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nLab > Latest Changes: perfectoid space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2014

    added some minimum content to perfectoid space. Also added a corresponding paragraph to function field analogy.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 5th 2017

    Since my chapter in What is a mathematical concept? appears next to Michael Harris’s ’The perfectoid concept: test case for an absent theory’, when I get a chance to read that it would fun to see what sense I can make of the idea.

    A throw-away thought: if, as Scholze says, perectoids were devised to mediate between ℚ p\mathbb{Q}_p and 𝔽 p((t))\mathbb{F}_p((t)), and

    Although these two fields have formally ’the same’ elements, the basic addition and multiplication operations are different: In ℚ p\mathbb{Q}_p, one computes with carry, but in 𝔽 p((t))\mathbb{F}_p((t)) without carry,

    is it interesting that carrying is a form of cohomology?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 5th 2017
    • CommentRowNumber4.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 10th 2022

    Added explicit descriptions of the fields in the Fontaine-Wintenberger theorem.

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 10th 2022

    Added definition of a perfectoid ring.

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 10th 2022

    Added the word “tilt”.

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 10th 2022

    Added definition of perfectoid field and perfectoid space.

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2022
    • (edited Jul 10th 2022)

    If you could add a sentence to the Idea-section on the actual Idea behind perfectoid spaces, that might be most useful.

    By the way, for better or worse the rendering of := comes out with the wrong amount of whitespace on the nLab. Best to always use \coloneqq, instead (I have made the change on this page now).

    diff, v15, current

    • CommentRowNumber9.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 23rd 2022

    Added section “generalizations”, mentioning diamonds and v-sheaves.

    diff, v16, current

    • CommentRowNumber10.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 23rd 2022

    Added applications section.

    diff, v16, current

    • CommentRowNumber11.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 29th 2022

    Made the Weinstein reference more pinpoint by specifying “section 5”. Currently planning to add a section on properties and/or a (sub) section on tilting.

    diff, v17, current

    • CommentRowNumber12.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 29th 2022

    Added tilting.

    diff, v18, current

    • CommentRowNumber13.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 30th 2022

    Added reference to Ana Caraiani’s talk on her work in progress with James Newton proving modularity for elliptic curves over quadratic imaginary fields.

    diff, v19, current

    • CommentRowNumber14.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    Made the definitions and propositions appearing in the right way. I don’t really understand this bug which happens often. It seems related to line breaks.

    I replaced:

    “\begin{definition}

    ABC DEF

    \end{definition}”

    by

    “\begin{definition} ABC DEF \end{definition}”

    (with line breaks but without the two empty lines at the beggining and at the end)

    and now it works.

    diff, v20, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2022

    Maybe I don’t actually know that version of the bug. Could you add a minimal demonstration of the issue to the Sandbox? Thanks!

    (What I am aware of is that these environments misbehave when they end right on a list item, without a blank line after the list. But what you mention seems different, in fact opposite.)

    • CommentRowNumber16.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    The bug was present in perfectoid space (Rev #19).

    I put the exact source code of this page in the Sandbox and the bug no longer appears, so it seems that it’s necessary to go deeper to understand what’s going on. It looks ununderstandable at this level.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    Okay, so I can’t see what you are alluding to. Maybe when you see the bug next time, you could try to copy into the Sandbox (not the entire entry but) just the lines that exhibit the bug. That would allow us to inspect the issue.

    • CommentRowNumber18.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    What I say is that in perfectoid space (Rev #19) there is the bug (you see \begin{definition} instead of the formatting for a definition) but when I copy the whole page to the sandbox there is no longer the bug.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    Ah, that’s just (“just”) because the page history works with an old version of the rendering-engine, while the LaTeX-style environments are only supported by the “new” engine which renders the main pages.

    That’s a bug, in a way (though, strangely enough, it was introduced and kept fully deliberately), but as things stand this counts as normal behaviour for the nLab — until the technical team amasses enough karma to tackle the Big Rewrite of the nlab software.

    But in #14 it sounded like you saw a bug in a live page?! Maybe you didn’t. If you ever spot a bug in a in a live page, you’ll do me a favor by dropping a minimal example into the Sandbox, so that we can inspect it and maybe fix it.

    • CommentRowNumber20.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 1st 2022
    • (edited Dec 1st 2022)

    Yes, the bug was on the live page exactly as in the history. But I don’t succeed to reproduce it in the sandbox! The sandbox doesn’t reproduce the bug when I put the exact same code.

    • CommentRowNumber21.
    • CommentAuthorJ-B Vienney
    • CommentTimeDec 1st 2022

    Oh probably I went on v19 instead of current, I understand, you should be right. It does the same when I go on v20.

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