Uniform sheafy Tate rings that are not stably uniform
Buzzard–Verberkmoes and Mihara show that a stably uniform Tate Huber ring is sheafy. We construct examples of uniform and strongly sheafy Tate rings which are not stably uniform, answering Question 7 of Kedlaya’s Nonarchimedean Scottish Book. In the first example a rational localisation is non-reduced, whilst in the second the rational localisation is an integral domain but is not uniform. The two main results are due to ChatGPT 5.6 Sol. Both constructions, the rational localisations witnessing failure of stable uniformity, and their strong sheafiness have been formalised in Lean 4
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