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Results tagged with vector-bundles
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user 39304
A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
11
votes
Accepted
Can non-split extension be isomorphic to the split one as objects
$\newcommand{\cO}{\mathcal{O}}$Consider exact sequence of trivial vector bundles $$0\to\cO\xrightarrow{\left(\begin{matrix}x \\ y\end{matrix}\right)}\cO\oplus\cO\xrightarrow{\left(\begin{matrix}y & -x …
- 7,377
4
votes
Is the vector bundle over a vector bundle, a vector bundle over the base scheme?
$\newcommand{\Spec}{\mathrm{Spec}\,}\newcommand{\cO}{{\cal{O}}}\newcommand{\cE}{{\cal{E}}}\DeclareMathOperator{\Sect}{Sect}$Here is an example where $\pi:E'\to X$ cannot be given a structure of a vect …
- 7,377
4
votes
Accepted
Vector bundles that are fixed under pull-back by the absolute Frobenius
For a finite flat cover $\pi:Y\to X$ the pushforward $E:=\pi_*\mathcal{O}_Y$ comes with a morphism $F^*E\to E$ induced by the Frobenius on $Y$. If $\pi$ is etale this morphism is an isomorphism: over …
- 7,377
5
votes
1
answer
479
views
Isomorphism classes of sheaves which arise as extensions
Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in t …
- 7,377
8
votes
Accepted
Pullback along Frobenius morphism
Consider the $p$-th tensor power $\mathcal{M}^{\otimes p}$. The group $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic permutations on it. Denote its generator by $\sigma$. There is a map from coinvariants to …
- 7,377