Subsection 2.3.1. (00MS) · ScalingStacks

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2.3.1. Basic constructions

Affinoid algebras are $k$ -Banach algebras that are quotient algebras of Tate algebras. Among them are strict affinoid algebras which have good finiteness properties such as Noetherianity. Some good properties pass to general affinoid algebra by a technique enlarging the base valued field which makes the affinoid algebra strict.

00IA

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

k\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}=\{f=\sum_{J\in\mathbb{N}^{n}}^{\infty}a_{J}\boldsymbol{T}^{J}:a_{J}\in k,\lvert a_{J}\rvert\boldsymbol{r}^{J}\to 0\text{ as }|J|\to\infty\}

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

\vvvert f\vvvert_{\mathcal{T}_{n}(\boldsymbol{r})}=\max_{J}\lvert a_{J}\rvert\boldsymbol{r}^{J}

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

00IB

Remark 2.39. This Gauss norm is obviously sub-multiplicative. It is in fact multiplicative by an argument as in the proof of Gauss Lemma.

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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.

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Remark 2.41. An affinoid algebra norm and the quotient algebra norm of Gauss algebra norm by the defining admissible surjective homomorphism are just equivalent but not necessarily equal.

One can construct new affinoid algebras out of old ones by various algebraic operations.

00IE

Example 2.42. The quotient Banach algebra of an affinoid algebra is an affinoid algebra.

00IF

Proposition 2.43. Let $\mathcal{C}$ be a $k$ -Banach algebra which is finite over an affinoid algebra $\mathcal{A}$ , then $\mathcal{C}$ itself is an affinoid algebra. If $\mathcal{A}$ is strict, then $\mathcal{C}$ is strict.

00IG

Proof. Let $\{c_{i}\}_{i\in\{1,\dots,m\}}\subset\mathcal{C}$ be a finite set of generators of $\mathcal{C}$ over $\mathcal{A}$ , then consider an $\mathcal{A}$ -Tate algebra $\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}$ where $r_{i}\geq\vvvert c_{i}\vvvert_{\mathcal{C}}$ . There is a surjective $k$ -algebra homomorphism defined by

\gamma:\mathcal{A}\{\boldsymbol{r}^{-1}\boldsymbol{T}\}\to\mathcal{C},\text{ }T_{i}\mapsto c_{i}

which is bounded as there exists $C>0$ such that

\vvvert\gamma(\sum_{J}a_{J}\boldsymbol{T}^{J})\vvvert_{\mathcal{C}}\leq\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\boldsymbol{c}^{J}\vvvert_{\mathcal{C}}\leq C\max_{J\in\mathbb{N}^{m}}\vvvert a_{J}\vvvert_{\mathcal{A}}\cdot\boldsymbol{r}^{J}

By Corollary 2.5, $\gamma$ is admissible, the norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{C}}$ is equivalent to the quotient norm of the $\mathcal{A}$ -Tate norm. Hence $\mathcal{C}$ is an affinoid algebra. The strictness is obtained by choosing $r_{i}\in\lvert k^{\times}\rvert$ (see Lemma 2.47). ∎

00IH

Proposition 2.44. Let $\mathcal{B}$ be a Banach $k$ -algebra. Suppose that there exists a finitely generated $k$ -algebra $A$ which is dense in $\mathcal{B}$ , then there exists an affinoid algebra $\mathcal{A}$ in which $A$ is a dense $k$ -sub-algebra and a homomorphism of Banach $k$ -algebras $\mathcal{A}\rightarrow\mathcal{B}$ which extends the identiy homomorphism on $A$ .

00II

Proof. Let $\{a_{i}\}_{i\in\{1,\dots,m\}}$ be a set of generators of $A$ . For each $i\in\{1,\dots,m\}$ , let $r_{i}$ denote $\vvvert a_{i}\vvvert_{\mathcal{B}}$ and let $\boldsymbol{r}$ denote the multi-radius consisting of $\{r_{i}\}_{i\in\{1,\dots,m\}}$ . Consider the Tate algebra $\mathcal{T}_{\boldsymbol{r}}$ and the homomorphism of $k$ -algebras

k[T_{1},\dots,T_{m}]\rightarrow\mathcal{B},\quad T_{i}\mapsto a_{i}

By the ultra-metricity of $\vvvert\mathord{\cdot}\vvvert_{\mathcal{B}}$ and the definition of $\boldsymbol{r}$ , one has

\forall n\in\mathbb{N},\forall J\in\mathbb{N}^{m},\forall f_{J}\in k,\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{B}}\leq\vvvert\sum_{J\in\mathbb{N}^{m}}f_{J}\cdot\boldsymbol{a}^{J}\vvvert_{\mathcal{T}_{\boldsymbol{r}}}

so by a density argument one can extend it to a homomorphism of Banach $k$ -algebras

\mathcal{T}_{\boldsymbol{r}}\rightarrow\mathcal{B},\quad T_{i}\mapsto f_{i}

Let $\mathscr{I}$ be the kernel ideal of this homomorphism. To conclude it suffices to take $\mathcal{A}$ as $\mathcal{T}_{\boldsymbol{r}}/\mathscr{I}$ . ∎

00IJ

Proposition 2.45. Let $(B,\vvvert\mathord{\cdot}\vvvert)$ be a normed algebra and let $\mathcal{B}$ be its separated completion. Let $A$ be a sub- $k$ -algebra of $B$ , equipped with the restriction algebra norm of $\vvvert\mathord{\cdot}\vvvert$ , and let $\mathcal{A}$ be the separated completion of $(A,\vvvert\mathord{\cdot}\vvvert)$ . Assume that $\mathcal{A}$ is an affinoid algebra. If $B$ is integral and is finite over $A$ , then $\mathcal{B}$ is Banach finite over $\mathcal{A}$ . Therefore $\mathcal{B}$ is an affinoid algebra.

00IK

Proof. By assumption, there exists $j\in\mathbb{N}$ and a homomorphism of $k$ -algebras and elements $\{e_{i}\}_{i\in\{1,\dots,j\}}\subseteq B$ such that

F:\bigoplus_{i\in\{1,\dots,j\}}A\rightarrow B,1_{i}\mapsto e_{i}

Moreover, $F$ is bounded

\vvvert\sum_{i\in\{1,\dots,j\}}a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\cdot e_{i}\vvvert\leq\max_{i\in\{1,\dots,j\}}\vvvert e_{i}\vvvert\cdot\max_{i\in\{1,\dots,j\}}\vvvert a_{i}\vvvert

So $F$ extends to a homomorphism of Banach $\mathcal{A}$ -modules

\mathcal{F}:\bigoplus_{i\in\{1,\dots,j\}}\mathcal{A}\rightarrow\mathcal{B},1_{i}\mapsto e_{i}

Let $\mathcal{B}^{-}$ be the image of $\mathcal{F}$ , it is a Banach finite $\mathcal{A}$ -module with the quotient norm $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ induced by $\mathcal{F}$ . As $\mathcal{B}^{-}$ is Banach finite over $\mathcal{A}$ , it is an affinoid algebra with an affinoid algebra spectral norm $\vvvert\mathord{\cdot}\vvvert^{-}$ , which is equivalent to $\lVert\mathord{\cdot}\rVert_{\mathcal{F}}$ . Now on $\mathcal{B}^{-}$ , $\vvvert\mathord{\cdot}\vvvert$ is bounded with respect to $\vvvert\mathord{\cdot}\vvvert^{-}$ by the continuity of $\mathcal{F}$ . To show the reverse, note that $\mathcal{B}^{-}$ is dense in $\mathcal{B}$ , so by Theorem 2.30 one has for any $b\in\mathcal{B}^{-}$

\vvvert b\vvvert^{-}=\max_{z\in\mathfrak{M}(\mathcal{B}^{-})}\lvert b(z)\rvert=\max_{z\in\mathfrak{M}(\mathcal{B})}\lvert b(z)\rvert=\vvvert b\vvvert_{\mathrm{sp}}\leq\vvvert b\vvvert

Therefore $\vvvert\mathord{\cdot}\vvvert$ and $\vvvert\mathord{\cdot}\vvvert^{-}$ are equivalent norms on $\mathcal{B}^{-}$ , so $\mathcal{B}^{-}$ is closed in $\mathcal{B}$ , hence coincides with it. ∎

To make an affinoid algebra strict, one can enlarge the base field.

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Lemma 2.46. Let $\boldsymbol{r}=(r_{1},\dots,r_{n})$ be a multi-radius such $\{\alpha(\log r_{i})\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$ -linearly independent. Then the $k$ -affinoid algebra

K_{\boldsymbol{r}}:=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{T}^{-1}\}=k\{\boldsymbol{r}^{-1}\boldsymbol{T},\boldsymbol{r}\boldsymbol{S}\}/(T_{1}S_{1}-1,\dots,T_{n}S_{n}-1)

is a field. ([Ber, Definition 2.1.1])

00IM

Lemma 2.47. Let $\mathcal{T}_{n}(\boldsymbol{r})$ be a $k$ -Tate alegbra. It is strict if and only if $r_{i}\in\sqrt{\lvert k^{\times}\rvert}$ for all $i$ . ([Ber, Corollary 2.1.6])

00IN

Corollary 2.48. Let $\mathcal{T}_{n}(\underline{r})$ be a $k$ -Tate alegbra. Let $I\subseteq\{1,\dots,n\}$ be a subset of indices such that $\{\alpha(\log r_{i})\}_{i\in I}$ are $\mathbb{Q}$ -linearly independent and $|I|$ is maximal for this independence property. Let $\boldsymbol{r}_{I}=(r_{i_{1}},\dots,r_{i_{1}})$ , then $K_{\boldsymbol{r}_{I}}\widehat{\otimes}_{k}\mathcal{T}_{n}(\boldsymbol{r})$ is a strict $K_{\boldsymbol{r}_{I}}$ -Tate algebra.

00IP

Corollary 2.49. For any $k$ -affinoid algebra $\mathcal{A}$ , there exists a multi-radius $\boldsymbol{r}_{I}=(r_{i})_{i\in I}$ such that $\{\alpha(\log r_{i})\}_{i\in I}$ are $\mathbb{Q}$ -linearly independent and $K_{\boldsymbol{r}_{I}}\widehat{\otimes}_{k}\mathcal{A}$ is a $K_{\boldsymbol{r}_{I}}$ -strict affinoid algebra. ([Ber, Proposition 2.1.2])