Tag Archives: p-adic geometry

Superfectoid spaces!?

Note (6/21): There is an issue with the proof of Lemma 1.7 below, but I’m pretty confident Kiran and I will find a fix.  In any case, you should never take blog math too seriously! In this post I want to announce some … Continue reading

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Period morphisms in p-adic Hodge theory

In classical Hodge theory, variations of Hodge structure give rise to varying periods, and then to period morphisms.  In p-adic Hodge theory, there are plenty of periods but (until recently) not so many period morphisms.  It turns out these exist … Continue reading

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Thoughts from Oberwolfach, Part I

Just back from a week-long workshop at Oberwolfach on “Nonarchimedean geometry and applications”. Some random comments: During the workshop, Kiran Kedlaya started a nonarchimedean Scottish book, to record open problems in the foundations of nonarchimedean geometry, especially in the theory … Continue reading

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It’s good to be spectral

Here’s a nice little theorem in rigid analytic geometry. Theorem. Let be an affinoid rigid space, with a closed analytic subset and an admissible open subset containing .  Then for some , the “tube” around is contained in . To quote … Continue reading

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Perfectoid universal covers of curves

In a previous post I mumbled a bit about perfectoid covering spaces of rigid analytic varieties.  In this post, I want to sketch a fun special case. Fix an algebraically closed nonarchimedean field. Recall the following from Scholze-Weinstein: given a … Continue reading

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The product of two diamonds

Let be the category of perfectoid spaces in characteristic , equipped with its pro-étale topology. Any gives rise to a sheaf on . By definition, a diamond is a sheaf on which admits a relatively representable surjection which pulls back … Continue reading

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Perfectoid uniformization, part one (ed.: and part two)

Let be a complete algebraically closed extension of , and let be a nice rigid analytic space/locally Noetherian adic space over .  What should it mean for to be covered by a perfectoid space?  Here’s one possible definition: Definition. A perfectoid covering space  of is … Continue reading

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