Subsection 2.3.3. (00MU) · ScalingStacks

2.3.3. Topological structures: the spectral norm

The Gauss norm on Tate algebra is equal to its spectral norm. For a general strict redueced affinoid algebra, the Banach algebra norm is equivalent to its spectral seminorm, thanks to the compatibility of Banach algebra norms with algebraic structures.

One studies the spectral norm of the Tate algebra case by direct calculation.

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Proposition 2.56. For any $f\in\mathcal{T}_{n}$ , there exists $z\in\mathrm{Max}(\mathcal{T}_{n})$ such that $\lvert f(z)\rvert_{z}=\vvvert f\vvvert_{\mathcal{T}_{n}}$ ([BGR, Proposition 5.1.4.3]). On $\mathcal{T}_{n}$ , the three norms are equal: $\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\text{sp}}=\vvvert\mathord{\cdot}\vvvert_{\mathcal{T}_{n},\mathrm{spM}}$ .

One then uses Noether normalization to investigate the spectral seminorm of general affinoid algebra.

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Proposition 2.57. Let $\mathcal{A}$ be a reduced strict affinoid algebra. Then its spectral norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{A},\text{sp}}$ is a complete norm on $\mathcal{A}$ . It is equivalent to the Banach algebra norm $\vvvert\mathord{\cdot}\vvvert_{\mathcal{A}}$ . ([FvdP, Theorem 3.4.9], [BGR, Theorem 6.2.4.1])

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Corollary 2.58. Let $\mathcal{A}$ be a reduced general affinoid algebra. Then there exists $C>0$ such that $\vvvert f\vvvert\leq C\vvvert f\vvvert_{\mathrm{sp}}$ for all $f\in\mathcal{A}$ . In particular, $\vvvert\mathord{\cdot}\vvvert_{\mathrm{sp}}$ is complete on $\mathcal{A}$ , and is equivalent to $\vvvert\mathord{\cdot}\vvvert$ . ([Ber, Proposition 2.1.4.ii])

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Remark 2.59. The constant $C$ here does not depend on $f\in\mathcal{A}$ , it is uniform.