla numbers. But, an explicit model of $\widehat{\mathcal{O}_{X,x}}$ can be given as:

$\displaystyle \widehat{\mathcal{O}_{X,x}}=\left\{(x_n)\in\prod_n \mathcal{O}_{X,x}/\mathfrak{m}_x^n:x_m\equiv x_n\mod \mathfrak{m}_x^m,\text{ }m\leqslant n\right\}$

(this is just an explicit model for the inverse limit).

Thus, we see that $\widehat{\mathcal{O}_{X,x}}$ can be thought of as “the set of all consistent differential data”. In other words, elements of $\widehat{\mathcal{O}_{X,x}}$ are just tuples $(f_n)$ where $f_n$ is $(n-1)^\text{st}$-order data, and such that if you take $f_{n+1}$, which is $n^\text{th}$-order data, and forget the highest order term, the $(n-1)^\text{st}$-order data you obtain is just $f_n$.

The natural map $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}$ can then be interpreted as taking a germ $f$ to the tuple which, in $n^\text{th}$-coordinate, has the $(n-1)^\text{st}$-differential data $f+\mathfrak{m}_x^n$ (in manifold land, this would send $f$ to its $(n-1)^\text{st}$ Taylor polynomial).

Now, the map $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}$ is injective (this is essentially the Krull intersection theorem), but far from surjective in general. For example, if we take $X=\text{Spec}(\mathbb{Z})$, and $x=(p)$, then $\mathcal{O}_{X,x}=\mathbb{Z}_{(p)}$, and $\widehat{\mathcal{O}_{X,x}}=\mathbb{Z}_p$. The map $\mathbb{Z}_{(p)}\to\mathbb{Z}_p$ is far, far from surjective. For example, $\mathbb{Z}_p$ has a solution to $x^2-q=0$ for all primes $q\ne p$, whereas $\mathbb{Z}$ as no such solution.

In essence, one thinks about the elements of $\widehat{\mathcal{O}_{X,x}}$ as being ‘power series’. The non-surjectivity of $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}$ can then be interpreted as saying that not every power series defines a germ of a function at $x$. 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, $\widehat{\mathcal{O}_{X,x}}$ 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 $\widehat{\mathcal{O}_{X,x}}$ as being functions on a ‘smaller than Zariski’ neighborhood of $x$.

Regardless of whether these intuitions are clear to you, one thing is certainly true. The ring $\widehat{\mathcal{O}_{X,x}}$ 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 $\mathcal{O}_{X,x}/\mathfrak{m}_x^n$). Explicitly, things like equations should be solvable, if and only if one can produce consistent solutions in each of the $n^\text{th}$-order data rings.

To sum up what we just said, we think of $\widehat{\mathcal{O}_{X,x}}$ as being a ring which captures the differential data of germs at $x$. A fact should be true about $\widehat{\mathcal{O}_{X,x}}$ if and only if there is no obstruction to checking that this fact is consistently true in all of the $n^\text{th}$-order. This allows us to think about $\widehat{\mathcal{O}_{X,x}}$ 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 $\widehat{\mathcal{O}_{X,x}}$ 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 $\widehat{\mathcal{O}_{X,x}}$ contains–the differential data of $X$ at $x$. 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 $k$, let us define $\mathsf{Art}(k)$ to be the category of local Artin rings with residue field $k$, for which a map $A\to A'$ is ring map such that

$\begin{matrix}A & &\\ \downarrow & \searrow & \\ k & \leftarrow & A'\end{matrix}$

commutes, where the maps $A\to k$ and $A'\to k$ are the projection maps.

In a similar vein, let us define $\mathscr{C}$ to be the category of schemes equipped with a map from $\text{Spec}(k)$, and where a map $(X,f:\text{Spec}(k)\to X)\to (Y,g:\text{Spec}(k)\to Y)$ is an morphism $\alpha:X\to Y$ such that $\alpha\circ f=g$. In some circles, this would be called ‘the coslice category of $\mathbf{Sch}$ over the object $\text{Spec}(k)$‘, but this is not one of those circles.

Note then that $\mathscr{C}$ has a full-subcategory $\mathscr{A}$ consisting of objects of the form $(\text{Spec}(A),\pi)$, where $A$ is a local Artin ring with residue field $k$, and $\pi:\text{Spec}(k)\to\text{Spec}(A)$ is induced from the quotient map $A\to k$. Then, this subcategory is anti-equivalent to $\mathsf{Art}(k)$.

Now, let’s suppose that $X$ is a scheme, and $x\in X$ is such that $k(x)=k$. Then, we obtain an object of $\mathscr{C}$ by letting $x:\text{Spec}(k)\to X$ be the natural map (where we abusively write $x$ for both the point of $X$, and the associated map). With this, we consider the contravariant Hom functor $h_{(X,x)}:\mathscr{C}^\text{op}\to\mathbf{Set}$.

Now, Yoneda’s lemma tells us precisely what information is contained inside the functor $h_{(X,x)}$, namely the isomorphism type of $(X,x)$ (the whole shebang, one might say). But, the premise of this post was to understand what information was contained in all infinitesimal neighborhoods of $x\in X$. And, note that, by definition, an infinitesimal neighborhood of $x\in X$ is an element of $h_{(X,x)}((\text{Spec}(A),\pi))$ for some $(\text{Spec}(A),\pi)\in\mathscr{A}$. Thus, a slightly more rigorous formulation of ‘what information is contained in infinitesimal neighborhoods’ is ‘what information is contained in $h_{(X,x)}\mid_{\mathscr{A}}$?”

To put this on more easy to understand (read ‘ring theoretic’) footing, let us first see what an element of $h_{(X,x)}((\text{Spec}(A),\pi))$ looks like. The first thing to notice, is that from basic algebraic geometry, since $\text{Spec}(A)$ is local, that any element of $h_{(X,x)}((\text{Spec}(A),\pi))$ factors uniquely as

$\begin{matrix}(\text{Spec}(A),\pi) & & \\ \downarrow & \searrow & \\ (\text{Spec}(\mathcal{O}_{X,x},x) & \rightarrow & (X,x)\end{matrix}$

where $x:\text{Spec}(k)\to \text{Spec}(\mathcal{O}_{X,x})$ and $(\text{Spec}(\mathcal{O}_{X,x},x)\to (X,x)$ are the obvious maps. This then defines a natural isomorphism $h_{(\text{Spec}(\mathcal{O}_{X,x}),x)}\to h_{(X,x)}$. Thus we shall only concern ourselves with $h_{(\text{Spec}(\mathcal{O}_{X,x}),x)}$.

But, now we can turn $h_{(\text{Spec}(\mathcal{O}_{X,x}),x)}\mid_\mathscr{A}$ into a purely algebraic functor. Namely, under the anti-equivalence $\mathsf{Art}(k)\to \mathscr{A}$, the functor $h_{(\text{Spec}(\mathcal{O}_{X,x}),x)}$ is carried to the covariant functor $F_{(X,x)}:\mathsf{Art}(k)\to\mathbf{Set}$ which takes $A$ to the set of maps $\mathcal{O}_{X,x}\to A$ such that

$\begin{matrix}\mathcal{O}_{X,x} & & \\ \downarrow & \searrow & \\ k & \leftarrow & A\end{matrix}$

commutes.

More generally, one can show that for any affine object $(\text{Spec}(R),f)\in\mathscr{C}$, one obtains a functor $F_{(\text{Spec}(R),f)}:\mathsf{Art}(k)\to\mathbf{Set}$ defined similarly to the above. For reasons that may be clear to some of the more knowledgable readers, we denote $F_{(\text{Spec}(\widehat{\mathcal{O}_{X,x}},x)}$ by $\widehat{F_{(X,x)}}$.

Thus, with this set-up, we can begin to state why the information contained in infinitesimal neighborhoods is ‘the same’ as that contained in $\widehat{\mathcal{O}_{X,x}}$. The first part of this assertion is the following.

Theorem 1: The map $\widehat{F_{(X,x)}}\to F_{(X,x)}$, induced by the natural map

$(\text{Spec}(\widehat{\mathcal{O}_{X,x}}),x)\to (\text{Spec}(\mathcal{O}_{X,x}),x)$

is an isomorphism.

The second part is the real ‘oomph’ of this post:

Theorem 2: Let $(X,x)$ and $(Y,y)$ be objects of $\mathscr{C}$. Then, $F_{(X,x)}\cong F_{(Y,y)}$ if and only if $\widehat{\mathcal{O}}_{X,x}\cong \widehat{\mathcal{O}_{Y,y}}$ as rings, with compatible projections to $k$ (i.e. $(\text{Spec}(\widehat{\mathcal{O}_{X,x}}),x)\cong (\text{Spec}(\widehat{\mathcal{O}_{Y,y}}),y)$ as objects of $\mathscr{C}$).

Let us begin by proving Theorem 1:

Proof(Theorem 1): Let’s turn this entirely into a theorem about algebra. Namely, the map $\widehat{F_{(X,x)}}\to F_{(X,x)}$ being an isomorphism, essentially means that if $A$ is an Artin local ring with residue field $k$, and we have a map $f:\mathcal{O}_{X,x}\to A$, for which the projections to the residue field commute, then there exists a unique map $\widehat{\mathcal{O}_{X,x}}\to A$, commuting with projections again, such that $f:\mathcal{O}_{X,x}\to A$ is $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}\to A$.

To see firstly that such a map exists, let us note that since $A$ is Artinian, that $\mathfrak{m}_xA$ is nilpotent, and so $\mathcal{O}_{X,x}\to A$ must contain $\mathfrak{m}_x^n$ in its kernel for some $n$. This gives rise to a factorization

$\begin{matrix}\mathcal{O}_{X,x} & & \\ \downarrow & \searrow^{\overline{f}} & \\ \mathcal{O}_{X,x}/\mathfrak{m}^n & \to & A\end{matrix}$

We then define $\widetilde{f}:\widehat{\mathcal{O}_{X,x}}\to A$ by

$\widehat{\mathcal{O}_{X,x}}\to \mathcal{O}_{X,x}/\mathfrak{m}_x^n\xrightarrow{\overline{f}}A$

It’s clear then that $f$ is the same as

$\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}\to\mathcal{O}_{X,x}/\mathfrak{m}_x^n\xrightarrow{\overline{f}}\to A$

since $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}\to\mathcal{O}_{X,x}/\mathfrak{m}_x^n$ is the same as $\mathcal{O}_{X,x}\to\mathfrak{m}_x^n$.

For unicity, let’s suppose that we have a factorization of $f$ as $\mathcal{O}_{X,x}\to\widehat{\mathcal{O}_{X,x}}\xrightarrow{g} A$. By the same argument as above, $g$ must factor as $\widehat{\mathcal{O}_{X,x}}\to \mathcal{O}_{X,x}/\mathfrak{m}_x^n\to A$ for some $\overline{g}:\mathcal{O}_{X,x}/\mathfrak{m}_x^n\to A$. But, since

$\mathcal{O}_{X,x}\to \widehat{\mathcal{O}_{X,x}}\to\mathcal{O}_{X,x}/\mathfrak{m}_x^n\xrightarrow{\overline{g}} A$

is supposed to $f$, this forces $\overline{g}$ to be $\overline{f}$. Thus, the only possibility for non-unicity of factorizations, is if perhaps $\mathfrak{m}_x^n$ and $\mathfrak{m}_x^m$ are both in the kernel, for different $m$ and $n$, and the resulting factorizations (as described above) are different. But, the fact that these induce the same map $\widehat{\mathcal{O}_{X,x}}\to A$ is elementary. $\blacksquare$

So, let us now prove Theorem 2. Before we do, we need to prove a lemma which essentially says that all isomorphisms $F_{(X,x)}\to F_{(Y,y)}$ are induced by a map $(X,x)\to (Y,y)$.

Lemma 3: Let $\eta:\widehat{F_{(X,x)}}\xrightarrow{\approx}\widehat{F_{(Y,y)}}$ be an isomorphism of functors. Then, there exists a morphism $\psi:(\widehat{\mathcal{O}_{X,x}},x)\to (\widehat{\mathcal{O}_{Y,y}},y)$ such that $\eta$ is the map induced by $\psi$.

Proof: Since Theorem 1 For each $n\in\mathbb{N}$, we have a bijection $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$ from $\widehat{F_{(X,x)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$ to $\widehat{F_{(Y,y)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$. Let us define $\varphi_n:\widehat{\mathcal{O}_{X,x}}\to\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n$ to be the unique element of $\widehat{F_{(X,x)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$ which maps, by $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$, to the projection maps $\pi_n:\widehat{\mathcal{O}_{(Y,y)}}\to\widehat{\mathcal{O}_{Y,y}}/\mathfrak{m}_Y^n$, which is an element of $\widehat{F_{(Y,y)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)$.

We claim that the maps $\varphi_n$ glue together to give a map $\varphi:\widehat{\mathcal{O}_{X,x}}\to\widehat{\mathcal{O}_{Y,y}}$, commuting with projections to $k$, such that $\pi_n\circ\varphi=\varphi_n$. To see this, we need only show that for $n\geqslant m$, $\pi_{n,m}\circ\varphi_n=\varphi_m$, where $\pi_{n,m}:\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n\to\mathcal{O}_{Y,y}/\mathfrak{m}_Y^m$ is the natural map. To do this, we merely appeal to the naturality of the isomorphism $\eta$. Namely, since $\pi_{n,m}$ is a morphism in $\mathsf{Art}(k)$, we have by assumption that

$\begin{matrix}\widehat{F_{(X,x)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n) & \to & \widehat{F_{(X,x)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^m)\\ \downarrow & & \downarrow\\ \widehat{F_{(Y,y)}}(\mathcal{O}_{(Y,y)}/\mathfrak{m}_Y^n) & \to & \widehat{F_{(Y,y)}}(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^m)\end{matrix}$

commutes. But, taking $\varphi_n$, an element of the set on the top left, and following it along the top, and then the right arrow, gives $\pi_{n,m}\circ \varphi_n$, and then $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^m)(\pi_{n,m}\circ\varphi_n)$. But, following it along the left, and then bottom arrow, gives $\pi_{n,m}\circ \eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)(\varphi_n)$. But, by very definition $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n)(\varphi_n)=\pi_n$, and so this second map is just $\pi_m$. But, then since, as we have just shown, $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_{Y,y}/\mathfrak{m}_Y^m)(\pi_{n,m}\circ\varphi_n)=\pi_m$, and $\varphi_m$ was the unique preimage of $\pi_m$ under $\eta(\mathcal{O}_{Y,y}/\mathfrak{m}_Y^m)$, we may conclude that $\pi_{n,m}\circ \varphi_n=\varphi_m$ as desired. Thus, we do obtain the desired map $\varphi$.

We now claim that $\alpha^{-1}$ is given by the map $\widehat{F_{(Y,y)}}\to\widehat{F_{(X,x)}}$ given by sending $f:\widehat{\mathcal{O}_{Y,y}}\to A$ to $f\circ\varphi$. To see this, let $A$ in $\mathsf{Art}(k)$ be arbitrary. We must show that $\alpha^{-1}(f)=f\circ\varphi$. To see this, note that, as we have remarked multiple times before, $f$ must factor as

$\widehat{\mathcal{O}_{Y,y}}\xrightarrow{\pi_n}\mathcal{O}_{Y,y}/\mathfrak{m}_Y^n\xrightarrow{\overline{f}}A$

for some $\overline{f}$, an arrow in $\mathsf{Art}(k)$. But, by naturality of $\alpha^{-1}$, we see that

$\alpha^{-1}(f)=\alpha^{-1}(\overline{f}\circ \pi_n)=\overline{f}\circ\alpha^{-1}(\pi_n)=\overline{f}\circ\varphi_n$

But, note that

$f\circ\varphi=\overline{f}\circ\pi_n\circ\varphi=\overline{f}\circ\varphi_n$

from where the conclusion follows. $\blacksquare$

$\blacksquare$

Let us now prove Theorem 2:

Proof(Theorem 2): If we have isomorphisms $F_{(X,x)}\cong F_{(Y,y)}$, then by Theorem 1, we obtain isomorphisms $\widehat{F_{(X,x)}}\cong\widehat{F_{(Y,y)}}$. By Lemma 3, this isomorphism is induced by some map $\varphi:\widehat{\mathcal{O}_{X,x}}\to \widehat{\mathcal{O}_{Y,y}}$, commuting with projections to $k$. Playing the exact same game with the inverse isomorphism $F_{(Y,y)}\cong F_{(X,x)}$, we obtain a morphism $\psi:\widehat{\mathcal{O}_{Y,y}}\to\widehat{\mathcal{O}_{X,x}}$. It suffices to show that $\varphi\circ\psi$ is an isomomorphism. But, this corresponds to a map giving an isomorphism $F_{(X,x)}\cong F_{(X,x)}$. And, so it suffices to show that if $\phi:\widehat{\mathcal{O}_{X,x}}\to\widehat{\mathcal{O}_{X,x}}$ is a ring map, commuting with projections to $k$ for which $f\mapsto f\circ \phi$ induces a bijection

$\left\{\begin{matrix}\widehat{\mathcal{O}_{X,x}} & &\\ \downarrow & \searrow ^f & \\ k & \leftarrow & A\end{matrix}\right\}\to \left\{\begin{matrix} \widehat{\mathcal{O}_{X,x}} & & \\ \downarrow & \searrow ^g & \\ k & \leftarrow & A\end{matrix}\right\}$

where $A$ ranges over elements of $\mathsf{Art}(k)$, and the morphisms to $k$ are projections.

But, since $\widehat{\mathcal{O}_{X,x}}$ is Noetherian, any ring endomorphism, which is surjective, is actually a ring isomorphism. So, it suffices to show that our morphism $\phi$ is surjective. But, by using the associated property with the projection maps $\pi_n:\widehat{\mathcal{O}_{X,x}}\to\mathcal{O}_{X,x}/\mathfrak{m}_x^n$, this easily follows.

Conversely, if we have isomorphisms $(\widehat{\mathcal{O}_{X,x}},x)\to (\widehat{\mathcal{O}_{Y,y}},y)$, then (by Theorem 1) we obviously obtain isomorphisms $F_{(X,x)}\cong F_{(Y,y)}$ as desired.$\blacksquare$

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 $\widehat{\mathcal{O}_{X,x}}$ (made formal by Theorem 2). But, by the previous discussion, this is precisely the ‘differential data’ of $X$ at $x$. 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.