Timeline for Homology of the étale homotopy type
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 21, 2021 at 17:45 | comment | added | Z. M | OK, this is answered here. | |
| Jul 20, 2021 at 10:53 | comment | added | Z. M | Sorry, is there a reason (or better, a reference) for the left adjoint $f_\natural$ being called "relative homology", i.e. how does that coincide with the classical notion of homology? | |
| Mar 17, 2021 at 13:46 | comment | added | D.-C. Cisinski | Ah, I think I follow you now: if we take $\mathbb{F}_l$ coefficients say, what I describe above would fit with the Artin-Mazur kind of construction (literally) whereas the left adjoint in the pro-étale setting would only fit with the actual Artin-Mazur construction, only after solidification (if your guess is correct!). | |
| Mar 17, 2021 at 13:36 | comment | added | Peter Scholze | Right -- this is the left adjoint I was referring to in the beginning of my comments. But if you take the one on general pro-etale $\mathbb Q_\ell$ (or even $\mathbb F_\ell$)-sheaves, it gets more complicated. So you have three options: Restrict to nice $X$ and nice coefficients, and get a good left adjoint; take general $X$ and solid sheaves, and get a good left adjoint; take general $X$ and all pro-etale sheaves, and get a weird left adjoint. | |
| Mar 17, 2021 at 13:32 | comment | added | D.-C. Cisinski | Sorry to insist, but the left adjoint of $f^*$ exists whever $f$ is a morphism of finite presentation with $0$-dimensional codomain, at least if we restrict to constructible $l$-adic sheaves (with $l$ prime to the residue characteristics). This left adjoint sends the constant sheaf to $f_! f^!$ of the constant sheaf. | |
| Mar 17, 2021 at 13:26 | comment | added | Peter Scholze | No, I think it's impossible to control the left adjoint to $f^\ast$ even in the best possible cases, see my edited answer. | |
| Mar 17, 2021 at 13:25 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 17, 2021 at 13:21 | comment | added | D.-C. Cisinski | The solidification is needed only in the absence of finiteness assumptions then, is that what you claim? I would expect that the "classical" left adjoint of $f^*$ is well behaved whenever the formation of $f_*$ is compatible with arbitrary base change: for instance, if $f$ is proper or if the target of $f$ is $0$-dimensional (and $f$ of finite presentation say). | |
| Mar 17, 2021 at 13:14 | comment | added | Peter Scholze | Ah, I see what you say. The problem is that if you apply the left adjoint on the full derived category of pro-etale $\mathbb Q_\ell$-sheaves, I don't think you can compute the answer: You need to solidify to get a sensible answer. On the other hand, you are right that it's not necessary for the comparison! But if you want this notion of homology to agree with the Artin--Mazur one, you need to solidify. | |
| Mar 17, 2021 at 13:11 | comment | added | D.-C. Cisinski | This fits with homology à la Artin-Mazur: they define homology as the pro-group obtained by applying the (left derived) pro-adjoint of the constant sheaf functor. | |
| Mar 17, 2021 at 13:09 | comment | added | D.-C. Cisinski | The pullback functor $\pi^*$ is well defined for $l$-adic sheaves and it has a left adjoint exactly as (in fact because it exsists in the case of) sheaves of anima. I would define homology by applying this left adjoint $\pi_\sharp$. I do not think and finitess assumtion is needed. | |
| Mar 17, 2021 at 12:47 | comment | added | Peter Scholze | I should maybe mention that one advantage of using the condensed anima $\pi_\natural(X)$ in place of the Artin--Mazur pro-homotopy type is that $\pi_\natural(X)$ can see general $\mathbb Q_\ell$-local systems (as in the argument above), which I think is not the case for the latter (whose $\pi_1$ is given by the SGA3 $\pi_1$ which is too small for say nodal curves of positive genus). But then to define the homology of a condensed anima, I think I really need the solid formalism. | |
| Mar 17, 2021 at 12:37 | comment | added | Peter Scholze | On the other hand, I wouldn't be completely sure how to define homology of the pro-etale homotopy type with coefficients. I guess usually one would use coefficients that come from stage in the pro-limit, and then take the projective limit of the homologies? I think this implicit passage to the projective limit is what's mirrored here by the use of solid modules, specifically $\mathbb Z[S]^\blacksquare = \varprojlim_i \mathbb Z[S_i]$ for profinite $S=\varprojlim_i S_i$. | |
| Mar 17, 2021 at 12:35 | comment | added | Peter Scholze | As long as $X$ is nice, you are right, the left adjoint exists on the usual level, but not for general $X$. If $X=\mathrm{Spec} \mathbb Q$ for example, $H^1(X,\mathbb F_2)$ is infinite, and dually $H_1(X,\mathbb F_2)$ ought to be an infinite product of $\mathbb F_2$'s. So you need some solid formalism, I think. | |
| Mar 17, 2021 at 12:30 | comment | added | D.-C. Cisinski | I do like very much everything you do on analytic rings and condensed mathematics, but I have trouble to see why solid modules are necessary here. I mean that the derived category of pro-étale l-adic sheaves, as documented in your joint paper with Bhatt, is sufficient to make sense of the left adjoint of the pullback functor: we can simply apply the $l$-adic version of $\pi_\sharp$ to $\mathbb{L}$ to get homology with coefficients. Could you explain what to expect from the consideration of solid modules everywhere? | |
| Mar 17, 2021 at 12:08 | vote | accept | curious math guy | ||
| Mar 17, 2021 at 10:43 | history | answered | Peter Scholze | CC BY-SA 4.0 |