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A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.

See for instance Defn 2.6 here.

What is the (geometric) significance of requiring $R^+ $ to be integrally closed?

On a related note, what is the difference (geometrically) between total integral closure and integrally closed. It is surprisingly hard to find anything about total integral closures...

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The adic spectrum of $(R,R^+)$ is a set of continuous valuations on $R$ having norm $\leq 1$ on $R^+$. It is clear from the definition and basic properties of integral closures that relaxing the integral closedness of $R^+$ buys you nothing new: a valuation on $R$ has norm $\leq 1$ on a subring $R_0 \subset R$ if and only if it has norm $\leq 1$ on the integral closure $R^+$ of $R_0$ in $R$. I believe this is the main reason one often assumes $R^+$ is integrally closed.

The difference between total integral closures and integral closures, roughly, corresponds to the difference between rank $1$ valuations and all valuations. But it might be better to ask a separate question about this as it is pure commutative algebra.

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