Timeline for Algebraic analog of a geometric result
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| May 15, 2020 at 4:40 | comment | added | Joshua Mundinger | @R.vanDobbendeBruyn Yes, you're right; representations of $\pi_1$ are local systems, not vector bundles. | |
| May 15, 2020 at 4:30 | comment | added | abx | Focus, did you see that the comment by @skd completely answers your question? Why did you edit it? | |
| May 15, 2020 at 3:05 | history | edited | Focus | CC BY-SA 4.0 |
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| May 14, 2020 at 17:00 | comment | added | R. van Dobben de Bruyn | @JoshuaMundinger: "A vector bundle of rank 2 is equivalent to a representation" ― this doesn't seem quite right. On $\mathbf P^1$ there are interesting vector bundles such as $\mathcal O \oplus \mathcal O(1)$, but $\pi_1(\mathbf P^1)$ is trivial... | |
| May 14, 2020 at 16:43 | comment | added | Joshua Mundinger | Here's a concrete example: let $E/\mathbb C$ be an elliptic curve. A vector bundle of rank $2$ is equivalent to a representation $\pi_1(E) \to GL_2(\mathbb C)$ up to conjugacy, that is, two commuting matrices up to conjugacy. These have a common eigenvector, but the vector bundle does not split unless the commuting matrices are simultaneously diagonalizable. | |
| May 14, 2020 at 16:36 | comment | added | Mohan | Serre proved his result for arbitrary varieties over an infinite filed. If $E$ is a globally generated vector bundle of rank greater than the dimension, a general section is nowhere vanishing. Of course, this does not give a splitting in general, except for affine varieties. | |
| May 14, 2020 at 16:29 | comment | added | Focus | @skd are there some global results on schemes？ | |
| May 14, 2020 at 16:25 | comment | added | Focus | @skd, Thanks a lot | |
| May 14, 2020 at 16:25 | history | edited | Focus | CC BY-SA 4.0 |
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| May 14, 2020 at 16:24 | comment | added | Focus | @abx,sure you are right.I‘ve edited, thanks. | |
| May 14, 2020 at 16:24 | comment | added | skd | Serre proved that any vector bundle on an affine scheme X of rank larger than the dimension of X has a trivial bundle as a summand; this is Theoreme 1 in numdam.org/article/SD_1957-1958__11_2_A9_0.pdf. | |
| May 14, 2020 at 16:11 | comment | added | abx | You want $A$ to be a direct summand of $M$. Since a projective module is torsion free, any nonzero element gives rise to an injection $A\hookrightarrow M$. | |
| May 14, 2020 at 16:11 | history | edited | Amir Sagiv | CC BY-SA 4.0 |
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| May 14, 2020 at 16:07 | review | First posts | |||
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| May 14, 2020 at 16:06 | history | asked | Focus | CC BY-SA 4.0 |