Timeline for What advantages do perverse sheaves provide over D-modules? (or vice versa)
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| when toggle format | what | by | license | comment | |
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| May 23 at 16:06 | comment | added | R. van Dobben de Bruyn | Riemann–Hilbert is harder in positive characteristic, because the 'topological' (étale) theory is an $\ell$-adic theory while the 'coherent'/'crystalline' side (I think this is arithmetic D-modules?) is a $p$-adic theory. But each of these objects separately are indeed studied, and maybe people have even thought about connections between them. I am far from an expert, but I do know that going from $p$ to $\ell$ is hard. | |
| May 23 at 1:51 | comment | added | Andrea B. | @R.vanDobbendeBruyn That makes sense. I'm not sure why I had the notion that D-modules were more analytic. Still, given that it is possible (albeit tedious) to define perverse sheaves in the étale setting, and there is a R-H correspondence there, it does still make sense to pursue generalizations of perverse sheaves in other contexts, such as crystalline or prismatic, right? Especially to establish some sort of R-H there. | |
| May 22 at 13:22 | comment | added | R. van Dobben de Bruyn | I think a slightly better way to state the difference is that perverse sheaves are topological in nature, while D-modules more or less live in the (quasi)coherent world. That may partially explain why algebraic constructions are more likely to use the D-module language. | |
| May 22 at 11:19 | comment | added | Gabriel | Your question surely has many possible answers. Let me give you the first that comes to mind: perverse sheaves are related to regular holonomic D-modules. But we also have a six-functor formalism, nearby / vanishing cycles, etc... for more general holonomic D-modules. (For example, the exponential D-module $(\mathcal{O},d+dx)$ on $\mathbb{A}^1$ is very interesting and appears in a lot of places. But it's irregular and has no perverse sheaf analog.) And that's not saying anything about even more general D-modules. | |
| May 22 at 10:03 | history | asked | Andrea B. | CC BY-SA 4.0 |