Timeline for Why is the inertia stack of a smooth Deligne-Mumford stacks called inertia?
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| Oct 10, 2019 at 1:46 | comment | added | Will Chen | @YuhangChen That's just a case where the terminology breaks down. When I said "the inertia groups of the Galois covering $\mathcal{H}\rightarrow\mathcal{H}/SL(2,\mathbb{Z})$, I meant the inertia groups in the traditional sense, which are subgroups of $PSL(2,\mathbb{Z})$ because the Galois group of that covering is $PSL(2,\mathbb{Z})$. In general the traditional inertia groups are a measure of how much the generalized/stacky inertia groups jump. | |
| Oct 10, 2019 at 1:17 | comment | added | Yuhang Chen | This is very intuitive. So attaching an inertia group to a point is like assigning a mass to it. I have another question regarding the example of moduli stack of elliptic curves. Why do you take images in $PSL(2,\mathbb{Z})$ when you consider inertia groups of the Galois covering $\mathcal{H} \to \mathcal{H}/SL(2,\mathbb{Z})$? Every generic point has an inertia group $\{I,-I\} \cong \mu_2$, right? By a generic point, I mean a point not in the orbits of $i$ or $e^{2\pi i/3}$. | |
| Oct 9, 2019 at 19:49 | comment | added | Will Chen | As for the use of inertia in physics, I think of points with large inertia groups as carrying additional "weight" compared to those with smaller inertia. Perhaps a picture I have in mind is that near a point with nontrivial inertia, the group action moves things around less and less, as if there's additional "physical" mass near that point. Maybe the person who first used the word inertia in the context of fixed points had a similar picture in mind. | |
| Oct 9, 2019 at 19:48 | comment | added | Will Chen | The inertia group from number theory is essentially the same as the one described above, except one needs to be more careful about exactly what a fixed point is when the residue field is not algebraically closed. | |
| Oct 9, 2019 at 19:03 | vote | accept | Yuhang Chen | ||
| Oct 9, 2019 at 19:02 | comment | added | Yuhang Chen | Thanks for the answer and sharing with us this interesting example. I thought "inertia" has something to do with the "inertia" in physics: the resistance to change; apparently that's not the case. I was not familiar with the notion of "inertia group", which I think indeed justifies the name "inertia stack" from your explanation. But why do people call the stabilizer subgroup of a group action "inertia group"? Now I recall a little bit the "inertia group" from algebraic number theory. I wouldn't be surprised if the person who coined up the term "inertia group" got inspiration from physics. | |
| Oct 9, 2019 at 3:39 | history | answered | Will Chen | CC BY-SA 4.0 |