Together with some help from my friend, I translated FAC into English. I didn't have so much time to proofread it, so probably there are some mistakes. It can be found here: FAC, Source.

Let $a_n$ be an increasing sequence of positive integers which grows really fast, say $a_{n+1} > \exp(a_n)$. Take $A = \{10^{-1}, \sum 10^{-a_n}\}$. Then $d_A(a_n) \leq 2\cdot 10^{- a_{n+1}} \leq 2\...

Here is an example mentioned in passing by user ali's answer, but I think it is cute (and powerful) enough to be worth fleshing out the details. Lifting from characteristic $p$ to characteristic zero ...

You might be rediscovering Zariski's Main Theorem, which implies your statement in case $X$ is normal (or just weakly normal) and the projection from the graph $\Gamma$ to $X$ is proper and separable. ...

Quick answer. I. See M. Olsson On Faltings’ method of almost etale extensions, chapter 5. He discusses there a version of this fact over a dvr, but I think you can easily extract what you want, if ...

I don't think you can draw something meaningful - I would be surprised if someone made a good drawing of the Frobenius morphism ;). That being said, here is an example (possibly misleading or ...

I'm not an expert, but here is how I would plan my trip: There are obviously two parts: rationality + functional equation + comparison with Betti numbers (which follow from the construction of etale ...

Even if you're only interested in say cohomology with coefficients in the constant sheaf, working with constructible sheaves gives you extra flexibility and is more amenable to inductive proofs. Here ...

I think the usual interpretation is this (see S. Landsburg's comment): The classical proof that $\mathbb{P}^n$ is proper uses elimination theory: we need to prove that for a ring $R$, $\mathbb{P}^n_R\...

Here is an example from Bhargav Bhatt's talk "Using DAG" at MSRI last week. Needless to say, any mistakes are mine. Theorem. Let $X$ be a coherent (quasi-compact and quasi-separated) scheme, let $A$ ...

I know nothing about $\mathbf{A}^1$-homotopy. I will describe how one can get a functor from smooth varieties over $\mathbf{C}((t))$ to ${\rm Top}_{/\mathbf{S}^1}$ using log smooth proper models and ...

My proposal: a usable tool for working with line bundles on toric varieties and their cohomology. There are some tools (Polymake, Latte) for working with polyhedra, but I haven't seen a library ...

No. The picture over a general base is this: let $A_0\to S_0$ be an ordinary abelian variety over a characteristic $p$ scheme $S_0$, and let $S_n$ ($n\geq 0$) be compatible flat liftings of $S_0$ over ...

I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth varieties over ...

Completing the discussion under Will Sawin's answer. The question has been answered completely and affirmatively by Ben-Bassat and Temkin in their paper "Berkovich spaces and tubular descent" (Adv. ...

TL;DR: Your expectation is right. In fact, there is a third object to compare with $R[1/p]$ and $\hat R[1/p]$, the affinoid rigid space ${\rm Spf}(\hat R)^{\rm rig}$. The cohomology comparison is ...

The answer is yes (in char. 0). Indeed, it suffices to show that for an infinitesimal deformation $\mathcal{X}$ of $X$ over an artinian algebra $A$, the cohomology $H^0(\mathcal{X}, \omega_{\mathcal{X}...

Yes, if $X$ is proper (by GAGA). Otherwise, this is false for covers of degree divisible by $p$. See Example 51 in Ducros's survey "Étale Cohomology of Schemes and Analytic Spaces", though ...

It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such a push-forward is a direct sum of line ...

I believe these are called 'hypersimplices'. See 1) Gelfand, Manin "Methods of homological algebra", Ex. IV.2 1(c), p. 260. 2) Belinson, Bernstein, Deligne "Faisceaux pervers", Remarque 1.1.14, p. ...

Take a singular quadric cone $C$ , then a ruling $L$ of the cone is not a Cartier divisor, hence $\mathcal{O}_C(L)$ is not locally free, but it becomes locally free after removing the vertex of the ...

The answer to the first question is no. Let $k=\mathbb{R}$ be the real numbers. Then every cubic in $\mathbf{P}^2_k$ has a rational point. Take the genus $1$ curve $C$ in $\mathbf{P}^3_k$ obtained by ...

(1) Can a condition not involving the existential quantifier and equivalent to good reduction be given for general smooth proper $K$-schemes? Not in terms of their "topology" in general. For abelian ...

Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, g\colon X\to B$ sending a base point $x\in X$ to $0\in B$. Every such map ...

Such a moduli problem for stacks is expected to be a $2$-stack. For example, consider the stack of line bundles on $X$, whose objects are parameterized by $H^1(X, \mathbb{G}_m)$. This is a (trivial) ...

The answer is no even for $G=\mathbb{Z}/2$ acting on $R=k[x,y]$ by swapping $x$ with $−x$ and $y$ with $−y$. In this case $R$ is finite, but not flat, over $R^G=k[x^2,xy,y^2]$, for example because the ...

Let $X$ be a variety (a separated integral scheme) with function field $K = k(X)$, maybe assumed normal. Let $L$ be a finite separable extension of $K$. From this data, we can construct a variety $Y$ ...

Yes. This follows from the Białynicki-Birula decomposition (see Theorem 4.4 in the original paper).

Here are some fairly recent and general references I like: The Handbook of Moduli http://intlpress.com/site/pub/pages/books/items/00000399/index.html The article on logarithmic geometry by ...

I'm thinking about the following argument. It's not entirely precise, I will be grateful for comments how to improve it. Suppose such an $X$ is toric. Because the exceptional curves $E_i$ do not move ...