Geometrically multiplicative non-archimedean normsThanks: This research was supported by MPIM-Bonn and ISF Grant 1203/22

All rings in this paper are assumed to be commutative and all seminorms are non-archimedean. Banach rings and their spectra are building blocks of Berkovich analytic geometry. Our goal is to study finer properties of norms, not preserved by equivalence of Banach rings. In addition it is often convenient to work with non-complete rings and apply completion only when needed, so this paper is written in the generality of normed or even seminormed rings over (non-archimedean) real valued fields. One can always divide a seminormed ring or module by the kernel of the seminorm and pass to the normed world, but seminorms show up naturally as outcomes of such constructions as the spectral seminorm and the tensor seminorm. For this reason we prefer to consider the seminormed case too.

Here are three very important properties a seminorm $|\ |$ can satisfy:

• (i)

  $|\ |$ is multiplicative if $|ab|=|a|\cdot|b|$ for any $a,b\in{{\mathcal{A}}}$.

• (ii)

  $|\ |$ is power-multiplicative or spectral if $|a^{n}|=|a|^{n}$ for any $a\in{{\mathcal{A}}}$ $n\geq 0$.

• (iii)

  $|\ |$ is spectrally multiplicative if $|\ |_{\rm sp}$ is multiplicative.

Recall that the spectral seminorm $|\ |_{\rm sp}$ defined by $|a|_{\rm sp}=\lim_{n}|a^{n}|^{1/n}$ is the minimal power-multiplicative seminorm dominated by $|\ |$, so condition (ii) means that $|\ |=|\ |_{\rm sp}$.

When working over a real valued ground field $k$ it is also natural to consider geometric or universal variants of these notions. Namely, we say that a seminorm on a $k$-algebra ${{\mathcal{A}}}$ is geometrically multiplicative, spectral or spectrally multiplicative if the tensor seminorm on ${{\mathcal{A}}}_{l}={{\mathcal{A}}}\otimes_{k}l$ is geometrically multiplicative, spectral or spectrally multiplicative, respectively, for any real valued field extension $l/k$. Geometrically multiplicative norms were used by Berkovich to study group actions on analytic spaces, and he called the points they define peaked points in [Ber90]. Poineau called them universal norms in [Poi13] and proved that any multiplicative norm over an algebraically closed field is universal. To the best of our knowledge no further study of universality was done in the literature. The goal of this paper is to fill in this gap and develop the theory of geometric spectrality and spectral multiplicativity quite analogous to the classical theory of geometric reducedness and irreducibility of schemes. The combination of these two yields the desired criterion of geometric multiplicativity.

We will work in the category of normed (or even seminormed) rings (and modules) with non-expansive homomorphisms. Seminormed rings will be denoted by calligraphic letters, e.g. ${{\mathcal{A}}}=({{\mathcal{A}}},|\ |)$, and we say that ${{\mathcal{A}}}$ is spectral, multiplicative or spectrally multiplicative if its seminorm is so. In fact, these properties should be viewed as extensions to the theory of seminormed rings of the classical properties from commutative algebra – reducedness, being a domain and having an integral reduction. This analogy will be used throughout the paper, but here are a couple of its instances.

• (o)

  A ring $A$ is reduced (resp. integral) if and only if the norm sending all non-zero elements to 1 is power-multiplicative (resp. multiplicative).

• (i)

  A norm on ${{\mathcal{A}}}$ is power-multiplicative (resp. multiplicative) if and only if the associated graded ring ${{\mathcal{A}}}_{\rm gr}=\oplus_{r>0}{{\mathcal{A}}}_{\leq r}/{{\mathcal{A}}}_{<r}$ is reduced (resp. integral).

• (ii)

  Each normed ring ${{\mathcal{A}}}$ possess a universal homomorphism $\phi\colon{{\mathcal{A}}}\to{{\mathcal{A}}}^{\rm sp}$ whose target is a spectral normed ring ${{\mathcal{A}}}^{\rm sp}$ that will be called the spectralization of ${{\mathcal{A}}}$. Furthermore, ${\mathcal{M}}({{\mathcal{A}}}^{\rm sp})={\mathcal{M}}({{\mathcal{A}}})=X$, $|\ |_{{{\mathcal{A}}}^{\rm sp}}=\max_{x\in X}|\ |_{x}$ and the kernel of $\phi$ consists of all quasi-nilpotent elements, i.e. elements with $|f|_{\rm sp}=\max_{x\in X}|f|_{x}=0$. This is the analogy of the reduction homomorphism $A\to{\widetilde{A}}=A/{\rm Rad}(A)$ in commutative algebra and of the fact that ${\rm Rad}(A)=\cap_{p\in{\rm Spec}(A)}p$ and the reduction induces homeomorphism of spectra.

• (iii)

  A seminormed ring ${{\mathcal{A}}}$ is spectrally multiplicative if and only if ${{\mathcal{M}}}({{\mathcal{A}}})$ contains a unique maximal point $|\ |_{x}$. A seminormed ring is multiplicative if and only if it is spectral and spectrally multiplicative. This is the analogue of the fact that a ring $A$ is a domain if and only if it is reduced and ${\rm Spec}(A)$ is irreducible.

We will study criteria for normed $k$-algebras to be geometrically spectral and geometrically spectrally multiplicative. It turns out that the theory is very analogous to its classical commutative algebra counterpart. For instance, if $k$ is of equal characteristic $p$, then the seminorm of ${{\mathcal{A}}}$ is geometrically spectral if and only if the tensor seminorm on ${{\mathcal{A}}}\otimes k^{1/p}$ is spectral. Moreover, one can replace $k^{1/p}$ by its deformations, called $p$-versal extensions of $k$, and then a similar criterion also applies in the fixed characteristic, see Theorem 5.3.1. Also, we prove that ${{\mathcal{A}}}$ is geometrically spectrally multiplicative if and only if it is spectrally multiplicative and ${\widehat{k}}$ is separably closed in the completion of ${\rm Frac}({{\mathcal{A}}}^{\rm sp})$, see 4.3.1. Combining these two results one deduces in Theorem 5.4.1 a criterion for ${\mathcal{A}}$ to be geometrically multiplicative. In particular, we reprove the result of Poineau and show the stronger result that over a perfectoid base $k$ any spectral $k$-algebra ${{\mathcal{A}}}$ is geometrically spectral and geometric multiplicativity holds if and only if ${{\mathcal{A}}}$ is multiplicative and $k$ is algebraically closed in the completion of ${\rm Frac}({{\mathcal{A}}})$.

We will see that the main case to study is when $l/k$ is finite. If $l/k$ is defectless, then the graded reduction completely controls the ground field extension and can be used to formally reduce the problem to the classical commutative algebra. However in the general case, instead of this we will have to work with reductions of a small but non-zero thickness ${\mathcal{A}}^{\circ}/\pi{\mathcal{A}}^{\circ}$ for a pseudo-uniformizer $\pi\in k$ whose valuation is close enough to 1. This imposes mild technical complications as we have to track the thickness (e.g. under Frobenius), but allows us to exploit the analogy with the classical results, and construct the proofs along the same general lines.

For simplicity, we will omit the word “generalized” in what one often calls generalized Gauss valuation or extension. By default, we provide tensor products of seminormed rings and modules with tensor product seminorm. If $l/k$ is a real valued field extension and ${{\mathcal{A}}}$ is a seminormed $k$-algebra, we will use the notation ${{\mathcal{A}}}_{l}={{\mathcal{A}}}\otimes_{k}l$.

Berkovich introduced analytic spectrum only for Banach rings, though this is mainly a matter of taste. As in the theory of Huber’s adic spectra, the definition makes sense more generally, but the outcome depends only on the completion viewed as a Banach ring. Thus, by Berkovich spectrum of a seminormed ring ${{\mathcal{A}}}$ we mean the set $X={{\mathcal{M}}}({{\mathcal{A}}})$ of all bounded real semivaluations on ${{\mathcal{A}}}$ with the weakest topology making all functions $|f|\colon X\to{{\mathbb{R}}}_{\geq 0}$ with $f\in{{\mathcal{A}}}$ continuous. This is a contravariant functor on the category of seminormed rings.

The points of $X$ will denoted $x$ or $|\ |_{x}$. The completed residue field ${{\mathcal{H}}}(x)$ is the completed field of fractions of the multiplicative normed ring ${{\mathcal{A}}}/{\rm Ker}(|\ |_{x})$. Given a homomorphism ${{\mathcal{A}}}\to{{\mathcal{B}}}$ of seminormed rings we say that the induced map ${{\mathcal{M}}}({{\mathcal{B}}})\to{{\mathcal{M}}}({{\mathcal{A}}})$ is an isomorphism¹¹ 1 The notion spectral isomorphism would be more precise, but we do not provide ${{\mathcal{M}}}({{\mathcal{A}}})$ with any finer structure in this paper. if it is a homeomorphism and induces isomorphism of completed residue fields. For example, the map ${{\mathcal{M}}}({\widehat{\mathcal{A}}})\to{{\mathcal{M}}}({{\mathcal{A}}})$ is an isomorphism because of the factorizations ${{\mathcal{A}}}\to{\widehat{\mathcal{A}}}\to{\mathcal{H}}(x)$. Thus Berkovich’s theorem [Ber90, Theorem 1.2.1] implies that ${{\mathcal{M}}}({{\mathcal{A}}})$ is compact and non-empty whenever ${{\mathcal{A}}}\neq 0$.

By the spectralization of ${{\mathcal{A}}}$ we mean the normed ring ${{\mathcal{A}}}^{\rm sp}={{\mathcal{A}}}/{\rm Ker}(|\ |_{\rm sp})$, whose norm is induced by the spectral seminorm $|\ |_{\rm sp}$ of ${{\mathcal{A}}}$ and denoted by the same notation. This procedure is the analogue of reduction in commutative algebra. Recall that $|\ |_{\rm sp}=\max_{x}|\ |_{x}$ by [Ber90, Theorem 1.3.1], which is the analogue of the classical fact that the nilradical equals the intersection of all prime ideals.

As a consequence, studying geometric spectral multiplicativity reduces to the case of fields.

Our choice to work in the generality of non-complete and even non-henselian valued fields forces us to distinguish abstract algebraic extensions of real valued fields, which may have a few extensions of valuations, and real valued extensions, where an extension is fixed. We say that an algebraic extension $l/k$ is unibranch if the extension is unique. Also, we say that a finite extension of real valued fields $l/k$ is defectless if $[l:k]=e_{l/k}f_{l/k}$. In particular, in this case $l/k$ is unibranch. This should not be confused with the more usual definition that an abstract extension $l/k$ is defectless when $[l:k]=\sum_{i}e_{i}f_{i}$, where the sum is over all extensions of the valuation.

Recall that a vector space $V$ over a real valued field $k$ is called cartesian if it possesses an orthogonal basis $\{e_{i}\}_{i\in I}$, which means that $\|\sum_{i}c_{i}e_{i}\|=\max_{i}\|c_{i}e_{i}\|$. A finite real-valued extension $l/k$ is defectless if and only if $[{\widetilde{l}}_{\rm gr}:{\widetilde{k}}_{\rm gr}]=[l:k]$ if and only if $l$ is cartesian, where $k_{\rm gr}=\oplus_{r>0}{\widetilde{k}}_{r}=\oplus_{r>0}k_{\leq r}/k_{<r}$ is the graded reduction of $k$ as defined in [Tem04, §2]. Recall that ${\widetilde{k}}_{\rm gr}$ is a graded field (i.e. 0 and (1) are the only homogeneous ideals). More generally, note that for any normed $k$-vector space $V$ the graded reduction ${\widetilde{V}}_{\rm gr}$ is a graded ${\widetilde{k}}_{\rm gr}$-vector space and the same argument as for $l/k$ shows that if $V$ is finite-dimensional, then it is cartesian if and only if the fundamental inequality ${\rm dim}_{{\widetilde{k}}_{\rm gr}}({\widetilde{V}}_{\rm gr})\leq{\rm dim}_{k}(V)$ is an equality.

As a consequence one can easily control the norms under finite defectless base changes. This case already covers discrete valuations and finite tame extensions.

The definition makes sense for seminormed rings and modules, but let us restrict the generality to a seminormed vector space $V$ over a real valued field $k$. A family of elements $\{e_{i}\}_{i\in I}$ is called $r$-orthogonal for $r\in(0,1]$ if for any linear combination $v=\sum_{i\in I}a_{i}e_{i}$ (with almost all $a_{i}$ zeros) one has that $|v|\geq r\max_{i}\|a_{i}e_{i}\|$. We say that $V$ is $r$-cartesian if it possesses an $r$-orthogonal basis, and weakly cartesian if any finite dimensional subspace possesses such a basis. In particular, $V$ is normed. As earlier, if $r=1$ we simply say orthogonal and cartesian. The notion of $r$-cartesianity is especially important in the finite-dimensional case when the valuation is not discrete. In particular, one has the following classical result.

Given a normed $k$-vector space $V$ we will use the notation $V^{\circ}=V_{\leq 1}$ to denote its unit ball. As usual, we say that a $k^{\circ}$-module $M$ almost vanishes if $|k^{\times}|$ is dense and $\pi M=0$ for any $\pi\in k^{\circ\circ}$.

Unlike claims (i) and (ii) of the lemma, one can not completely remove the adjective “almost” in claim (iii), and this is so even when the vector spaces are strict (i.e. $\|U\|=|k|$) or, moreover, $|k^{\times}|={\mathbb{R}}_{\geq 0}$. Here are some examples.

There is a standard trick in Berkovich geometry which reduces many questions to the strict case – apply a base change with respect to an appropriate extension $K/k$ with a large $|K^{\times}|$ (typically a Gauss extension) and descend the results. We will only need to use it when $k$ is discretely valued and Lemma 3.2.1(iii) cannot be applied as it is.

As a toy application let us reprove the following simple well known statement, e.g. see [Poi13, Lemme 3.1].

As another application let us prove the following simple result, which of course admits more straightforward proofs too.

As a corollary we obtain a criterion for failure of geometric multiplicativity. Consider a henselian real valued field $k$ (so, the valuation extends to $k^{a}$ uniquely). For an algebraic element ${\alpha}\in k^{a}$ let $d_{{\alpha}/k}=\inf_{c\in k}|c-{\alpha}|$ denote the distance between ${\alpha}$ and $k$.

We can also use $r$-orthogonality to provide a criterion when the multiplicativity is preserved by a base change.

This general criterion can be made quite explicit in many practical situations. One such case was already established in Lemma 2.5.1. Here is a subtler case when defect can happen. In fact, the claim below is not covered by Lemma 2.5.1 precisely when $l/k$ has defect.

We can summarize this in the following example.

Consider the following valuative analogue of purely transcendental extensions in commutative algebra. A $k$-valuation $|\ |_{l}$ on $l=k(t)$ is called almost $k$-split if it is an infimum of translated Gauss valuations $|\ |_{a_{i},r_{i}}$, where $|\ |_{a,r}$ is defined by $|\sum_{j}c_{j}(t-a)^{j}|_{a,r}=\max_{j}r^{j}|c_{j}|$. If $k$ is algebraically closed, then $k(t)$ is automatically almost split. For example, if $d_{t/k}>0$, then this follows from the classification of points on Berkovich affine line over ${\widehat{k}}$ in [Ber90, §1.4]. Otherwise $l\subset{\widehat{k}}$, and the image $a\in{\widehat{k}}$ of $t$ defines a classical point $|\ |_{a,0}$ of ${\mathbb{A}}^{1}_{\widehat{k}}$, which is the semivaluation on ${\widehat{k}}[t]$ with kernel $(t-a)$, and of course $|\ |_{a,0}$ is also an infimum of Gauss valuations $|\ |_{a,r}$ dominating it. The restriction of $|\ |_{a,0}$ onto $k[t]$ is the valuation induced from $l$.

Next we study how spectral multiplicativity behaves in the case of algebraic ground field extensions.

A couple of words about the reasons which forced us to choose the oddly looking formulation of the lemma.

At this stage we can already prove the main result about geometric spectral multiplicativity. Which is very similar to the usual theory of geometric irreducibility, up to the nuance with the completion or henselization.

It remains to establish the result we have used above.