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Thinking about passing from functions to algebra can partially explain where the $A^+$ comes from. In practice, we impose bounds on functions. DoTo prove theorems, we replace these bounds with bounds on the smallest possible ringssmallest possible rings containing those functionslargest ring containing thosethe functions and for which the bound imposed automatically extends. That ring turns out to be one of these open and integrally closed subrings $A^+$. (See Section 10.3 of Conrad's notes in his perfectoid seminar.) But, it has no reason to be $A^{\circ}$ itself. For instance, in $\mathrm{Cont}(\mathbb{C}_p\langle X\rangle)$ we do have the automatic bound $|n|\leq 1$ for all $n \in \mathbb{Z}$ and the corresponding smallest ring $A^+$ will be something smaller than $A^{\circ} = \mathcal{O}_{\mathbb{C}_p}\langle X \rangle$. (See Example 11.3.14 in the 11th lecture of Conrad's perfectoid seminar.)
Thinking about passing from functions to algebra can partially explain where the $A^+$ comes from. In practice, we impose bounds on functions. Do prove theorems, we replace these bounds with bounds on the smallest possible rings containing those functions. That ring turns out to be one of these open and integrally closed subrings $A^+$. (See Section 10.3 of Conrad's notes in his perfectoid seminar.) But, it has no reason to be $A^{\circ}$ itself. For instance, in $\mathrm{Cont}(\mathbb{C}_p\langle X\rangle)$ we do have the automatic bound $|n|\leq 1$ for all $n \in \mathbb{Z}$ and the corresponding smallest ring $A^+$ will be something smaller than $A^{\circ} = \mathcal{O}_{\mathbb{C}_p}\langle X \rangle$. (See Example 11.3.14 in the 11th lecture of Conrad's perfectoid seminar.)
Thinking about passing from functions to algebra can partially explain where the $A^+$ comes from. In practice, we impose bounds on functions. To prove theorems, we replace these bounds with bounds on the smallest possible rings containing those functionslargest ring containing the functions and for which the bound imposed automatically extends. That ring turns out to be one of these open and integrally closed subrings $A^+$. (See Section 10.3 of Conrad's notes in his perfectoid seminar.) But, it has no reason to be $A^{\circ}$ itself. For instance, in $\mathrm{Cont}(\mathbb{C}_p\langle X\rangle)$ we do have the automatic bound $|n|\leq 1$ for all $n \in \mathbb{Z}$ and the corresponding smallest ring $A^+$ will be something smaller than $A^{\circ} = \mathcal{O}_{\mathbb{C}_p}\langle X \rangle$. (See Example 11.3.14 in the 11th lecture of Conrad's perfectoid seminar.)
