Timeline for An fppf cover with trivial Picard group
Current License: CC BY-SA 3.0
20 events
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| Oct 15, 2017 at 6:16 | vote | accept | user2831784 | ||
| Oct 15, 2017 at 6:16 | comment | added | user2831784 | Thanks. In Proposition 1, in order to see that each irreducible component $Y_{i}$ of $Y$ has constant relative dimension over $X$, I guess we can use e.g. math.stanford.edu/~conrad/249BW17Page/handouts/fiberdim.pdf. | |
| Oct 9, 2017 at 12:41 | comment | added | Jason Starr | @user2831784. Those are good points. I edited the answer (with "edit" in boldface preceding each edit) to address those points. | |
| Oct 9, 2017 at 12:40 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| S Oct 8, 2017 at 1:47 | history | suggested | user2831784 | CC BY-SA 3.0 |
typos and clarification
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| Oct 7, 2017 at 23:04 | comment | added | user2831784 | (4) Regarding the use of Krull's Hauptidealsatz in Proposition 1: I guess the local flatness criterion gives us that multiplication by $t$ will be injective on $\mathcal{O}_{Y \times_{X} \operatorname{Spec} \mathcal{O}_{X,x}}$. Then since everything is Noetherian, we should be able to take $W$ small enough that the lift $t_{W}$ is also a nonzerodivisor on $\mathcal{O}_{Y \times_{X} W}$. | |
| Oct 7, 2017 at 22:58 | comment | added | user2831784 | (3) Regarding Proposition 1: I guess it doesn't matter for the counterexample to my question since we are working over an algebraically closed field, but wouldn't we need to know that the field $k$ in $x : \text{Spec}\ k \to X$ is infinite, in order to ensure that we can take $t_{x}$ to be linear? In any case, unless I am mistaken, it seems to me that the linearity of $t_{x}$ is not used in the proof. | |
| Oct 7, 2017 at 22:52 | review | Suggested edits | |||
| S Oct 8, 2017 at 1:47 | |||||
| S Oct 7, 2017 at 22:46 | history | suggested | user2831784 | CC BY-SA 3.0 |
typos and clarification
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| Oct 7, 2017 at 22:22 | review | Suggested edits | |||
| S Oct 7, 2017 at 22:46 | |||||
| Oct 7, 2017 at 22:16 | comment | added | user2831784 | Thanks very much for the extended answer. Can I ask some clarifying questions? (1) In Proposition 1, I guess we may assume that $X$ is also affine (by taking an affine open cover of the image of $p$ in the beginning), in order to get the closed immersion $e$? (2) In Proposition 1, why is $Y_{\text{equi}}$ nonempty? | |
| Oct 4, 2017 at 12:26 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Oct 3, 2017 at 11:43 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Oct 3, 2017 at 11:37 | history | edited | R.P. | CC BY-SA 3.0 |
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| Oct 2, 2017 at 22:46 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Oct 2, 2017 at 14:49 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Oct 2, 2017 at 10:41 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Oct 2, 2017 at 9:49 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| S Oct 2, 2017 at 9:43 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Oct 2, 2017 at 9:43 | history | made wiki | Post Made Community Wiki by Jason Starr |