Timeline for Blow-ups in Motivic Homotopy Theory
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 28, 2014 at 10:11 | comment | added | Marc Hoyois | Ah, that's a great example. In this case the blow-up square is trivially a pushout (without suspending), but there's no reason for it to remain a pushout when you remove a point... I missed this point before. | |
| Apr 25, 2014 at 21:42 | comment | added | Jesse Wolfson | I think it's unlikely that it will ever be an $\mathbb{A}^1$-equivalence. Consider the blow up of $\mathbb{A}^2$ at a point. This is the total space of the tautological line bundle over $\mathbb{P}^1$. If we remove a point from the exceptional divisor, we get something whose topological realization looks like $S^3\vee S^2$. FWIW, the situation I was considering turns out to involve blow-ups of affine space where we remove the fiber over a hyperplane section of the exceptional divisor (I only saw the hyperplane at first). After removing the whole fiber, we get an $\mathbb{A}^1$-equivalence. | |
| Apr 24, 2014 at 10:41 | comment | added | Marc Hoyois | If you remove a hyperplane section from the exceptional divisor, then I agree that it looks like the map could be an $\mathbb{A}^1$-equivalence, but you need a finer topology than the Nisnevich one. Maybe this will be useful: Unstable motivic homotopy categories in Nisnevich and cdh-topologies. | |
| Apr 24, 2014 at 3:21 | vote | accept | Jesse Wolfson | ||
| Apr 24, 2014 at 3:21 | comment | added | Jesse Wolfson | Actually, this intuition is wrong, and this is not the question I should have asked. Thanks for helping clear this up. | |
| Apr 23, 2014 at 23:38 | comment | added | Jesse Wolfson | Thanks for the answer, and you're right that the question I wrote requires codimension 2. My intuition is that $Bl_Z(X)\setminus\sigma(Z)$ is obtained from X by gluing in an affine bundle over $Z$. Since affine spaces are $\mathbb{A}^1$-contractible, my intuition was that we could contract away the fibers over $Z$ and recover $X$. In higher codimension, I need to delete a hyperplane section over $Z$ to get the right picture, but this is the situation I had in mind. Do you know if there is reason to be suspicious of this intuition? | |
| Apr 23, 2014 at 13:44 | history | answered | Marc Hoyois | CC BY-SA 3.0 |