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I should point out that Joshi's paper does not falsify Remark 9 of our note. In Joshi's Theorem 4.8 (which he claims to falsify our Remark 9) the curve $X/E$ stays the same (and hence of course its te …

This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions". But I wanted to elaborate: Faltings regards the almost puri …

Let me start with the second question first: The usual de Rham complex is not locally acyclic in positive degrees, in any of the topologies (analytic (= of rational subsets), étale, pro-étale, ...). …

Good question! We've been trying to figure this out as we went along, but so far unsuccessfully. Some more precise points: For many (but definitely not all) applications to geometry over the real num …

For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified …

We finally have finished our paper, detailing the conjecture! We have also included an extensive introduction that I hope gives some impression of why one might hope for such a statement, and I'll sim …

Here's a rule of thumb: As long as the construction is nonarchimedean and does not involve noncompleted tensor products, it's solid. More precisely, anything you can build from discrete abelian groups …

When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually …

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 th …

Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the co …

I don't know much about this stuff, so instead of answering the question, I try to formulate more precise questions, in the hope someone else will take up these questions: One of the main reason to l …

Classically, the functor $f^!$ is indeed not a right adjoint in general. Clausen and I have recently found a way to make it a right adjoint in general, by enlarging the category of modules to that of …

Lol @"varying degrees of enthusiasm" ;-). And sorry for the late answer... Let me try to answer your questions. First, for any connected analytic adic space $X$, say, with a geometric point $\overline …

Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many …

Let $k$ be some field, algebraically closed and of characteristic $0$, if you like. Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of $U_ …