# Krull dimension of completions in non-noetherian setting (especially completed perfections)

What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology?

An example of the sort of "nice" topological ring I'm looking for is a perfection of an affinoid algebra over a perfectoid (probably just complete non-archimedean with a non-discrete valuation is good enough) field $$K$$ in characteristic $$p > 0$$.

So let $$R$$ be a quotient of a Tate algebra $$K\langle T_1, \ldots, T_n \rangle$$ of power series which converge on the adic closed (i.e. the coefficients tend to $$0$$). The topology is induced from the non-archimedean valuation topology on $$K$$ and is given by the Gauss norm $$\|\sum_\alpha a_\alpha T^\alpha\| = \sup_\alpha |a_\alpha|$$. So in particular $$R$$ is a noetherian topological ring which is "Tate" (meaning its topology is defined by a topologically nilpotent unit $$\varpi$$, and there's an open subring on which the topology is $$\varpi$$-adic).

We can form its perfection $$R^{\mathrm{perf}} = \varinjlim_m R^{1/p^m}$$, which is a quotient of the perfection of the Tate algebra $$\cup_m K \langle T_1^{1/p^m}, \ldots, T_n^{1/p^m} \rangle$$. The perfection comes equipped with the direct limit topology, also compatible with the Gauss norm and "induced from the topology on $$K$$" in an appropriate sense. The completion of $$R^{\mathrm{perf}}$$ with respect to this topology is what is sometimes called the "completed perfection" of $$R$$.

I'd like to show that the Krull dimension of this ring agrees with that of $$R^{\mathrm{perf}}$$, and therefore with $$R$$.