10
votes
Accepted
Cycle type in Galois group from ramified primes
It does not imply there is such a permutation in the Galois group, and your question has an interesting history.
First of all, I would say the question you ask is arguably the wrong one: the more ...
- 50.6k
9
votes
Accepted
Is there a separable isogeny between any two isogenous abelian varieties?
The answer is no. I think Asvin's example in the comments is correct but there may be a few things to check. I will give a different example that is easier to check, using Moret-Bailly's famous ...
- 30.5k
7
votes
Frobenius and regular scheme
This is true Zariski-locally by Popescu's desingularisation theorem [Tag 07GC]. Indeed, any regular $\mathbf F_p$-algebra $A$ is geometrically regular [Tag 0381], so by Popescu's theorem it can be ...
- 28.7k
7
votes
Frobenius pushforward of an equivariant tautological bundle on the flag variety
EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment.
In general, the push-forward of a line bundle on the flag variety $G/B$ will not ...
- 16.1k
4
votes
Accepted
When is $R$ a direct summand of Frobenius pushforwards?
If there exists one $e > 0$ so that $R \to F^e_* R$ splits, then by composing splittings one sees that $R \to F^{ne}_* R$ splits for all $n > 0$. Ie, if $\phi : F^e_* R \to R$ is a splitting (...
- 20.5k
3
votes
Accepted
Length of a module and Frobenius map
This is false. The noetherian local ring $R =
\mathbb{F}_3[[X,Y]]/(Y^2 - X^3)$ has dimension one, and if $x,y$ are the images of $X,Y$ in $R$ then the sequence
$$
R \supseteq (x,y) \supseteq (x^2,y) ...
- 7,249
3
votes
Accepted
The Frobenius and Dold–Puppe
$\newcommand{\Sym}{\mathrm{Sym}}$Let's first consider the case $i=p$, and I'll assume for simplicity that $A$ is an ordinary commutative ring flat over $\mathbb{Z}_{(p)}$. We will show that the 1st ...
- 7,377
2
votes
Accepted
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
(1) is elementary deformation theory: if $\alpha\colon \mathcal{O}_{\tilde U}\to \mathcal{O}_{\tilde U}$, then $\alpha-{\rm id}\colon \mathcal{O}_{\tilde U}\to \mathcal{O}_{\tilde U}$ vanishes modulo $...
- 16.1k
1
vote
Frobenius action on the trivial connection
Since both connections are defined on the same domain, I managed to confuse myself with the definition of the pullback connection.
The pullback connection extends via the Leibniz rule on a local base.
...
- 2,907
1
vote
Accepted
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
@uno, you are correct in the comments above, $F^e_*(-)$ is not $R$-linear. In particular, the mistake is the line right here:
The exact sequence $0\to R \xrightarrow{x} R\to R/xR\to 0$ then gives ...
- 20.5k
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