ScalingStacks

Definition 2.1. Let $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be seminorms on $V$. We say that $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent if there exist two constants $C_{1}>0$ and $C_{2}>0$ such that $C_{1}\lVert\mathord{\cdot}\rVert_{1}\leq\lVert\mathord{\cdot}\rVert_{2}\leq C_{2}\lVert\mathord{\cdot}\rVert_{1}$. Note that this condition holds if and only if the seminorms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ induce the same topology on the vector space $V$ ([Bou, Corollaire I.3.3.1])(note that the absolute value $\lvert\mathord{\cdot}\rvert$ is supposed to be non-trivial).

Definition 2.2. Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$. If $W$ is a vector subspace of $V$, then map $(x\in W)\mapsto\lVert x\rVert$ defines a seminorm on $W$, called the restriction of $\lVert\mathord{\cdot}\rVert$ on $W$. If $Q$ is a quotient vector space of $V$ and $\pi:V\rightarrow Q$ is the quotient map, then the map $(q\in Q)\mapsto\inf_{x\in\pi^{-1}(\{q\})}\lVert x\rVert$ defines a seminorm on $Q$, called the quotient of $\lVert\mathord{\cdot}\rVert$ on $Q$.

Definition 2.3. Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ and $(W,\lVert\mathord{\cdot}\rVert_{W})$ be seminormed vector spaces over $k$, and $f:V\rightarrow W$ be a $k$-linear map. We say that $f$ is bounded if there exists a constant $C>0$ such that $\lVert f(x)\rVert_{W}\leq C\lVert x\rVert_{V}$ for any $x\in V$. Note that this condition holds if and only if $f$ is continuous with respect to the topologies on $V$ and $W$ induced by the seminorms $\lVert\mathord{\cdot}\rVert_{V}$ and $\lVert\mathord{\cdot}\rVert_{W}$ respectively. We say that $f$ is admissible if it is bounded and if on the image of $f$, the quotient seminorm of $\lVert\mathord{\cdot}\rVert_{V}$ and the restriction of $\lVert\mathord{\cdot}\rVert_{W}$ are equivalent.

Definition 2.7. Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space. The dual norm of $\lVert\mathord{\cdot}\rVert^{\vee}$ on the dual vector space $V^{\vee}$ is defined by

Definition 2.9. Let $(V,\lVert\mathord{\cdot}\rVert)$ be a normed vector space. Let $(k^{\prime},\lvert\mathord{\cdot}\rvert^{\prime})$ be a complete valued field extension of $(k,\lvert\mathord{\cdot}\rvert)$. Set $V_{k^{\prime}}$ to be $V\otimes_{k}k^{\prime}$, which can be identified with $\mathrm{Hom}_{k}(\mathrm{Hom}_{k}(V,k),k^{\prime})$. The norm

defined via this identification is called the scalar extension of $\lVert\mathord{\cdot}\rVert$.

Definition 2.12. Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space over $k$. A basis $\{e_{i}\}_{i\in\{1,\dots,n\}}$ of $V$ is called orthogonal (with respect to $\lVert\mathord{\cdot}\rVert$) if

Moreover, it is said to be orthonormal if in addition $\lVert e_{i}\rVert=1$ for all $i\in\{1,\dots,n\}$.

Definition 2.16. Let $A$ be a $k$-algebra (the unit of which is denoted by $\mathbf{1}$) and $\lVert\mathord{\cdot}\rVert$ be a seminorm on $A$ (viewed as a vector space over $k$).

1. (1)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is said to be sub-multiplicative if for any $(a,b)\in A\times A$ one has $\lVert ab\rVert\leq\lVert a\rVert\cdot\lVert b\rVert$.

2. (2)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is called power-multiplicative if $\lVert a^{n}\rVert=\lVert a\rVert^{n}$ for any $a\in A$ and any $n\in\mathbb{N}\setminus\{0\}$.

3. (3)

   The seminorm $\lVert\mathord{\cdot}\rVert$ is called multiplicative if $\lVert ab\rVert=\lVert a\rVert\cdot\lVert b\rVert$ for any $(a,b)\in A^{2}$.

A $k$-algebra seminorm (resp. $k$-algebra norm) on $A$ is defined to be a sub-multiplicative seminorm (resp. sub-multiplicative norm) $\lVert\mathord{\cdot}\rVert$ on $A$ such that $\lVert\mathbf{1}\rVert=1$. We denote by $\vvvert\mathord{\cdot}\vvvert$ an algebra seminorm. Any $k$-algebra equipped with a complete $k$-algebra norm is called a Banach $k$-algebra.

We use calligraphic letters to denote Banach algebras and Banach modules (defined below) and use the corresponding capital letters to denote the underlying $k$-algebra or the underlying module of a $k$-algebra. For example, a Banach $k$-algebra $(A,\vvvert\mathord{\cdot}\vvvert)$ is denoted by $\mathcal{A}$. If $A^{\prime}$ is a sub-$k$-algebra of $A$, then the restriction of $\vvvert\mathord{\cdot}\vvvert$ on $A^{\prime}$ is a $k$-algebra norm. If this norm is complete, we say that $\mathcal{A}^{\prime}$ ($A^{\prime}$ equipped with the restricted norm) is a Banach $k$-sub-algebra of $\mathcal{A}$. Similarly, if $Q$ is a quotient $k$-algebra of $A$, then the quotient of the norm $\vvvert\mathord{\cdot}\vvvert$ on $Q$ is a sub-multiplicative seminorm. If it is a complete norm, we say that $\mathcal{Q}$ ($Q$ equipped with the quotient norm) is a Banach quotient $k$-algebra of $\mathcal{A}$.

Definition 2.18. Let $\mathcal{A}_{1},\mathcal{A}_{2}$ be two Banach $k$-algebras, and $\phi:A_{1}\to A_{2}$ be a homomorphism of $k$-algebras. We say that $\phi$ is a homomorphism of Banach $k$-algebras if it is bounded as a $k$-linear map. A homomorphism of Banach $k$-algebra $\phi$ is often denoted by $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$. A homomorphism of Banach $k$-algebra $\phi$ is called an isomorphism of Banach $k$-algebras if there exists a homomorphism of Banach $k$-algebras $\psi:\mathcal{A}_{2}\to\mathcal{A}_{1}$ such that $\phi\circ\psi=\mathrm{Id}_{\mathcal{A}_{2}}$ and $\psi\circ\phi=\mathrm{Id}_{\mathcal{A}_{1}}$.

Definition 2.19. Let $\mathcal{A}$ be a $k$-Banach algebra. The Berkovich spectrum $\mathfrak{M}(\mathcal{A})$ is the following topological space: the points, denoted by $z$, are bounded multiplicative $k$-algebra seminorms $\vvvert\mathord{\cdot}\vvvert_{z}$ on $\mathcal{A}$, and the topology is the weakest topology on this set of points, for which all $\mathbb{R}_{\geq 0}$-valued functions of the form $z\mapsto\vvvert f\vvvert_{z}$ are continuous for any $f\in A$. This topology is called the canonical topology. For any subset $V$ of $\mathfrak{M}(\mathcal{A})$, we denote by $\text{Int}^{\mathrm{top}}(V)$ the topological interior of $V$. This topological interior is to be compared with the notion of interior of an affinoid subdomain in an affinoid domain (see [Ber, Definition 2.5.7]), which we do not use in this article.

Definition 2.22. Let $\mathcal{A}$ be a Banach $k$-algebra. A character $\chi$ of $\mathcal{A}$ is a homomorphism of Banach $k$-algebra from $\mathcal{A}$ to some complete valued field extension $(K,\lvert\mathord{\cdot}\rvert_{K})$ of $(k,\lvert\mathord{\cdot}\rvert_{k})$. Two characters $\chi_{1}:\mathcal{A}\to(K_{1},\lvert\mathord{\cdot}\rvert_{K_{1}})$ and $\chi_{2}:\mathcal{A}\to(K_{2},\lvert\mathord{\cdot}\rvert_{K_{2}})$ are said to be equivalent if there exist a character $\chi:\mathcal{A}\to(K,\lvert\mathord{\cdot}\rvert_{K})$ and valued field extensions $\iota_{1}:K\to K_{1}$ and $\iota_{2}:K\to K_{2}$ which preserve norms such that $\chi=i_{1}\circ\chi_{1}=i_{2}\circ\chi_{2}$. Let $[\chi]$ be the equivalence class of $\chi$.

Definition 2.24. The Gelfand transform of $\mathcal{A}$ is the homomorphism of Banach $k$-algebras

Definition 2.28. The spectral algebra seminorm $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ of an algebra seminorm $\vvvert\mathord{\cdot}\vvvert$ on a $k$-algebra $A$ is the one defined by

Note that the triangle inequality for $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ follows from sub-multiplicativity of $\vvvert\mathord{\cdot}\vvvert$. In general, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is only a seminorm even if $\vvvert\mathord{\cdot}\vvvert$ is a norm.

Definition 2.31. Let $\mathcal{A}$ be a Banach $k$-algebra. The radical of $\mathcal{A}$ is the null-space of its spectral seminorm $\mathfrak{n}(\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}})$. A Banach $k$-algebra with radical equal to $\{0\}$ is called semi-simple. Elements in the radical are said to be quasi-nilpotent (or topological nilpotent).

Definition 2.34. Let $\mathcal{M}$ be a Banach $\mathcal{A}$-module. It is called a Banach finite $\mathcal{A}$-module if there exists $l\in\mathbb{N}_{+}$ and a surjective homomorphism of Banach $\mathcal{A}$-modules $\mathcal{A}^{\oplus l}\to\mathcal{M}$ where $\mathcal{A}^{\oplus l}$ is the Banach $\mathcal{A}$-module corresponding to the $A$-module $A^{\oplus l}$ equipped with the norm $(a_{1},\dots,a_{l})\mapsto\max\vvvert a_{i}\vvvert$. (Note that such a homomorphism is necessarily admissible.)

Definition 2.36. Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism between Banach $k$-algebras. It is called Banach finite if $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$-module. In this case $\mathcal{A}_{2}$ is called a Banach finite $\mathcal{A}_{1}$-algebra.

Definition 2.38. For a multi-radius $\boldsymbol{r}=(r_{1},\dots,r_{n})\in\mathbb{R}^{n}$, the algebra

is called the Tate algebra over $k$ with multi-radius $\boldsymbol{r}$. Denote it by $\mathcal{T}_{n}(\boldsymbol{r})$. It is a $k$-Banach algebra with respect to the Gauss norm of multi-radius $\boldsymbol{r}$ defined by

One can define Tate algebra over other complete ultra-metric valued fields.

Definition 2.40. A $k$-Banach algebra $\mathcal{A}$ is called an affinoid algebra if there exists an admissible surjective homomorphism from some Tate algebra $k\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ to $\mathcal{A}$. The Banach algebra norm on an affinoid algebra $\mathcal{A}$ is called an affinoid algebra norm. If one can take $\boldsymbol{r}$ with $r_{i}=1$ for all $i\in\{1,\dots,n\}$, then $\mathcal{A}$ is called a strict affinoid algebra. One may define affinoid algebra similarly over other complete ultrametric valued field.

Definition 2.50. An element $f\in\mathcal{T}_{n}$ with $\vvvert f\vvvert_{\mathcal{T}_{n}}=1$ is said to be regular in $z_{n}$ of degree $d$ if its reduction $\bar{f}=\lambda(z_{n})^{d}+\sum_{0\leq i\leq d-1}c_{i}(z_{n})^{d-i}$ in $\bar{\mathcal{T}_{n}}$ where $\lambda\in k^{\times}$ and $c_{i}\in\bar{k}[z_{1},\dots,z_{n-1}]$.