Timeline for existence of global good filtration for D-modules?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2011 at 2:36 | comment | added | Moosbrugger | @algchen: The associated gradeds can indeed be different, but in both cases the set-theoretic support of the modules is the zero section of the cotangent bundle. (Note that in the VHS case the action of vector fields is nilpotent on the associated graded.) | |
| Nov 12, 2011 at 20:35 | vote | accept | genshin | ||
| Nov 12, 2011 at 20:34 | comment | added | genshin | Great! thanks a lot. Also a question to Moosbrugger's comment to the original question: if $M$ is the flat connexion associated to a variation of Hodge structure $\mathcal{V}$ on $X$ (complex algebraic variety), what about the Hodge filtration as a good filtration for the $D_X$-module structure? I'm confused as the graded module of the Hodge filtration is in general different from the trivial graded quotient mentioned above in your comment. | |
| Nov 12, 2011 at 13:34 | comment | added | Moosbrugger | Sorry -- my problem was that I was trying to choose the coherent guy $M_0$ in a stupid way. But objection retracted! | |
| Nov 12, 2011 at 6:44 | comment | added | Sam Gunningham | It just occurred to me that you might be asking about a filtration on the global sections $\Gamma (M)$ rather than a globally defined filtration on $M$... | |
| Nov 12, 2011 at 6:19 | comment | added | Sam Gunningham | I am not sure I understand your objection. When I say a submodule $M_0$ which generates $M$, I mean $M_0$ is a subsheaf of $M$ such that the map $D_X \otimes {\mathcal O_X} M_0 \to M$ is a surjective map of sheaves (so that any section of $M$ can be locally written as $Pm_0$ for some local sections $P$ of $D_X$ and $m_0$ of $M$). Similarly, when I write $F_nD_X(M_0)$, I mean everything as sheaves. I never need to talk about global sections of anything. Certainly theorem 2.1.3 applies in general (D-affineness is not needed). What am I missing? Have I misunderstood the question? | |
| Nov 12, 2011 at 4:11 | comment | added | Moosbrugger | This seems only to work in the case when $M$ is generated by global sections (e.g., the D-affine case). | |
| Nov 12, 2011 at 3:24 | history | answered | Sam Gunningham | CC BY-SA 3.0 |