Boyarsky
  • Member for 11 years, 7 months
Counterexamples in algebra?
63 votes

The ring $A = \prod_{n=1}^{\infty} \mathbb{F}_2$ has some interesting/disturbing properties. For example, the affine scheme $X := {\rm{Spec}}(A)$ has non-open connected components (since it has ...

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separated schemes
16 votes

The necessary and sufficient condition on a schemes $\{U_i\}$ and gluing isomorphisms $\varphi_{ij}:U_{ij} \simeq U_{ji}$ between opens $U_{ij} \subset U_i$ ($i, j \in I$) that the gluing $X$ be ...

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coherent analytic cohomology vanishes for q > 2dim
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13 votes

Note that it is not necessary to say to avoid GAGA, as GAGA has no relevance in the absence of compactness assumptions. Anyway, something much more general (and satisfying) is true: all topological ...

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Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free?
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13 votes

The reference is "Auslander-Buchsbaum formula". What matters is that $X$ and $Y$ are smooth of some common pure dimension $n$ (which forces flatness due to the quasi-finiteness, by the way), not that ...

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Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields
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13 votes

Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-...

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Dimension of central simple algebra over a global field "built using class field theory".
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12 votes

To paraphrase Igor Pak: OK, this I know. It is remarkable how difficult it is to track down a reference which gives an actual proof for this fact (moreover applicable to all global fields). The ...

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étale fundamental group of projective space
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11 votes

It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper ...

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Is it possile for all real algebraic numbers to have continued fractions with bounded partial quotients ?
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10 votes

There is no example of an algebraic number of degree $> 2$ for which the boundedness or not of the entries of the continued fraction has been determined. In a 1976 Annals of Math paper, Baum & ...

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Why do generic polynomials work in reality?
9 votes

Serre introduced a notion of "thin set" in the $k$-rational points of a $k$-variety (such as $k^n$ viewed as the $k$-rational points of affine $n$-space, or likewise for a Zariski-dense open locus in ...

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Are there applications of category theory to countable sets?
8 votes

Etale cohomology, whose definition and study would be inconceivable without the use of categorical methods, underlies the construction of many interesting Galois representations as well as the entire ...

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How should the Math Subject Classification (MSC) be revised or improved?
8 votes

To be honest, I have never in my life paid any attention to this stuff (except for its sieving application when reading postdoc job applications). I read papers because of the title/abstract, or ...

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What is the family derived from the absolute Frobenius on the Hilbert scheme?
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7 votes

Indeed since the map on the base is not an isomorphism (apart from silly cases such as when $X$ is empty: the Frobenius endomporphism of a locally finite type $\mathbb{F}_p$-scheme is an isomorphism ...

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Elliptic curves over proper variety over $\mathbf{F}_q$ isotrivial
6 votes

I give an argument using the correct definition of isotriviality (relating it to the weak definition), and the reader who is familiar with Deligne-Mumford stacks can see that the method is actually ...

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Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set?
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6 votes

It sounds like the question you mean to ask is the following: if $X$ is an integral noetherian scheme with generic point $\eta$ and $C^{\bullet}$ is a finite complex of coherent sheaves on $X$ such ...

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A question about iterated quotients in riemannian geometry
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5 votes

The group $D$ is the preimage of $E$ in $N(F)$, so it is as you expect. The finiteness hypothesis can be weakened, which is important for many applications. Things become clearer if one thinks ...

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What is Grothendieck associated points(Prime cycles) of coherent sheaves on noetherian scheme?
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4 votes

EGA IV$_4$, 3.1.1 for any quasi-coherent sheaf on any (pre)scheme: the commutative algebra definition on stalks. So one needs 3.1.2 and 3.1.3 there to get useful alternative formulations of this ...

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The fiber of the sheaf of invariants
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3 votes

It all works out as well as you could want in every possible sense because of the freeness of the action. As you know, you can make a cover by $G$-stable affine opens, so the real work is in that case....

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Functoriality of base change
3 votes

Yes. Inject $f_ {!}$ into $f_ {\ast}$ to convert it into a claim concerning equality among "base change morphisms" (not generally isomorphisms) relating topological pushforward and pullback. Then it ...

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