Piotr Achinger
  • Member for 11 years, 11 months
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Serre's FAC in English
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82 votes

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.

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Massive cancellations
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34 votes

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\...

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What is the cotangent complex good for?
31 votes

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 ...

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Is there an algebraic geometry analogue of the closed graph theorem?
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26 votes

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. ...

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A theorem of M. Artin
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23 votes

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 ...

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Intuitive pictures in characteristic p
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22 votes

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 ...

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Learning path for the proof of the Weil Conjectures
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21 votes

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 ...

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How to motivate constructible sheaves
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18 votes

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 ...

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Why A. Weil considered elimination theory to be eliminated?
16 votes

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\...

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Results relying on higher derived algebraic geometry
15 votes

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 $A$ ...

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Is there a geometric realization of $\mathbf{C}((t))$-varieties?
15 votes

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 ...

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What algorithm in algebraic geometry should I work on implementing?
15 votes

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 ...

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Canonical lift of the deformation of an ordinary abelian variety
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14 votes

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 ...

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The main idea in the proof of Artin's vanishing
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14 votes

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 ...

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Beauville-Laszlo for schemes
13 votes

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. ...

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Etale cohomology of localizations of henselian rings
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13 votes

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 ...

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Deformations of Calabi-Yau manifolds
12 votes

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}...

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Étale fundamental group of rigid analytification
11 votes

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 ...

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Pushforward of line bundle under "toric isogeny"
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11 votes

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 ...

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Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
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11 votes

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. ...

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Vector Bundles on normal surfaces
11 votes

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 ...

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Embedding torsors of elliptic curves into projective space
10 votes

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 ...

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Confusion about good reduction
10 votes

(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 ...

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Can distinct morphisms between curves induce the same morphism on singular cohomology?
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10 votes

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 ...

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Moduli 'space' of stacks?
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10 votes

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) ...

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Are quotients of affine schemes by finite groups faithfully flat?
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10 votes

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 ...

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meaning of normalization
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10 votes

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 $Y$ ...

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Does a torus action with isolated fixed points imply rational?
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9 votes

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

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Soft question: beginners reference to moduli spaces
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9 votes

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 ...

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What are the general techniques for proving a variety is not toric?
9 votes

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 ...

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