Adic Spaces and Perfectoid Spaces

At the end of Formal Schemes we hinted at the concept of adic spaces, which subsumes both formal schemes and rigid analytic spaces (see also Rigid Analytic Spaces). In this post we will define what these are, give some examples, and introduce and discuss briefly a very special type of adic spaces, the perfectoid spaces, which generalizes what we discussed in Perfectoid Fields.

We begin by discussing the rings that we will need to construct adic spaces. A topological ring (see also Formal Schemes) $A$ is called a Huber ring if it contains an open subring $A_{0}$ which is adic with respect to a finitely generated ideal of definition $I$ contained in $A_{0}$ . This means that the nonnegative powers of $I$ form a basis of open neighborhoods of $0$ . The subring $A_{0}$ is called a ring of definition for $A$ .

Here are some examples of Huber rings:

Any ring $A$ , equipped with the discrete topology, with the ring of definition $A_{0}=A$ , and the ideal of definition $I=(0)$ .
The p-adic numbers $A=\mathbb{Q}_{p}$ , with the p-adic topology, with the ring of definition $A_{0}=\mathbb{Z}_{p}$ , and the ideal of definition $I=(p)$ .
The field of formal Laurent series $A=k((x))$ over some field $k$ , with the metric topology given by the nonarchimedean valuation defined by the order of vanishing at $0$ , with the ring of definition $A_{0}=k[[x]]$ , and the ideal of definition $I=(x)$ .
Generalizing the previous two examples, any nonarchimedean field $K$ is an example of a Huber ring, with ring of definition $A_{0}=\lbrace x:\vert x\vert\leq 1\rbrace$ and ideal of definition $I=(\varpi)$ for some $\varpi$ satisfying $0<\vert\varpi\vert <1$ .

A subset $S$ of a Huber ring, or more generally a topological ring, is called bounded if, for any open neighborhood $U$ of $0$ , we can always find another open neighborhood $V$ of $0$ such that all the products of elements of $V$ with elements of $S$ are contained inside $U$ . An element of a Huber ring is called power bounded if the set of all its nonnegative powers is bounded. For a Huber ring $A$ we denote the set of power bounded elements by $A^{\circ}$ . Any element of the ring of definition will always be power bounded.

With the definition of power bounded elements in hand we give two more examples of Huber rings:

Let $K$ be a nonarchimedean field as in the previous example, and let $\varpi$ again be an element such that $0<\vert\varpi\vert<1$ . Its set of power bounded elements is given by $K^{\circ}=\lbrace x:\vert x\vert\leq 1\rbrace$ . Now let $A=K^{\circ}[[T_{1},\ldots, T_{n}]]$ with the $I$ -adic topology (see also Formal Schemes), where $I$ is the ideal $(\varpi, T_{1},\ldots,T_{n})$ . Then $A$ is a Huber ring with ring of definition $A_{0}=A$ and ideal of definition $I$ .
Let $K$ , $K^{\circ}$ , and $\varpi$ be as above. Consider the Tate algebra $A= K\langle T_{1},\ldots,T_{n}\rangle$ (see also Rigid Analytic Spaces), a topological ring whose topology is generated by a basis of open neighborhoods of $0$ given by $\varpi^{n} A$ . Then $A$ is a Huber ring with ring of definition given by $A_{0}=K^{\circ}\langle T_{1},\ldots,T_{n}\rangle$ and ideal of definition given by $(\varpi)$ .

A subring $A^{+}$ of a Huber ring $A$ which is open, integrally closed, and power bounded is called a ring of integral elements. A Huber pair is a pair $(A,A^{+})$ consisting of a Huber ring $A$ and a ring of integral elements $A^{+}$ contained in $A$ . Note that the set of power bounded elements, $A^{\circ}$ , is itself an example of a ring of integral elements! In fact, in many examples that we will consider the relevant Huber pair will be of the form $(A,A^{\circ})$ .

Now we introduce the adic spectrum of an Huber pair $(A,A^{+})$ , denoted $\mathrm{Spa}(A,A^{+})$ . They will form the basic building blocks of adic spaces, like affine schemes are to schemes or affinoid rigid analytic spaces are to rigid analytic spaces. We will proceed in the usual manner; first we define the underlying set, then we put a topology on it, and then construct a structure sheaf – except that in the case of adic spaces, what we will construct is merely a structure presheaf and may not always be a sheaf! Then we will define more general adic spaces to be something that locally looks like the adic spectrum of some Huber pair.

The underlying set of the adic spectrum $\mathrm{Spa}(A,A^{+})$ is the set of equivalence classes of continuous valuations $\vert\cdot\vert$ on $A$ such that $\vert a\vert\leq 1$ whenever $a$ is in $A^{+}$ . From now on we will change our notation and let $x$ denote a continuous valuation, and we write $f$ for an element of $A$ , so that we can write $\vert f(x)\vert$ instead of $\vert a\vert$ , to drive home the idea that these (equivalence classes of) continuous valuations are the points of our space, on which elements of our ring $A$ are functions.

The underlying topological space of $\mathrm{Spa}(A,A^{+})$ is then obtained from the above set by equipping it with the topology generated by the subsets of the form

$\displaystyle \lbrace x: \vert f(x)\vert \leq \vert g(x)\vert \neq 0\rbrace$

for all $f,g\in A$ .

Let us now define the structure presheaf. First let us define rational subsets. Let $T$ be a subset of $A$ such that the set consisting of all products of elements of $T$ with elements of $A$ is an open subset of $A$ . We define the rational subset

$\displaystyle U\left(\frac{T}{s}\right):=\lbrace x:\vert t(x)\vert \leq\vert s(x)\vert\neq 0\rbrace$

for all $t\in T$ . If $U$ is a rational subset of the Huber pair $(A,A^{+})$ , then there is a Huber pair $(\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }(U),\mathcal{O}_{ \mathrm{Spa}(A,A^{+}) }^{+}(U))$ such that the map $\mathrm{Spa}(A,A^{+})\to \mathrm{Spa}(\mathcal{O}_{X}(U),\mathcal{O}_{X}^{+}(U))$ factors through $U$ and this map is universal among such maps.

Now we define our structure presheaf by assigning to any open set $W$ the Huber pair $(\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W) , \mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W))$ where $\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}(W):=\varprojlim \mathcal{O}(U)$ where the limit is taken over all inclusions of rational subsets $U\subseteq W$ , and $\mathcal{O}_{ \mathrm{Spa}(A,A^{+} )}^{+}(W)$ is similarly defined.

Again, the structure presheaf of $\mathrm{Spa}(A,A^{+})$ may not necessarily be a sheaf; in the case that it is, we say that the Huber pair $(A,A^{+})$ is sheafy. In this case we will also refer to $\mathrm{Spa}(A,A^{+})$ (the underlying topological space together with the structure sheaf) as an affinoid adic space. We can now define more generally an adic space as the data of a topological space $X$ , a structure sheaf $\mathcal{O}_{X}$ , and for each point $x$ of $X$ , an equivalence class of continuous valuations on the stalk $\mathcal{O}_{X,x}$ , such that it admits a covering of $U_{i}$ ‘s giving rise to the data of a structure sheaf and a collection of valuations, all of which is isomorphic to that given by an affinoid adic space.

Recall that we said above that the set of power-bounded elements, $A^{\circ}$ , is an example of a ring of integral elements. Therefore $(A,A^{\circ})$ is an example of a Huber pair. It is convention that, if our Huber pair is given by $(A,A^{\circ})$ we write $\mathrm{Spa}(A)$ instead of $\mathrm{Spa}(A,A^{\circ})$ . Let us now look at some examples of adic spaces.

Consider $\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ (which by the previous paragraph we may also simply write as $\mathrm{Spa}(\mathbb{Q}_{p})$ , since $\mathbb{Z}_{p}$ is the set of power-bounded elements of $\mathbb{Q}_{p}$ ). Then the underlying topological space of $\mathrm{Spa}(\mathbb{Q}_{p},\mathbb{Z}_{p})$ consists of one point, corresponding to the usual p-adic valuation on $\mathbb{Q}_{p}$ .

Next we consider $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ (which by the same idea as above we may write as $\mathrm{Spa}(\mathbb{Z}_{p}$ ). The underlying topological space of $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ now consists of two points, one of which is open, and one of which is closed. The open (or “generic”) point corresponds once again to the usual p-adic valuation $\mathbb{Q}_{p}$ restricted to $\mathbb{Z}_{p}$ . The closed point is the valuation which sends any $\mathbb{Z}_{p}$ which contains a power of $p$ to $0$ , and sends everything else to $1$ .

More complicated is $\mathrm{Spa}(\mathbb{Q}_{p}\langle T\rangle,\mathbb{Z}_{p}\langle T\rangle)$ , also known as the adic closed unit disc. We can compare this with the closed unit disc discussed in Rigid Analytic Spaces. In that post we the underlying set of the closed unit disc was given by the set of maximal ideals of $\mathbb{Q}_{p}\langle T\rangle$ . But every such maximal ideal gives rise to a continuous valuation on $\mathbb{Q}_{p}\langle T\rangle$ . So every point of the rigid analytic closed unit disc gives rise to a point of the adic closed unit disc. But the adic closed unit disc has more points!

An example of a point of the adic closed unit disc is as follows. Let $\Gamma$ be the ordered abelian group $\mathbb{R}_{>0}\times \gamma^{\mathbb{Z}}$ , where $\gamma$ is such that $a<\gamma<1$ for all real numbers $a<1$ in this order. Define a continuous valuation $\vert\cdot\vert_{x^{-}}$ on $\mathbb{Q}_{p}\langle T\rangle$ as follows:

$\displaystyle \vert \sum_{n=0}a_{n}T^{n}\vert_{x^{-}}=\sup_{n\geq 0}\vert a_{n}\vert\gamma^{n}$

This valuation defines a point $x^{-}$ of the adic closed unit disc. This valuation sees $T$ as being infinitesimally less than $1$ , i.e. $\vert T(x^{-})\vert=\vert T\vert_{x^{-}}<1$ , but $\vert T(x^{-})\vert>a$ for all $a<1$ in $\mathbb{Q}_{p}$ . This point $x^{-}$ serves a useful purpose. Recall in Rigid Analytic Spaces that we were unable to disconnect the closed unit disc into two open sets (the “interior” and the “boundary”) because of the Grothendieck topology. In this case we do not have a Grothendieck topology but an honest-to-goodness actual topology, but still we will not be able to disconnect the adic closed unit disc into the analogue of these open sets. This is because the disjoint union of the open sets $\cup_{n\geq 1}\vert T^{n}(x)\vert<\vert p\vert$ and $\vert T(x)\vert=1$ will not miss the point $x^{-}$ , so just these two will not cover the adic closed unit disc.

Finally let us consider $\mathrm{Spa}(\mathbb{Z}_{p}[[ T]],\mathbb{Z}_{p}[[T]])$ . This is the adic open unit disc. This has a map to $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ , and the preimage of the generic point of $\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$ is called the generic fiber (this generic fiber may also be thought of as the adic open unit disc over $\mathbb{Q}_{p}$ , which makes it more comparable to the example of the adic closed unit disc earlier). The adic open unit disc has many interesting properties (for example it is useful to study in closer detail if one wants to study the fundamental curve of p-adic Hodge theory, also known as the Fargues-Fontaine curve) but we will leave this to future posts.

Let us now introduce a very special type of adic space. First we define a very special type of Huber ring. We say that a Huber ring $A$ is Tate if it contains a topologically nilpotent unit (also called a pseudo-uniformizer). An element $\varpi$ is topologically nilpotent if its sequence of powers $\varpi, \varpi^{2},\ldots$ converges to $0$ . For example, the Huber ring $\mathbb{Q}_{p}$ (as discussed above) is Tate, with pseudo-uniformizer given by $p$ .

If, in addition to being Tate, the Huber ring $A$ is complete, uniform (which means that $A^{\circ}$ is bounded in $A$ ), and contains a pseudo-uniformizer $\varpi$ such that $\varpi^{p}\vert p$ in $A^{\circ}$ and the p-th power map map $A/\varpi\to A/\varpi^{p}$ is an isomorphism, then we say that $A$ is perfectoid. As can be inferred from the name, this generalizes the perfectoid fields we introduced in Perfectoid Fields. We recall the important property of perfectoid fields (which we can now generalize to perfectoid rings) – if $R$ is perfectoid, then the category of finite etale $R$ -algebras is equivalent to the category of finite etale $R^{\flat}$ -algebras, where $R^{\flat}$ is the tilt of $R$ . For fields, this manifests as an isomorphism of their absolute Galois groups, which generalizes the famous Fontaine-Wintenberger theorem.

A perfectoid space is an adic space which can be covered by affinoid adic spaces $\mathrm{Spa}(A,A^{+})$ , where $A$ is perfectoid. If $X$ is a perfectoid space, we can associate to it its tilt $X^{\flat}$ , by taking the tilts of the affinoid adic spaces that cover $X$ and gluing them together. In fact, for a fixed perfectoid space $X$ , there is an equivalence of categories between perfectoid spaces over $X$ , and perfectoid spaces over $X^{\flat}$ . This is the geometric version of the equivalence of categories of finite etale algebras over a perfectoid ring and its tilt. In addition, although we will not do it in this post, one can define the etale sites of $X$ and $X^{\flat}$ , and these will also be equivalent.

To end this post, we mention some properties of perfectoid spaces that make it useful form some applications. It turns out that if $X$ is a smooth rigid analytic space, it always has a pro-etale cover by affinoid perfectoid spaces. A pro-etale map $U\to X$ may be thought of as a completed inverse limit $\varprojlim_{i} U_{i}\to X$ , where each $U_{i}\to X$ is an etale map. An example of a pro-etale cover is as follows. If we let $\mathbb{Q}_{p}^{\mathrm{cycl}}$ be the perfectoid field given by the completion of $\cup_{n}\mathbb{Q}_{p}(\mu_{p^{n}})$ (this is somewhat similar to the example involved in the Fontaine-Wintenberger theorem in Perfectoid Fields), then $\mathrm{Spa}(\mathbb{Q}_{p}^{\mathrm{cycl}})$ is a pro-etale cover of $\mathrm{Spa}(\mathbb{Q}_{p})$ . To see why this is pro-etale, note that a finite separable extension of fields is etale, and $\mathbb{Q}_{p}^{\mathrm{cycl}}$ is the completion of the infinite union (direct limit) of such finite separable extensions $\mathbb{Q}_{p}(\mu_{p^{n}})$ of $\mathbb{Q}_{p}$ , but looking at the adic spectrum means the arrows go the other way, which is why we think of it as an inverse limit.

Another property of perfectoid spaces is the following. If $U$ is a perfectoid affinoid space over $\mathbb{C}_{p}$ , then for all $i>0$ $H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+})$ (this is the cohomology of the sheaf of functions bounded by $1$ on the etale site of $X$ ) is annihilated by the maximal ideal of $\mathcal{O}_{\mathbb{C}_{p}}$ . We also say that $H^{i}(U_{\mathrm{et}},\mathcal{O}_{X}^{+})$ is almost zero.

Together, what these two properties tell us is that we can compute the cohomology of a smooth rigid analytic space via the Cech complex associated to its covering by perfectoid affinoid spaces. This has been applied in the work of Peter Scholze to the mod p cohomology of the rigid analytic space associated to a Siegel modular variety, in order to relate it to Siegel cusp forms (see also Siegel modular forms). In this case the covering by perfectoid affinoid spaces is provided by a Siegel modular variety at “infinite level”, which happens to have a map (called the period map) to a Grassmannian (the moduli space of subspaces of a fixed dimension of some fixed vector space), and there are certain properties that we can then make use of (for instance, the line bundle on the Siegel modular variety whose sections are cusp forms can be obtained via pullback from a certain line bundle on the Grassmannian) together with p-adic Hodge theoretic arguments to relate the mod p cohomology to Siegel cusp forms.

All this has the following stunning application. Recall that in we may obtain Galois representations from cusp forms (see for example Galois Representations Coming From Weight 2 Eigenforms). This can also be done for Siegel cusp forms more generally. These cusp forms live on a modular curve or Siegel modular variety, which are obtained as arithmetic manifolds, double quotients $\Gamma\backslash G(\mathbb{R})/K$ of a real Lie group $G$ (in this case the symplectic group) by a maximal compact open subgroup $K$ and an arithmetic subgroup $\Gamma$ . But they are also algebraic varieties, so can be studied using the methods of algebraic geometry (see also Shimura Varieties). For example, we can use etale cohomology to obtain Galois representations.

But not all arithmetic manifolds are also algebraic varieties! For instance we have Bianchi manifolds, which are double quotients $\Gamma\backslash\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2)$ , where $\Gamma$ can be, say, a congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z}[i])$ (or we can also replace $\mathbb{Z}[i]$ with the ring of integers of some other imaginary quadratic field). The groups involve look complex, but the theory of algebraic groups and in particular the method of Weil restriction allows us to look at them as real Lie groups. This is not an algebraic variety (one way to see this is that $\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}(2)$ is hyperbolic 3-space, so a Bianchi manifold has 3 real dimensions and as such cannot be related to an algebraic variety the way a complex manifold can).

Still, it has been conjectured that the singular cohomology (in particular its torsion subgroups) of such arithmetic manifolds which are not algebraic varieties can still be related to Galois representations! And for certain cases this has been proved using the following strategy. First, these arithmetic manifolds can be found as an open subset of the boundary of an appropriate compactification of a Siegel modular variety. Then, methods from algebraic topology (namely, the excision long exact sequence) allow us to relate the cohomology of the arithmetic manifold to the cohomology of the Siegel modular variety.

On the other hand, by our earlier discussion, the covering of the (rigid analytic space associated to the) Siegel modular variety by affinoid perfectoid spaces given by the Siegel modular variety at infinite level, together with the period map of the latter to the Grassmannian, allows one to show that the mod p cohomology of Siegel modular varieties is related to Siegel cusp forms, and it is known how to obtain Galois representations from these. Putting all of these together, this allows us to obtain Galois representations from the cohomology of manifolds which are not algebraic varieties.

A deeper look at aspects of perfectoid spaces, as well as their generalizations and applications (including a more in-depth look at the application to the mod p cohomology of Siegel modular varieties discussed in the previous couple of paragraphs), will hopefully be discussed in future posts.

References:

Perfectoid space on Wikipedia

Adic spaces by Jared Weinstein

Berkeley lectures on p-adic geometry by Peter Scholze and Jared Weinstein

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

Lecture notes on perfectoid Shimura varieties by Ana Caraiani

On torsion in the cohomology of locally symmetric varieties by Peter Scholze

p-adic Hodge theory for rigid analytic varieties by Peter Scholze

Theories and Theorems

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Adic Spaces and Perfectoid Spaces

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