Tag Archives: Scholze

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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A remark on sheafy Huber rings

Let be a (complete) Huber ring, i.e. a complete topological ring containing an open subring which is adic with respect to a finitely generated ideal of definition . A Huber ring is Tate if contains a topologically nilpotent unit ; … Continue reading

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Diamonds in the rough

This post is an outgrowth of my (ongoing) attempt to understand Peter Scholze’s remarkable course at Berkeley.  The first couple of paragraphs below are a rather comically compressed summary of some portions of Jared Weinstein’s excellent notes (though of course all mistakes are my … Continue reading

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Hodge-Tate proliferation

Let be a complete algebraically closed extension, and let be an abelian variety, with p-adic Tate module and dual abelian variety . There are then at least three natural candidates for a “Hodge-Tate map” To wit: 1. (Scholze) For any … Continue reading

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Some little remarks on big Hecke algebras

Edit (7/22): The second paragraph in the proof of the proposition below is total nonsense.  I’ll put up a second version of this post soon, but I’m leaving the original as a warning to myself to not write blog posts when … Continue reading

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