Timeline for Can we use Mann's six-functor formalism with D-modules?
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26 events
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| Mar 15, 2023 at 8:20 | comment | added | Gabriel | As far as I understand, neither this paper nor this book says nothing about holonomic D-modules (and we don’t have the six-functors for more general D-modules). The beginning of chapter 4 in the aforementioned book even says that the pov of crystals seems to be ill suited for dealing with holonomicity. (I also didn’t found the tensor product that I’ve explained anywhere in there.) | |
| Mar 14, 2023 at 13:21 | comment | added | D.-C. Cisinski | @Gabriel You may have a look at Gaitsgory and Rozenblyum paper: arxiv.org/abs/1111.2087 Contrary to what I wrote earlier, we can everything with crystals: this follows from Prop. 2.5.6 and paragraph 5.5.2 in there. You may complete with chapter 4 in their book "A Study in Derived Algebraic Geometry" Vol. II (in which the tensor product you want is introduced). You may extract from there the answer you would have liked: use Mann's thesis instead of Part III "A Study in Derived Algebraic Geometry" Vol. II in order to get the six operations for D-modules. | |
| Mar 13, 2023 at 8:18 | vote | accept | Gabriel | ||
| Mar 13, 2023 at 8:18 | comment | added | Gabriel | Dear @D.-C.Cisinski, it is indeed the tensor product corresponding to the analytic one via RH. I indeed expect the fact that "with this tensor product, the six-funtor formalism of holonomic D-modules satisfies precisely the same properties that the one of l-adic perverse sheaves over finite fields" to be known to experts. However, away from literally two papers by Martin Gallauer and Brad Drew's thesis, I never saw this anywhere. Do you know a single other paper which uses this tensor product? | |
| Mar 12, 2023 at 20:15 | comment | added | D.-C. Cisinski | @Scholze the fact that $f_!$ can be obtained uniquely by universal property is well known since Deligne's construction in the 60's then (the story of the subject is then to promote this gradually in more abstract/structured settings). We could compare all this story with Kan extensions. We want to extend some functorialities to a $2$-functor from a suitable $2$-category of correspondences. Abstract constructions are nice and can provide existence easily, but we want to compute the resulting extension in elementary terms. What we call "construction of $f_!$" is in fact this computation. | |
| Mar 12, 2023 at 20:02 | comment | added | D.-C. Cisinski | @Gabriel Then I am afraid I do not know how to avoid what you call hard theorems. We will need to understand $f_!=f_*$ at least in the case of closed immersions and of the projection of the form $\mathbb P^n \times X\to X$ and I do not know how to avoid that, whatever formalism we use for $D$-modules. The tensor product you consider is well known: this is the one obtained from the one of ordinary sheaves through the Rienmann-Hilbert correspondence, isnt'it? | |
| Mar 12, 2023 at 19:26 | comment | added | Peter Scholze | Re "hard: I wasn't able to give a slick presentation of the usual proper base change theorem in etale cohomology (a proof would seem to require a further digression), so I'd consider that "hard". | |
| Mar 12, 2023 at 19:23 | comment | added | Peter Scholze | @D.-C.Cisinski About "constructing $f_!$ vs proving that it exists": In the construction principle I explain in lecture 4 of my notes, the functor $f_!$ is not additional data; rather its existence is really a proposition (in the HTT sense, if you want). I agree that the verification of the assumptions becomes the hard part, but that part is not anymore a construction (i.e., you don't have to supply any data), it's just proving a theorem. And let's see how me writing the notes goes, but I think it won't really be hard either. | |
| Mar 12, 2023 at 13:48 | comment | added | Gabriel | Dear @PeterScholze, as far as I know, there are no books or papers (except from Brad Drew's thesis and some papers of Martin Gallauer) which even consider the tensor product which I explained. (Which is, I think, absolutely necessary if we want to have a six-functor formalism satisfying the same formulas as usual.) Also there's not much on D-modules over singular schemes. So, the literature on D-modules would really benefit from you explaining this six-functor formalism. :) | |
| Mar 12, 2023 at 13:42 | comment | added | Gabriel | @D.-C.Cisinski just to clarify: Kashiwara's theorem is not what I meant by a "hard" theorem. Instead, it's "hard" to show that there's a natural transformation $f_!\to f_*$, which is an isomorphism for proper $f$, and it's "hard" to show that $f_*$ preserves the derived category of holonomic D-modules. | |
| Mar 12, 2023 at 12:31 | comment | added | D.-C. Cisinski | @Scholze I am also a little bit confused by some of your comments: what is the difference between constructing $f_!$ and proving it exists? Even if you consider Mann's thesis as way to obtain $f_!$, then you must prove that the assumptions of his theorem hold, the proof of which will then be the "hard part" of the construction of $f_!$ (e.g. the proper base change in étale cohomology). | |
| Mar 12, 2023 at 11:14 | comment | added | D.-C. Cisinski | @Scholze Since we want the theory to agree with the usual one for smooth schemes, if we do it the way I suggest above, we need Kashiwara's theorem (I am not sure what people call hard in there, but it seems to me that, simply to formulate the theorem, you need the "hard" part, and, once it is formulated, this is not "hard" to prove). I look forward to see how you will present the theory in your lectures, though! | |
| Mar 12, 2023 at 11:13 | comment | added | D.-C. Cisinski | @Scholze We should maybe define "hard" before discussing what it means to avoid "hard theorems". My answer above is not very precise about what should be proved about $f_!$ or not: I just wanted to emphazise that we should extend the theory to non smooth schemes. (to be continued) | |
| Mar 11, 2023 at 20:37 | comment | added | Peter Scholze | @Z.M Regarding your first comment: Yes, I think one could also set up this using the infinitesimal site (in characteristic 0), or the de Rham stack (which is essentially the same story), or also with solid modules (where one can treat formal completions differently, and accordingly gets a slightly different theory). I'm still a bit confused about the relations between all possible choices (I'm just learning about D-modules...), I hope to clear this up for myself while writing the notes... | |
| Mar 11, 2023 at 18:12 | comment | added | Peter Scholze | (By the way, when referring to it as Mann's definition, I'm aware that closely related definitions have been around (Liu-Zheng, Gaitsgory-Rozenblyum, etc), but as far as I'm aware Mann was the first to pin down this precise definition, which at least for my concerns is the "best" among many closely related definitions.) | |
| Mar 11, 2023 at 18:06 | comment | added | Peter Scholze | @D.-C.Cisinski I'm a bit confused by your answer. Most importantly, in Lucas Mann's construction principle, you don't have to construct $f_!$, you just check that it exists. Often the only nontrivial step is to check proper base change. I plan to write up some notes on $D$-modules for my course; I think one can give fairly direct proofs (not having to rely on hard theorems). Also, six functors as in Mann's definition don't have to satisfy excision, $\mathbb A^1$-invariance or cdh-descent, which is good as coherent sheaves don't. | |
| Mar 10, 2023 at 12:29 | comment | added | D.-C. Cisinski | In particular I answered your question and I do not believe there is anything else to say. | |
| Mar 10, 2023 at 12:24 | comment | added | D.-C. Cisinski | @Gabriel To my knowledge it is always hard to construct all of the operators $f*$, $\otimes$ and $f_!$ (there is always at least one of them that causes trouble) and no formalism will do that for you. Formalism is a way to make precise the kind of functoriality and of coherence you should provide and/or expect. | |
| Mar 10, 2023 at 12:19 | comment | added | D.-C. Cisinski | @Z.M. That is why I said the ground field would be of characteristic zero. In positive characteristic I mentioned the possibility to work with Berthelot's Arithmetic D-modules. They correspond to rigid cohology wich coincides with crystalline cohomology for proper and smooth schemes, while being homotopy invariant. | |
| Mar 10, 2023 at 9:16 | comment | added | Z. M | The de Rham cohomology is not $\mathbb A^1$-invariant in char p, either: the de Rham cohomology of $\mathbb F_p[T]$ over $\mathbb F_p$ is very nontrivial. | |
| Mar 10, 2023 at 9:08 | comment | added | Z. M | I was talking about things in char 0, thus it is homotopy invariant, I think. | |
| Mar 10, 2023 at 9:03 | comment | added | Gabriel | All in all, I have the impression that you're doing what I said in the end: using all the knowledge of the six-functors for holonomic D-modules to construct it in this new formalism. (Of course, we get more out of it: we now have more compatibilities, everything works for singular schemes as well, etc...) | |
| Mar 10, 2023 at 9:02 | comment | added | Gabriel | Dear @D.-C.Cisinski, if I understand correctly your construction, both the tensor product and the inverse image come from your hypersheaf of D-modules on $X$ supported on $Z$. I.e., we need to construct them as usual for smooth D-modules. (But those were the "hard" functors, that one would like to avoid constructing explicitly!) The same happens with your $g_!$. | |
| Mar 10, 2023 at 8:34 | comment | added | D.-C. Cisinski | @Z.M I do not think that your guess is correct: the infinitesimal site point of view leads the development of crystalline cohomology and the latter is not homotopy invariant. But the six operations imply cdh-descent, hence, using resolution of singularities, since de Rham cohomology is homotopy invariant for smooth schemes, any theory of six operations of $D$-modules defined on all schemes must lead to an homotopy invariant theory. | |
| Mar 10, 2023 at 8:07 | comment | added | Z. M | I guess that one could also work with Grothendieck's infinitesimal site (in Dix Exposés), or de Rham stacks with solid coefficients. | |
| Mar 10, 2023 at 7:56 | history | answered | D.-C. Cisinski | CC BY-SA 4.0 |