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The work on analytic geometry is all joint with Dustin Clausen! Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#...


2

There exist examples of commutative local rings of finite dimension whose completions have infinite dimension (all dimensions are Krull, of course). Take $A$ to be a complete valuation ring, whose valuation of non-discrete but of rank 1 (i.e., the value group can be embedded in $\mathbb{R}$, but is not cyclic), and consider the ring $A[X]$. Since $A$ has ...


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