Berkovich spaces I

Around 1990, Berkovich discovered a rather elegant refinement of the notion of scheme that is particularly well-suited to non-archimedean analytic geometry. I feel that the ideas here deserve more exposure, although I don’t have any specific applications in mind. Note: this post is really long.

We define a seminorm on a commutative ring $A$ to be a map $\Vert \cdot \Vert: A \to \mathbb{R}_{\geq 0}$ such that

$\Vert 0 \Vert = 0$

$\Vert 1 \Vert = 1$

$\Vert a + b \Vert \leq \Vert a \Vert + \Vert b \Vert$

$\Vert ab \Vert = \Vert a \Vert \Vert b \Vert$

The standard example of a seminorm is the trivial norm $\Vert \cdot \Vert_0$ on a domain, which is $1$ on all nonzero elements (zero divisors can’t be treated canonically). Seminorms naturally pull back along ring homomorphisms – this is not true of norms, because anything in the kernel has norm zero.

Exercise: Show that the only seminorm on a finite field is trivial.

I’l define two notions of spectrum, one of which appears in the literature. Given a commutative ring $A$ , its unrestricted spectrum $uSp(A)$ is the set of all seminorms on $A$ . If $A$ is equipped with a norm with respect to which it is complete (i.e. $A$ is a Banach algebra), then the Berkovich spectrum $Sp(A)$ is the set of all seminorms on $A$ that are bounded with respect to that norm, meaning the seminorm is at most equal to the norm.

For each element $a \in A$ , there are evaluation maps $ev_a: uSp(A), Sp(A) \to \mathbb{R}$ , and we endow $uSp(A)$ and $Sp(A)$ with the weakest topologies such that all evaluations are continuous, i.e., a basis of open sets is given by preimages of real open intervals. Forgetting the norm on a Banach algebra induces a natural embedding $Sp(A) \to uSp(A)$ , and I think it might be a homotopy equivalence, but I’m not sure.

Let’s try to compute $uSp(\mathbb{Z})$ . We first consider norms, which are known to have three forms:

Archimedean norms $\Vert \cdot \Vert_\infty^\alpha, 0 < \alpha \leq 1$

$p$ -adic norms $\Vert \cdot \Vert_p^\beta, 0 < \beta \leq +\infty$

The trivial norm $\Vert \cdot \Vert_0 = \Vert \cdot \Vert_p^0 = \Vert \cdot \Vert_\infty^0$

The remaining seminorms are pulled back from the trivial norms on $\mathbb{Z}/p\mathbb{Z}$ , and are written $\Vert \cdot \Vert_p^\infty$ . Now we consider evaluation maps. $ev_0$ maps to zero, $ev_1$ and $ev_{-1}$ map to one, and for any other $n$ , $ev_n$ takes $\Vert \cdot \Vert_\infty^\alpha$ to $|n|^\alpha$ and $\Vert \cdot \Vert_p^\beta$ to one if $(p,n)=1$ and $p^{-\beta v_p(n)}$ if $p | n$ . If we pull back intervals along $ev_p$ , we find that $\{ \Vert \cdot \Vert_p^\beta | 0 < \beta \leq \infty \}$ forms an embedded copy of the interval $[0,1)$ for each prime, and that $\{ \Vert \cdot \Vert_\infty^\alpha | 0 < \alpha \leq 1 \}$ forms an additional half-open interval homeomorpic to $(1,p]$ . These are glued together at the trivial norm, whose open neighborhoods contain all but finitely many of the intervals. You can imagine this as a broom, where the handle is given by archimedean norms, and the bristles are non-archimedean. The topology is metrizable, and you can embed this space in $\mathbb{R}^2$ by making the nonarchimedean bristles progressively shorter.

$\mathbb{Z}$ is only complete with respect to the trivial norm and powers of the archimedean norm, so we have a family of Banach algebra structures parametrized by a line segment. For any such structure, the nonarchimedean seminorms are bounded, so the Berkovich spectrum is the subset of the unrestricted spectrum given by sawing off part of the broom handle.

If we invert some primes, all of the norms remain norms, but some become unbounded, and the seminorms pulled back from quotients by those primes disappear, since $\Vert \frac{1}{p} \Vert_p^\infty$ doesn’t make sense. In the broom picture, we are removing endpoints of the bristles of $uSp$ . The situation with $Sp(\mathbb{Z}[1/N])$ is more subtle, since it depends on the norm we chose. If we are given a positive power of the Archimedean norm, then the whole thing collapses to a point. If we chose the trivial norm, then we remove the bristles corresponding to $p|N$ . If we invert all of the primes, we get $uSp(\mathbb{Q})$ as a broom without endpoints, while $Sp(\mathbb{Q})$ is a point. In general, there is a continuous natural transformation $uSp(A) \mapsto Spec(A)$ given by sending a seminorm to its kernel. This map endows $uSp(A)$ with the structure of a locally ringed space. We can also give $Sp(A)$ a locally ringed space structure, but we define the local rings to be the completions of the local rings pulled back from $uSp(A)$ .

I’ll try a few more examples: $uSp(\mathbb{Z}[i])$ also looks like a broom, but the archimedean norm comes from a complex embedding, and we have “more” bristles, since primes of the form $4k+1$ split into two. $uSp(\mathbb{Z}[5i])$ looks a lot like the previous space, but the bristles pulled back from the $(1 + 2i)$ -adic and $(1 - 2i)$ -adic valuations have been glued together at the tips. In general, the unrestricted spectrum of a ring is contractible if and only if the ring is Dedekind.

$uSp(\mathbb{F}_q[t])$ has spokes for each irreducible polynomial, corresponding to some $\epsilon \in [0,1)$ to the power of how many times a given element can be divided by it. There is also a spoke for positive powers of degree. We can make $\mathbb{P}^1_{\mathbb{F}_q}$ by adding a point at infinity, or gluing two copies of the affine line along the reciprocal map.

So far, the two notions of spectra did not seem to result in very different spaces. If we move to larger rings, this changes rather spectacularly. For example, the field of complex numbers admits a set of seminorms of cardinality at least $2^c$ , since we can pull back the archimedean norm along any ring-theoretic automorphism. $uSp(\mathbb{C})$ is still a contractible space, but it is more cumbersome than $Sp(\mathbb{C}$ , which is a point. In general, the unrestricted spectrum is better for studying global fields, since we can see more than one archimedean seminorm at a time, while the Berkovich spectrum is better for anything involving completion, like $p$ -adic geometry.

Recall from my $p$ -adic fields post that $\mathbb{C}_p$ is the completion of an algebraic closure of $\mathbb{Q}_p$ . We write $\mathbb{C}_p \langle t \rangle$ for the ring of power series $\sum_{n \geq 0} a_n t^n$ with the norms of $a_n \in \mathbb{C}_p$ converging to zero. This is the set of series which converge if we substitute any element of norm at most one for $t$ , and we endow the ring with the norm given by taking the supremum of all such substitutions. The Gauss Lemma implies this norm is multiplicative, and it gives $\mathbb{C}_p \langle t \rangle$ a Banach algebra structure.

The Berkovich unit disc is defined to be $Sp(\mathbb{C}_p \langle t \rangle)$ . It is a fundamental object, in the sense that it plays a role in rigid geometry analogous to that of $\mathbb{A}^1$ – varieties are defined by solutions to equations on products of discs.

Let’s try to find some points on the disc, i.e., seminorms on $\mathbb{C}_p \langle t \rangle$ . As noted above, the sup norm on the points of norm at most one gives a distinguished point, called the Gauss point. One can also take any element $a \in \mathbb{C}_p$ of norm at most one, and a real number $r$ between zero and one (not necessarily in the value group), and set $\Vert f \Vert_{B(a,r)} = sup \{ f(x) | \, |x-a| < r \}$ . More generally, one can take any nested sequence of balls of decreasing radius, and take the limit seminorm. Berkovich’s classification theorem asserts that all seminorms come from some nested sequence, and two such norms are equivalent if and only if the sequences are cofinal (meaning any term in one sequence has a successor in the other sequence). We can categorize the seminorms into four types:

Type I points correspond to nested sequences for which the radius goes to zero. Since $\mathbb{C}_p$ is complete, these are just evaluation at an element. These are called classical points.

Type II points correspond to nested sequences whose intersection is a ball whose radius lies in the value group, i.e., the seminorm is just sup over the ball. These are called rational points.

Type III points correspond to nested sequences whose intersection is a ball whose radius is not in the value group, but it is still given by sup over a ball. These points exist because absolute values are rational powers of $|p|$ .

Type IV points correspond to nested sequences whose intersection is empty. These points exist because $\mathbb{C}_p$ is not spherically complete.

How do we view this as a topological space? If one ball contains another, we connect the corresponding sup seminorms with a line segment, which is made of the seminorms of balls of intermediate radius. We get a sort of tree structure expressing the containment partial ordering on balls in the disc. Unlike an ordinary tree, this one has infinitely many branches coming out of a dense subset of any line segment (geometric group theorists call this an $\mathbb{R}$ -tree). In particular, the type II points are incident to “edges” that are in natural bijection with points on the projective line over the residue field $\overline{\mathbb{F}_p}$ . The connected open neighborhoods of any point $x$ are given by choosing finitely many points and excluding the parts of the tree for which a path to $x$ passes through those points, so they tend to be quite large – large enough that the unit disc
is compact.

If we glue two copies of the unit disc along the unit circle, we get another tree, but the Gauss point no longer plays a distinguished role. This space is called $\mathbb{P}^1_{Berk}$ , and it is an equivariantly compactified union of Bruhat-Tits buildings for $PGL_2$ over finite extensions of $\mathbb{Q}_p$ . If you switch to power series fields, restriction of scalars gives a natural bijection between type II points with integer valuation and points on the affine Grassmannian. You can get many natural representations of groups over local fields by choosing sheaves on these ind-buildings with good equivariance properties. Apparently the Harris-Taylor proof of local Langlands correspondence for $GL_n$ used étale cohomology on these spaces, although I must confess that I never got that far into their book.

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