Timeline for Katz's proof of Cartier's (descent) theorem
Current License: CC BY-SA 4.0
4 events
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| Jul 26, 2020 at 20:23 | comment | added | Joshua Mundinger | First, since $\partial$ generates the vector fields on $S$, $\nabla P = 0$ and $\nabla_\partial P = 0$ are equivalent. Yes, the main idea is that $P$ constructs sufficiently many horizontal sections. Evaluation means looking at the value of $e$ in the fiber. | |
| Jul 25, 2020 at 22:04 | comment | added | clarkkent | Thank you so much for the quick reply! A couple of stupid questions: 1. Assuming that flat sections are the same as horizontal sections, did you mean $\nabla P = 0$ instead of ${\nabla}_{\partial} P = 0$? If something is in the kernel of $\nabla$, then it is definitely in the kernel of ${\nabla}_{\partial}$, but is the converse true? 2. Having zero $p$-curvature means having "enough" horizontal sections. So is the "main idea" behind the construction of $P$ simply to provide sufficiently many horizontal sections? 3. What exactly do you mean by evaluating the section $e$ at 0? | |
| Jul 25, 2020 at 21:53 | history | edited | Joshua Mundinger | CC BY-SA 4.0 |
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| Jul 25, 2020 at 21:05 | history | answered | Joshua Mundinger | CC BY-SA 4.0 |