Timeline for Why are the open and closed adic discs defined the way that they are?
Current License: CC BY-SA 4.0
7 events
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| Apr 14, 2021 at 12:02 | comment | added | Ashwin Iyengar | One way to think about it is that the ring $\mathbb Q_p\langle T \rangle$ consists of power series which converge if you plug in numbers in the closed unit disk in $\mathbb C_p$. But if you allow all power series in $\mathbb Z_p[[T]]$ (or $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ for that matter), these series only converge when you plug in numbers in the interior of the disk. This should explain the "functions on a space" motivation | |
| Sep 3, 2018 at 4:47 | comment | added | Andrew NC | It sounds like what you're saying is that the functor of points applied to C_p produces the open/closed discs of radius 1. So it sounds like your point is that the reason that it's called the open/closed discs is primarily and perhaps solely because they represent functors of points that are nice. | |
| Sep 1, 2018 at 8:15 | comment | added | David Loeffler | May I suggest that you set aside for a moment all of this abstract nonsense with functors of points, analogies to $\mathbf{C}$, etc, and actually do the exercise I outlined in my previous comment? | |
| Sep 1, 2018 at 5:24 | history | edited | Andrew NC | CC BY-SA 4.0 |
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| Sep 1, 2018 at 5:23 | comment | added | Andrew NC | Let's take for a minute the open disc of radius 1 in $\mathbb{C}$ with the complex topology. Then in the category of complex manifolds, it doesn't make any sense that the functor of points of the open disc applied to some complex manifold is "the open disc of radius 1" of that manifold, because complex manifolds don't have a unique open disc. It just seems so far removed from the complex situation that either "open/closed disc" is disorientingly misleading, or I'm missing some hidden intuition... I'm hoping for the latter. | |
| Aug 31, 2018 at 12:25 | comment | added | David Loeffler | If you compute the $\mathbf{C}_p$ (or $\overline{\mathbf{Q}}_p$) points of these two spaces, then you might get some intuition for why these are natural candidates for closed and open discs. Note that these are supposed to be discs of radius one -- infinitesimally small discs are a red herring here. | |
| Aug 31, 2018 at 6:00 | history | asked | Andrew NC | CC BY-SA 4.0 |