Timeline for A vanishing condition for cup products in Galois cohomology
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 1, 2015 at 20:02 | answer | added | Yonatan Harpaz | timeline score: 2 | |
| Oct 1, 2015 at 13:53 | comment | added | Leonid Positselski | And for the same reason, one has $[a_1]\cup\dotsb\cup[a_n]=0$ for any $a_i$ such that $\prod_ia_i=1$ when $n$ is odd. | |
| Oct 1, 2015 at 13:47 | comment | added | Leonid Positselski | On the other hand, over any field $k$ one has $[x_1]\cup [x_2]\cup\dotsb\cup [x_{2m-1}]\cup[-x_1\dotsb x_{2m-1}]=$ $[x_1]\cup\dotsb\cup [x_{2m-1}]\cup[-x_1\cdot -x_2\cdot\dotsc\cdot-x_{2m-1}]=$ $[x_1]\cup \dotsb\cup [x_{2m-1}]\cup[-x_1]+\dotsb+[x_1]\cup\dotsb\cup[x_{2m-1}]\cup[-x_{2m-1}]=$ $0+\dotsb+0=$ $0$. So your assertion is true for $n$ even and false for $n$ odd. | |
| Oct 1, 2015 at 13:29 | comment | added | Yonatan Harpaz | Right. Thank you. So my claim is simply wrong. I guess I have some reassessing to do then. | |
| Oct 1, 2015 at 13:20 | comment | added | Leonid Positselski | Yes, over $\mathbb R$, one has $\prod_{i=1}^3-1=-1$ and $[-1]\cup[-1]\cup[-1]\ne 0$ in $H^3(\mathbb R,\mathbb Z/2)$. | |
| Oct 1, 2015 at 12:58 | comment | added | few_reps | And in fact, over $\mathbf R$, the quadratic form $<1,-a>^{\otimes s}$ is never hyperbolic ... for $a<0$, in particular for $a=-1$ and $s$ odd ... | |
| Oct 1, 2015 at 12:25 | comment | added | few_reps | Pernickety remark : you should add $n\geq 2$ in the claim. | |
| Oct 1, 2015 at 12:21 | history | edited | Yonatan Harpaz | CC BY-SA 3.0 |
added 15 characters in body
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| Oct 1, 2015 at 11:36 | history | asked | Yonatan Harpaz | CC BY-SA 3.0 |