Timeline for Etale cohomology of regular local rings
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27 events
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| Dec 20, 2010 at 21:56 | comment | added | Bhargav | I think what Dustin's referring to is Artin's theorem that any smooth pointed variety (X,x) admits a Zariski neighbourhood (U,x) which is the K(G,1) in the following sense: for every lcc sheaf M on U, the natural map $H^*(\pi_1(U),M) \to H^*(U,M)$ is an isomorphism, i.e., that $M \simeq R F_* F^* M$ in the notation used above; this does not mean that $F_*$ is exact. | |
| Nov 4, 2010 at 23:07 | comment | added | Leonid Positselski | My understanding of Clausen's comments is limited; I certainly cannot add anything to them. However, I think I now see that the functor $F_*$ is indeed not exact in the nonhenselian case. It suffices to take $R$ to be the local ring of a point on a curve over an algebraically closed field of characteristic zero. Then $\mathbb{Z}/n\to i_*\mathbb{Z}/n$ is an epimorphism of etale sheaves. The functor $F_*$ transforms it to the monomorphism $\mathbb{Z}/n\to \mathbb{Z}/n(G)$, where $\mathbb{Z}/n(G)$ is the group of locally constant $\mathbb{Z}/n$-valued functions on $G=\pi_1^{et}(R)$. | |
| Nov 4, 2010 at 16:06 | comment | added | BCnrd | Leonid, now I see what you mean. I wasn't fully using the henselian condition. OK, so I agree with the preservation of injectivity, and hence the compatibility of cohomology on lcc objects. Very good. I still wonder about Clausen's optimism without the henselian condition. | |
| Nov 4, 2010 at 12:22 | comment | added | Leonid Positselski | All the three functors F^*, F_=i^, i_* appear to be exact. Exactness of the functor F^* implies preservation of injectivity under the functor F_. Since the functor F^ is fully faithful, one has F_F^*=Id. This should be sufficient to conclude that F^ is fully faithful on the level of Ext's. | |
| Nov 4, 2010 at 12:17 | comment | added | Leonid Positselski | Dear BCnrd, could you point out the problem that you see with my Yoneda Ext argument for higher Ext's? Concerning the preservation of injectivity, let F denote the map from the etale site of Spec R to the site of finite etale morphisms into Spec R (i.e., finite G-sets), as discussed in my original question. Let i denote the embedding of the closed point into Spec R. Then the functor F_* can be identified with i^*. The functor F^* is left adjoint to F_=i^ and the functor i_* is right adjoint to F_=i^. The functor F^* is the embedding of lcc sheaves into all etale sheaves. (cont-d) | |
| Nov 4, 2010 at 2:44 | comment | added | BCnrd | Dear Leonid: I understand your argument for degree-1 Ext, but for higher Ext's I don't understand. Also, since an lcc sheaf on the henselian local scheme is not a pushforward from the closed point (when dimension positive), the argument with preservation of injectives seems to be in the wrong direction. (It would seem one wants that the stalk at closed point of an injective is injective, but that seems doubtful, as adjunction appears to be going the wrong way.) Maybe I am missing something? | |
| Nov 3, 2010 at 20:42 | comment | added | Leonid Positselski | Dear BCnrd: As I see it, the inverse and direct image functors, being exact, induce maps of the Yoneda Ext, and the composition in one direction is the identity because the composition of functors is. Now if I start with two G-modules and consider their Yoneda extension in the category of sheaves, and apply the direct and inverse image functors to it, then the extension I obtain is connected with the original one by the adjunction morphism, hence equivalent to it. Equivalently, one can say that the direct image preserves injectivity since it is right adjoint to an exact functor. | |
| Nov 3, 2010 at 19:59 | comment | added | BCnrd | Dear Leonid: Leonid, can you flesh out the comparison of Ext's more fully? It sounds like one is using a lemma about preservation of "injectivity" of objects, or something like that. Since in the henselian case the etale fundamental group is the Galois group of the residue field, comparison of cohomology on the henselian local scheme with that of the closed point (for suitable coefficient sheaves) would do the job if correct, but I don't see offhand what the full argument is (though it may be easy; I haven't had time to think it through). | |
| Nov 3, 2010 at 17:42 | comment | added | Leonid Positselski | BCnrd: I think here is the proof in the henselian case. The direct image functor is exact, since it can be identified with the inverse image for the embedding of the closed point. Now given an abelian category with a full abelian subcategory and a functor adjoint to its embedding, assuming that both the embedding and the adjoint functor are exact, the groups Ext in both categories are the same. | |
| Nov 3, 2010 at 16:44 | comment | added | Leonid Positselski | BCnrd: The idea is that any etale scheme over a henselian local scheme whose image contains the closed point has a connected component that is finite over the base, is that right? | |
| Nov 3, 2010 at 16:39 | comment | added | Dustin Clausen | Leonid: I'd imagine it's not important! I was just citing the version of Artin's theorem I remember from SGA, since I'm not comfortable enough with the details to confidently state it more generally. But yes, I feel that perhaps any Noetherian local ring is a K(pi,1). | |
| Nov 3, 2010 at 16:33 | comment | added | Leonid Positselski | Dustin: why is it important that the field be algebraically closed? The field itself is a K(pi,1) by definition, I would think. | |
| Nov 3, 2010 at 16:23 | comment | added | Dustin Clausen | (actually smooth might be unnecessary in the above) | |
| Nov 3, 2010 at 16:20 | comment | added | Dustin Clausen | Leonid: Artin's theorem says that if you've got a smooth variety over an algebraically closed field, then there is a fundamental system of neighborhoods of any point consisting of K(pi,1)'s (in fact, fibered iteratively in hyperbolic curves). In your case you might try proving that Spec(R) is a filtered limit of hyperbolic curves over another such R' of lower Krull dimension. | |
| Nov 3, 2010 at 16:07 | comment | added | Leonid Positselski | @Dustin: That's what I wanted to know. Is Spec(R) an etale K(pi,1), really? What is Artin's theorem on good covers? | |
| Nov 3, 2010 at 15:57 | comment | added | Dustin Clausen | corresponding G-module because, by Artin's theorem on good covers, such a Spec(R) will be a K(pi,1) (at least in the geometric case). | |
| Nov 3, 2010 at 15:57 | comment | added | Dustin Clausen | Perhaps to summarize the discussion: finite G-modules will embed fully faithfully into etale sheaves on Spec(R), the essential image being those etale sheaves which are locally finite constant. These are not all as one can have, e.g., the pushforward of the constant sheaf from the complement of the closed point, which will be constructible but not constant. For your concrete question I will disagree with the previous commenters and say that my guess is "yes": the etale cohomology of Spec(R) with coefficients in such a locally finite constant sheaf will equal the cohomology of the [cont'd] | |
| Nov 3, 2010 at 15:49 | comment | added | BCnrd | Perhaps one can use cohomological descent with hypercovers to prove an affirmative result in the henselian case (using that in such cases, (i) every finite etale cover is itself a finite disjoint union of henselian local schemes, and (ii) the cofinality feature in the henselian case, as noted in my first comment)? | |
| Nov 3, 2010 at 15:46 | comment | added | BCnrd | Oops, I see that Keerthi posted a similar comment as I was writing mine. I will leave it up since the part about degree-1 wasn't mentioned in his comment. | |
| Nov 3, 2010 at 15:45 | comment | added | BCnrd | For degree-1 cohomology everything is fine since one can identify the cohomologies with torsors and on the geometric side such torsors are automatically represented by finite etale schemes over the base. This is likely in Milne's book in the discussion of torsors. To have a chance at an affirmative answer in higher degree, you should assume that $R$ is henselian (so not the algebraic local ring at a point on a positive-dimensional variety over a field), as then finite etale covers are cofinal among all etale covers. After all, $G$ only "knows" about the finite etale covers. | |
| Nov 3, 2010 at 15:39 | comment | added | Keerthi Madapusi | I think the answer to your concrete question will be 'yes' only if you assume the ring to be henselian (which is unlikely for the cases you seem to require). Otherwise, the topology of finite etale covers (i.e. the one corresponding to the site of $G$-sets) is a lot coarser than the topology of general etale covers, and the direct image functor from the latter to the former can have non-trivial cohomology. | |
| Nov 3, 2010 at 15:21 | comment | added | Sebastian Petersen | Right of course - sorry | |
| Nov 3, 2010 at 15:08 | comment | added | Leonid Positselski | I think for $X=\mathbb P^1$ over an algebraically closed field of characteristic 0 one has $H^2_{et}(X,\mathbb Z/n)=\mathbb Z/n$. | |
| Nov 3, 2010 at 15:04 | comment | added | Sebastian Petersen | Why? If $X$ is the projective line over an algebraically closed field of characteristic zero, then $H^1(X, {\mathbb Z}/n)=Hom(\pi_1(X), \Zz/n)=0$. Maybe I do not understand what was meant. | |
| Nov 3, 2010 at 14:37 | comment | added | Leonid Positselski | Take $X=\mathbb P^1$ over an algebraically closed field of characteristic 0. Then the etale fundamental group of $X$ is trivial, but the etale cohomology with finite constant coefficients isn't. So the answer to my last question is negative in this case, and so is the answer to the question about (what I call) the direct image functor being exact. | |
| Nov 3, 2010 at 14:13 | comment | added | Kevin Buzzard | Does Appendix A of Freitag-Kiehl "etale cohomology and the Weil conjecture", or Milne's "etale cohomology", answer your questions? I am thinking that $R$ being a regular local ring might be a red herring. The constructions you allude to above give an equiv of abelian cats between lcc sheaves of ab gps on $X$ and continuous $\pi_1(X)$-modules for $X$ an arbitrary connected Noetherian (and perhaps even that isn't necessary but I've never thought about the non-Noeth case) scheme. | |
| Nov 3, 2010 at 14:02 | history | asked | Leonid Positselski | CC BY-SA 2.5 |