Timeline for Can we use Mann's six-functor formalism with D-modules?
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| Mar 18, 2023 at 20:14 | comment | added | Peter Scholze | The problem from the cited question disappears because I've renamed $f^!$ and $f_\ast$ into $f^\ast$ and $f_!$, and their right adjoint do exist (as well as internal Hom), so you get a 6-functor formalism. (It doesn't "feel" like one, but we have a definition, and it fits the bill!) It's just that the functors don't quite mean what you'd think they mean. But they do once you pass to opposite categories. (But then it's just a 3-functor formalism as the required right adjoints cease to exist; also, your categories are no longer presentable.) | |
| Mar 18, 2023 at 15:55 | comment | added | Peter Scholze | @Gabriel As I say towards the end, if you take the $\mathrm{Pro}$-perspective (and restrict to coherent $D$-modules), you get the standard $\otimes,f^\ast,f_!$ functors. You can then further restrict to holonomic $D$-modules, and then some hard theorems tell you that this category is stable under all six operations. But for the construction of the functors, you don't need the theorems. | |
| Mar 18, 2023 at 11:27 | comment | added | Gabriel | Also, on left D-modules, the "true" pullback $f^!$ has to be a shift of the "naive" one if we want it to be adjoint to $f_!$, I think... (Of course, you could shift $f_*$ then... but I'm not sure if that's what you want. For example, if $M$ is a holonomic D-module and $p:X\to \operatorname{pt}$ is the structure map, then $p_* M$ and $p_! M$ only have cohomologies in degrees from $-\dim X$ up to $\dim X$. This is very similar to what happens with perverse sheaves, and I guess we would like to keep it.) | |
| Mar 18, 2023 at 11:13 | comment | added | Gabriel | Dear @PeterScholze, I admit that I don't really understand what you did in there. The "hard theorems" that I said before all deal with problems of holonomicity, which you seem to avoid entirely. Actually, I was basically convinced that there was no six-functor formalism for quasi-coherent D-modules (I even thought that condensed mathematics could perhaps help here...). For example, as far as I understand, GR only deal with a 3-functor formalism. How does you formalism avoid this problem in here? mathoverflow.net/questions/405511/… | |
| Mar 18, 2023 at 5:29 | comment | added | David Roberts♦ | A very minor point: Remark 8.40 has a broken citation. | |
| Mar 18, 2023 at 3:49 | comment | added | Exit path | Isn’t it also possible to define a sheaf theory a la Gaitsgory-Rozenblyum which sends an affine scheme to its (Ind) category of holonomic D-modules? Unfortunately I don’t know how to define the latter on singular schemes without using embeddings into smooth schemes | |
| Mar 18, 2023 at 1:41 | history | answered | Peter Scholze | CC BY-SA 4.0 |