2.3.2. Algebraic structures: Noetherianity [00MT]

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2.3.2. Algebraic structures: Noetherianity

Let $\mathcal{A}$ be a Banach $k$ -algebra, one denotes by $\mathcal{A}^{\circ}$ the $k^{\circ}$ -algebra $\{f\in\mathcal{A}\text{ }|\text{ }\vvvert f\vvvert_{\mathcal{A},\text{sp}}\leq 1\}$ , and by $\mathcal{A}^{\circ\circ}$ the ideal of $\mathcal{A}^{\circ}$ constituting of elements $\vvvert f\vvvert_{\mathcal{A},\text{sp}}<1$ . The $\widetilde{k}$ -algebra $\mathcal{A}^{\circ}/\mathcal{A}^{\circ\circ}$ is called the reduction of $\mathcal{A}$ . It can be shown that $\widetilde{\mathcal{T}_{n}}$ is isomorphic to $\widetilde{k}[T_{1},\dots,T_{n}]$ . ([BGR, Proposition 5.1.2.2])

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 $c_{i}\in\bar{k}[z_{1},\dots,z_{n-1}]$ .

Proposition 2.51.

[Weierstrass division] Let $\mathcal{T}_{n}$ be the $k$ -Tate algebra of multiradius $\underline{r}=\underline{1}$ , then

(1)

Let $f\in\mathcal{T}_{n}$ be an distinguished element in $z_{n}$ of degree $d$ , and $g\in\mathcal{T}_{n}$ be any element. Then there exist unique $r\in\mathcal{T}_{n-1}[z_{n}]$ of degree less than $d$ in $z_{n}$ and $q\in\mathcal{T}_{n}$ such that $g=q\cdot f+r$ . Moreover $\vvvert g\vvvert_{\mathcal{T}_{n}}=\max\{\vvvert q\vvvert_{\mathcal{T}_{n}},\vvvert r\vvvert_{\mathcal{T}_{n}}\}$
(2)

Let $f\in\mathcal{T}_{n}$ with $\vvvert f\vvvert_{\mathcal{T}_{n}}=1$ . Then there exists a $k$ -algebra automorphism $\tau$ of $\mathcal{T}_{n}$ such that $\tau(f)$ is regular in $z_{n}$ .

([BGR, Theorem 5.2.1.2], [FvdP, Theorem 3.1.1])

Proposition 2.52.

The Tate algebra $\mathcal{T}_{n}$ is Noetherian. All of its ideals are closed. ([BGR, Theorem 5.2.6.1, Corollary 5.2.7.2], [FvdP, Theorem 3.2.1])

Corollary 2.53.

Any strict affinoid algebra is Noetherian. All of its ideals are closed ([BGR, Proposition 6.1.1.3], [FvdP, Theorem 3.2.1]). Any affinoid algebra is Noetherian. All of its ideals are closed ([Ber, Propositon 2.1.3]).

Proposition 2.54.

[Noether normalization] For strict affinoid algebra $\mathcal{A}$ , there exists an injective finite and admissible Banach algebra homomorphism $\mathcal{T}_{d}\to\mathcal{A}$ for some $d>0$ . Moreover, $d$ equals the Krull dimension of $\mathcal{A}$ . ([BGR, Theorem 6.1.2.1], [FvdP, Theorem 3.2.1])

Corollary 2.55.

Let $\mathfrak{m}$ be a maximal ideal of strict affinoid algebra $\mathcal{A}$ , then $\mathcal{A}/\mathfrak{m}$ is a finite extension of $k$ .