Timeline for Category of $\mathcal{D}$-modules on a singular variety
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Oct 7, 2019 at 2:56 | comment | added | FunctionOfX | yeah... i dont understand it, but it is a result we can quote. I only found it out a few days ago. Thank you so much for your answer. | |
| Oct 7, 2019 at 2:48 | comment | added | Avi Steiner | @FunctionOfX ah. That paper. I have it on my list of to-reads for if I ever learn DAG | |
| Oct 7, 2019 at 2:29 | comment | added | FunctionOfX | @ avi steiner, thank you. Yeah, the crossed out part was exactly my confusion. D Gaitsgory actually proved this in this paper CRYSTALS AND D-MODULES (prop 4.7.3). The paper uses crystals (which i am not very familiar with), but the proof itself actually doesn't deal with it too much (even though I don't claim I understand it fully). | |
| Oct 6, 2019 at 20:25 | comment | added | Avi Steiner | @FunctionOfX So, I was apparently thinking in the crossed-out part that $\Gamma_X$ fixes $\mathcal{D}_V$-injectives, or maybe that one can always resolve by $\mathcal{D}_V$-injectives for which this is true. However, I have no reason to believe either of those statements, hence the crossed-out things. Anyway, I hopefully salvaged things correctly in my latest edit. Let me know! | |
| Oct 6, 2019 at 20:22 | history | edited | Avi Steiner | CC BY-SA 4.0 |
Removed the incorrect part of my first edit, then corrected it.
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| Oct 4, 2019 at 5:04 | comment | added | FunctionOfX | thanks! I still don't quite see how preserve injectives (not necessarily exact) help with computing hom sets, if will be great if you can spell out the detail! | |
| Oct 3, 2019 at 18:16 | comment | added | Avi Steiner | @FunctionOfX $\Gamma_X(\mathcal{M})$ is a $\mathcal{D}_V$-submodule of $\mathcal{M}$. One way to see this (at least in the algebraic category, where I feel much more confident about all this stuff) is that $\Gamma_X(\mathcal{M})$ is the sheaf of sections of $\mathcal{M}$ killed by a power of the defining ideal of $X$; now use that this property is preserved under multiplication by elements of $\mathcal{D}_V$. | |
| Oct 3, 2019 at 1:21 | comment | added | FunctionOfX | @ Avi Steiner sorry to bother you again, how does D act on $\Gamma_X$ | |
| Aug 31, 2019 at 0:18 | comment | added | Avi Steiner | @FunctionOfX since $\Gamma_X$ preserves injectives, I can always resolve an object of $\mathcal C$ by injectives in $\mathcal D_V$-mod | |
| Aug 31, 2019 at 0:16 | comment | added | FunctionOfX | sorry one last question... surely we need preservation of injective resolutions rather than just injectives... | |
| Aug 30, 2019 at 20:06 | comment | added | Avi Steiner | @FunctionOfX ugh! You’re totally right! | |
| Aug 30, 2019 at 20:06 | history | edited | Avi Steiner | CC BY-SA 4.0 |
added 1 character in body
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| Aug 30, 2019 at 20:03 | comment | added | FunctionOfX | no problem, in fact thank you so much for your answer, one last question, should the word left adjoint be replaced by right adjoint? I believe it is the right adjoint for closed embedding at least in the smooth case. | |
| Aug 30, 2019 at 19:56 | comment | added | Avi Steiner | @FunctionOfX sorry again. I meant fully faithful. | |
| Aug 30, 2019 at 19:55 | history | edited | Avi Steiner | CC BY-SA 4.0 |
edited body
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| Aug 30, 2019 at 19:55 | comment | added | FunctionOfX | and do you mean right adjoint and exact rather than equivalence? | |
| Aug 30, 2019 at 16:00 | comment | added | Avi Steiner | @FunctionOfX shoot! Yes, I do! Good catch. It’s fixed now. | |
| Aug 30, 2019 at 15:59 | history | edited | Avi Steiner | CC BY-SA 4.0 |
edited body
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| Aug 30, 2019 at 13:26 | comment | added | FunctionOfX | oh you mean D_V? | |
| Aug 30, 2019 at 1:59 | comment | added | FunctionOfX | Sorry, I am not understanding why $\Gamma_X: \mathcal{D}_X$-mod$\to \mathcal{C}$ preserve injectives. In fact, what is $\mathcal{D}_X$ | |
| Aug 28, 2019 at 19:43 | comment | added | Avi Steiner | @FunctionOfX see my edit | |
| Aug 28, 2019 at 19:43 | history | edited | Avi Steiner | CC BY-SA 4.0 |
added 298 characters in body
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| Aug 27, 2019 at 1:26 | comment | added | FunctionOfX | how do you show fully faithfulness? is that automatic? | |
| Aug 23, 2019 at 23:08 | history | answered | Avi Steiner | CC BY-SA 4.0 |