Timeline for How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2022 at 20:14 | comment | added | user484289 | The ongoing book of Huybrechts would also be helpful. You can find a link on his website. | |
| Jul 1, 2022 at 16:46 | vote | accept | Zhaoting Wei | ||
| Jul 1, 2022 at 10:01 | comment | added | Sasha | @ZhaotingWei: I added an argument to the answer. | |
| Jul 1, 2022 at 10:01 | history | edited | Sasha | CC BY-SA 4.0 |
added 884 characters in body
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| Jun 30, 2022 at 15:40 | comment | added | Zhaoting Wei | @Sasha From Example 3.41 we can get $i^!=\mathcal{O}_X(X)[-1]\otimes i^*$ hence to prove $i^*i_*\mathcal{F}\to \mathcal{F}\to \mathcal{O}_X(-X)[2]\otimes \mathcal{F}$ is exact is equivalent to prove $\mathcal{F}\to i^!i_*\mathcal{F}\to \mathcal{O}_X(X)[-1]\otimes \mathcal{F}$ is exact. But I don't know how to prove either of them. | |
| Jun 30, 2022 at 15:04 | comment | added | Sasha | @ZhaotingWei: What were you saying about Cor. 3.40 and Ex. 3.41 then? | |
| Jun 30, 2022 at 13:15 | comment | added | Zhaoting Wei | @Sasha Where can I find the triangle for $i^!i_*$? | |
| Jun 30, 2022 at 10:11 | comment | added | crystalline | Or you can just use the triangle for the unit $id \to i_*i^*$ and the "zig-zag" identities for the adjunction $(i^*, i_*)$ to get the one for the counit. | |
| Jun 30, 2022 at 7:04 | comment | added | Sasha | I am sure you can use for $i^*$ the same argument that is used for $i^!$. Alternatively, you can use the fact that $i^!$ differs from $i^*$ by a line bundle twist and shift, and applying this to the triangle for $i^!i_*$, obtain the triangle for $i^*i_*$. | |
| Jun 30, 2022 at 4:40 | comment | added | Zhaoting Wei | I think his Corollary 3.40 and Example 3.41 are closed to the result I want but they are still on different topic. Actually they compute $i^!$ but I am not sure if it implies the result you mentioned. | |
| Jun 30, 2022 at 4:26 | comment | added | Sasha | I think you can find this in the book of Huybrechts (FM transforms ...). | |
| Jun 30, 2022 at 4:22 | comment | added | Zhaoting Wei | Is it well-known? How to see it? | |
| Jun 30, 2022 at 4:17 | history | answered | Sasha | CC BY-SA 4.0 |