Timeline for When can we cancel vector bundles from tensor products?
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| Apr 14, 2010 at 17:52 | comment | added | Hailong Dao | @Ulrich: I was indeed confused by the line bundle analogue, because there you have to tensor with $G^*$. But here you will need to know that $E\otimes G^{\perp} \cong F\otimes G^{\perp}$ as well! Is that easy to see? | |
| Apr 14, 2010 at 9:22 | comment | added | Ulrich Pennig | Sorry, I may have caused some confusion by my comparison with the case of line bundles: I did mean the direct sum and the isomorphism $E^n \simeq E \otimes \underline{\mathbb{C}^n} \simeq E \otimes (G \oplus G^\perp) \simeq (E \otimes G) \oplus (E \otimes G^\perp) \simeq (F \otimes G) \oplus (F \otimes G^\perp) \simeq F^n$. | |
| Apr 13, 2010 at 20:24 | comment | added | Adam Gal | I'm sure he meant direct sum, as evidenced by his orthogonal complement notation, where such examples come from. | |
| Apr 13, 2010 at 20:04 | comment | added | Hailong Dao | @Ulrich: Do you mean to say $G\otimes G^{\perp}$ is trivial? I am fairly certain this forces $G$ to split, and so have to be direct sum of $\mathcal O(n)$ for same $n$. | |
| Apr 13, 2010 at 19:23 | history | answered | Ulrich Pennig | CC BY-SA 2.5 |