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Triviality of vector bundles on affine open subsets of affine space

For your final question, the answer is that all vector bundles over $U$ are trivial. Sketch of proof: Let $R=k[x_1,\ldots,x_n]$. Let $L_1, L_2,\ldots, L_m$ be the equations of hyperplanes in $R$. ...
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1 vote
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Existence of rigid objects in the derived category of a smooth projective variety

I am writing up as one answer the comments by @Johan, by @Libli, and by myself. If either of them prefers to write an answer, I am happy to delete this answer. Let $A$ be an Abelian variety. For ...
1 vote

Stability of sheaves of non-constant rank

The rank of a coherent sheaf is defined as its rank at the general point (equivalently, as the rank on a dense open subset where the sheaf is locally free). So, yes, the definition applies.
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1 vote

Why is it useful to study vector bundles?

Calculus is extended from vector spaces to manifolds by way of the tangent bundle. This is a very specific kind of vector bundle. The Cartan calculus of differential forms is built over it as well as ...
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