# Formal Schemes

In Galois Deformation Rings we discussed the dual numbers as well as the concept of deformations. The dual numbers are numbers with an additional “tangent direction” or a “derivative” – we can further take into account higher order derivatives to consider deformations, which leads us to the concept of deformations.

In this post, we will consider spaces related to deformations, called formal schemes. Let us begin with a motivating example. Consider a field $k$. From algebraic geometry, we know that the underlying topological space of the scheme $\mathrm{Spec}(k)$ is just a single point. What about the dual numbers, which is the ring $k[x]/(x^{2})$. What is the underlying topological space of the scheme $\mathrm{Spec}(k[x]/(x^{2}))$? It turns out it is also just the point!

So as far as the underlying topological spaces go, there is no difference between $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k[x]/(x^{2}))$ – they are both just points. This is because they both have one prime ideal. For $k$ this is the ideal $(0)$, which is also its only ideal that is not itself. For $k[x]/(x^{2})$, its one prime ideal is the ideal $(x)$; note that the ideal $(0)$ in $k[x]/(x^{2})$ is not prime, which is related to this ring not being an integral domain. However a scheme is more than just its underlying topological space, one also has the data of its structure sheaf, i.e. its ring of regular functions, and in this regard $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k[x]/(x^{2}))$ are different.

We sometimes consider $\mathrm{Spec}(k[x]/(x^{2}))$ as a “thickening” of $\mathrm{Spec}(k)$ – they are both just the point, but the functions on $\mathrm{Spec}(k[x]/(x^{2}))$ have derivatives, as if there were tangent directions on the point that is the underlying space of $\mathrm{Spec}(k[x]/(x^{2}))$ on which one can move “infinitesimally”, unlike on the point that is the underlying space of $\mathrm{Spec}(k)$.

Just like in our discussion in Galois Deformation Rings, we may want to consider not only the “first-order derivatives” which appear in the dual numbers but also “higher-order derivatives”; we may even want to consider all of them together. This amounts to taking the inverse limit $\varprojlim_{n}k[x]/(x^{n})$, which is the formal power series ring $k[[x]]$. However, if we take $\mathrm{Spec}(k[[x]])$, we will see that it actually has two points, a “generic point” (corresponding to the ideal $(0)$, which is prime because $k[[x]]$ is an integral domain) and a “special point” (corresponding to the ideal $(x)$, which is also prime and furthermore the lone maximal ideal), unlike $\mathrm{Spec}(k)$ or $\mathrm{Spec}(k[x]/(x^{2}))$ (or more generally $\mathrm{Spec}(k[x])/(x^{n}))$, for any $n$, justified by similar reasons to the preceding argument).

This is where formal schemes come in – a formal scheme can express the “thickening” of some other scheme, with all the “higher-order derivatives”, where the underlying topological space is the same as that of the original scheme but the structure sheaf might be different, to reflect this “thickening”.

A topological ring is a ring $A$ equipped with a topology such that the usual ring operations are continuous with respect to this topology. In this post we will mostly consider the $I$-adic topology, for some ideal $I$ called the ideal of definition. This topology is generated by a basis consisting of sets of the form $a+I^{n}$, for $a$ in $A$.

An example of a topological ring with the $I$-adic topology is the formal power series ring $k[[x]]$ which we have discussed, together with the ideal of definition $(x)$. Another example is the p-adic integers $\mathbb{Z}_{p}$, together with the ideal of definition $(p)$. We note that all these examples are complete with respect to the $I$-adic topology.

More generally for higher dimension one can take, say, an affine variety cut out by some polynomial equation, say $y^{2}=x^{3}$, and consider the ring $k[x,y]/(y^{2}-x^{3})$. Note that $\mathrm{Spec}(k[x,y]/(y^{2}-x^{3}))$ is an affine variety. Now we can form a topological ring complete with respect to the $I$-adic topology, by taking the completion with respect to the ideal $I=(y^{2}-x^{3})$, i.e. the inverse limit of the diagram $\displaystyle k[x,y]/(y^{2}-x^{3})\leftarrow k[x,y]/(y^{2}-x^{3})^{2} \leftarrow k[x,y]/(y^{2}-x^{3})^{3} \leftarrow\ldots$

The formation of this ring is an important step in describing the “thickening” of the affine variety $\mathrm{Spec}(k[x,y]/(y^{2}-x^{3}))$, but as said above, it cannot just be done by taking the “spectrum”. Therefore we introduce the idea of a formal spectrum.

Let $A$ be a Noetherian topological ring and let $I$ be an ideal of definition. In addition, let $A$ be complete with respect to the $I$-adic topology. We define the formal spectrum of $A$, denoted $\mathrm{Spf}(A)$, to be the pair $(X,\mathcal{O}_{X})$, where $X$ is the underlying topological space of $\mathrm{Spec}(A/I)$, and the structure sheaf $\mathcal{O}_{X}$ is defined by setting $\mathcal{O}_{X}(D_{f})$ to be the $I$-adic completion of $A[1/f]$, for $D_{f}$ the distinguished open set corresponding to $f$. Applied to the examples earlier, this gives us what we want – a sort of “thickening” of some affine scheme, with an underlying topological space the same as that of the original scheme but with a structure sheaf of functions with “higher-order derivatives”.

More generally, to include the “non-affine” case a formal scheme is a topologically ringed space, i.e. a pair $(X,\mathcal{O}_{X})$ where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf of topological rings, such that for any point $x$ of $X$ there is an open neighborhood of $x$ which is isomorphic as a topologically ringed space to $\mathrm{Spf}(A)$ for some Noetherian topological ring $A$.

Aside from being useful in deformation theory, formal schemes are also related to rigid analytic spaces (see also Rigid Analytic Spaces). For certain types of formal schemes (“locally formally of finite type over $\mathrm{Spf}(\mathbb{Z}_{p})$“) one can assign (functorially) a rigid analytic space. For example, this functor will assign to the formal scheme $\mathrm{Spf}(\mathbb{Z}_{p}[[x]])$ the open unit disc (the interior of the closed unit disc in Rigid Analytic Spaces). This functor is called the generic fiber functor, which is an odd name, because $\mathrm{Spf}(\mathbb{Z}_{p}[[x]])$ has no “generic points”! However, there is a way to make this name make more sense using the language of adic spaces, which also subsumes the theory of both formal schemes and rigid analytic spaces, and also provides a natural home for the perfectoid spaces we hinted at in Perfectoid Fields. The theory of adic spaces will hopefully be discussed in some future post on this blog.

References:

Formal Scheme on Wikipedia

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Modular Curves at Infinite Level by Jared Weinstein