Timeline for D-modules as ind-coherent sheaves over positive characteristics?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2023 at 20:52 | vote | accept | Dat Minh Ha | ||
| May 2, 2022 at 19:24 | answer | added | Dat Minh Ha | timeline score: 5 | |
| Jan 18, 2022 at 9:20 | comment | added | Jon Pridham | @Z.M One hopefully final comment. For a $q$-analogue of $\hat{\mathbb{G}}_a$, you work on $\lambda$-rings, starting from free (not cofree) $\lambda$-rings over $\mathbb{Z}[q]$ rather than polynomial rings. As in the second part of Remark 1.4, $\nu^k(a):= [k]_q!(q-1)^k\lambda^k(\frac{a}{q-1})$ gives a $q$-analogue of $a^k$, the recursive formula $\nu^k(a)=\sum_{i>0}(q-1)^{i-1} \lambda^i(a)\frac{[k-1]_q!}{[k-i]_q!}\nu^{k-i}(a) $ showing it has no denominators. Instead of $T^n/n!$, you then want to include $\nu^n(T)/[n]_q!$. | |
| Jan 17, 2022 at 21:19 | comment | added | Jon Pridham | @Z.M I'm well aware of what $W$'s doing - it features heavily in section 3 of my paper (and the big Witt vectors earlier). My point was just that Prop 1.15 is set up to define a functor on $\lambda$- or $\delta$-rings, not ordinary rings, so $W$ isn't needed to make a comparison. (If you do, it sends a flat $\mathbb{Z}[q]$-algebra $A$ to the image of the Teichmueller map $[A]\to W(A)/(q-1)$; you have to modify by powers of $(q-1)$ to get the functor sending $A$ to the image of $[A] \to W_p(A)/(\Psi^p){-1}[p]_q$, and that map's locally surjective. ) | |
| Jan 17, 2022 at 20:19 | comment | added | Z. M | @JonPridham Thanks. For the appearance of $W$, it is used to define gadgets on ordinary rings instead of $\delta$-rings, and this works for prismatic cohomology after fpqc sheafification (Bhatt–Lurie should cover this). | |
| Jan 17, 2022 at 18:32 | comment | added | Jon Pridham | @Z.M I've just deleted my comments on plethories which I added too hastily; none of the rings in the resolution of $q$-de Rham is a plethory, as they aren't closed under the coadditivity map $x \mapsto x\otimes 1 +1\otimes x$. However, the functors there are all defined on $\lambda$- and $\delta$-rings, so (after modifying with powers of $(q-1)$) your last comment roughly holds if you remove $W$. | |
| Jan 13, 2022 at 22:26 | comment | added | Jon Pridham | @Z.M They don't coincide, because Prop 1.15 has extra powers of $(q-1)$ in the denominator. REmoving them needs the Anschuetz-Le Bras paper, which roughly shows that the $\mathbb{Z}_p[x][[q-1]]$-subalgebra generated by the elements $(q-1)^k\lambda^k(\frac{y-x}{q-1})$ is just the $\lambda$-ring generated by $\frac{y^p-x^p}{[p]_q}$. However, Th 2.8 proves an analogue for $X^{q,p}_{strat}$ and Th 3.11 uses the local equivalence $X^{q,p}_{strat} \simeq X^{q,p}$ on perfectoid input to establish a version with $p$-power roots of $q$. | |
| Jan 13, 2022 at 21:42 | comment | added | Z. M | @JonPridham Thanks for the clarifications. I wonder where do you prove that this $X^{q,p}$ is related to the "plethory" given by the subalgebra in Prop 1.15? In view of Bhatt–Lurie (and Drinfeld), given $X^{q,p}$, the functor defined on $\mathbb Z_p[[q-1]]$-algebras should be simply composing the cofree functor, i.e. taking Witt vectors. So which statement in your paper is essentially saying that $\operatorname{cofib}((\Psi^p)^{-1}([p]_qW(R))\to W(R))$, at least up to fpqc sheafification w.r.t. $R$, coincides with the one given in Prop 1.15? | |
| Jan 13, 2022 at 20:34 | comment | added | Z. M | @JonPridham My understanding is that, in Prop 1.15, your variables are of rank 1, and this does not seem to allow us to describe something like "the radical is stable under $q$-divided powers" (if you want to define a $q$-de Rham stack on the site of $\delta$-rings over $\mathbb Z_p[[q-1]]$). | |
| Jan 13, 2022 at 9:36 | comment | added | Jon Pridham | @Z.M In mixed characteristic (so working over $\mathbb{Z}_p$ rather than $\mathbb{Z}$), the divided powers are giving you nilpotence for free, as $n! \to 0$. For $q$-analogues, structure has to go on the whole ring, not just the ideal. You get these sorts of constructions in arxiv.org/abs/1608.07142, in particular Prop 1.15 - if you clear out powers of $(q-1)$ from the denominators in mixed characteristic, it turns out that you get the functor $X^{q,p}$ from Remark 3.15 (via Prop 4.9 of arxiv.org/abs/1907.10530), whose cohomology is essentially $q$-crystalline. | |
| Jan 12, 2022 at 22:14 | comment | added | Z. M | @JonPridham If I understand correctly, in char $p$ and mixed char, one usually does not Hodge-complete, which should be closely related to the nilpotency that you want to impose. On the other hand, this approach has advantage that the prismatic analogue is directly related. In particular, without that, I don't know how to work out a category of "rings with $q$-PD structure" for the stacky approach. | |
| Jan 12, 2022 at 21:18 | comment | added | Jon Pridham | To clarify, if the OP's sole objection to DP structures is their non-uniqueness, then yes, simplicial rings provide a solution by allowing derived quotients, so it doesn't matter that the map $\{\gamma \in I^{\mathbb{N}} ~:~ \binom{m+n}{n}\gamma_{m+n}=\gamma_m\gamma_n\}\to I$ sending $\gamma$ to $\gamma_1$ isn't injective, but I wouldn't regard that as avoiding divided powers. | |
| Jan 12, 2022 at 20:34 | comment | added | Jon Pridham | @Z.M You can get that sort of construction just by putting divided powers into the de Rham stack, though for consistency with characteristic $0$ I'd expect a nilpotence condition on the elements $T^n/n!$ as well (so some sort of pro-object like a power series). Then $\mathbb{G}_a^{\sharp}$ is essentially just parametrising DP structures on the radical. | |
| Jan 12, 2022 at 5:35 | comment | added | Z. M | @JonPridham One could simply work with the category of (animated) rings. The de Rham stack (outside characteristic 0) of $X$ is given by $R\mapsto X(\mathbb G_a^{\operatorname{dR}}(R))$, where $\mathbb G_a^{\operatorname{dR}}:=\operatorname{cofib}(\mathbb G_a^\sharp\to\mathbb G_a)$ and $\mathbb G_a^\sharp=\operatorname{Spec}(\mathbb Z[T,T^2/2!,T^3/3!,\dots])$. See Bhatt's talk youtube.com/watch?v=v2Jfk-NTjp4 | |
| Jan 12, 2022 at 5:27 | comment | added | Z. M | Arithmetic D-modules are different from crystals on the crystalline site, if I understand correctly, and the later does not have six functor formalism. | |
| Sep 28, 2021 at 20:19 | comment | added | Jon Pridham | For a smooth scheme $X$, ind-coherent sheaves are a much less significant ingredient than the stack $X_{dR}$, from section 7 of Simpson's arXiv:alg-geom/9604005v1. If you try to mimic Simpson's construction to get crystalline cohomology, you need to take the DP envelope of the diagonal ideal, and for that to represent a functor, you would have to work on a category of rings with DP structures on their radicals. | |
| Sep 28, 2021 at 19:07 | history | asked | Dat Minh Ha | CC BY-SA 4.0 |