Timeline for A few general questions about pre-sheaves and sheaves
Current License: CC BY-SA 4.0
33 events
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| Jun 14, 2018 at 12:05 | comment | added | BrianT | In any case, I’m happy my question has raised such an interesting discussion. Actually, I asked that because I’m trying to understand a result from a paper in which one computes the cohomology of a sheaf (actually a presheaf) knowing that, on a certain open cover, the presheaf looks like a direct sum of presheaves which are $\mathbb{C}$ or $0$ depending on the open set. The result should be that the cohomology is the direct sum of the cohomology of the “constant or 0” sheaves. If it interests someone I can ask this more specific question. | |
| Jun 14, 2018 at 11:40 | comment | added | Maxime Ramzi | @Malkoun you are right. I'll try to edit my answer in a reasonable way | |
| Jun 14, 2018 at 11:30 | comment | added | Malkoun | I don't know. I wish we could clean up this page, and remove many of the comments (including mine of course), to make it more readable. | |
| Jun 14, 2018 at 11:26 | comment | added | Malkoun | Max, I guess the statement you are making, namely that the direct sum of the sheafifications is the sheafification of the direct sum, but with the direct sum notion being taken in the corresponding category (presheaves/sheaves) could be confusing. In some sense, it can be taken as the definition of direct sums in the category of sheaves. So it can be viewed as a definition. | |
| Jun 14, 2018 at 10:28 | comment | added | Maxime Ramzi | I don't see how to get a B-sheaf from the $F(U_i)$ without constructing a presheaf then sheafifying it to finally take the B-restriction. But again, I'm not sure | |
| Jun 14, 2018 at 10:05 | comment | added | BrianT | But can’t we construct a B-sheaf from the data $F(U_i)$, and then define $G$ as the sheaf associated to this B-sheaf ? | |
| Jun 14, 2018 at 10:03 | comment | added | Maxime Ramzi | But the $F(U_i)$ do not form a B-sheaf; so who knows what $G$ might look like ? It need not be a sheaf at all, so you still have to sheafify it; and I don't think that its sheafification will look like $F$'s. | |
| Jun 14, 2018 at 9:55 | comment | added | BrianT | I don’t understand where the problem is with what you said. If we construct the sheaf $G$ from the B-sheaf obtained via the data $F(U_i)$, then $G$ need not coincide with $F$ on $U_i$, but with the sheafification of $F$. I may be totally confused, but isn’t it what it should be (the original claim was: take the restriction of a presheaf to a basis, and construct a B-sheaf; then the sheaf obtained from this B-sheaf is the sheafification of the presheaf) ? | |
| Jun 14, 2018 at 9:31 | comment | added | Maxime Ramzi | I don't know the answer to your last question. I would say "no", because as I said the data of a presheaf on a basis is not enough to recover the presheaf, and probably not enough to recover the sheafification of the presheaf. What you can do is define a certain sheaf $G$ from the values $F(U_i)$, but if $F$ is bad enough, $G$ and $F$ won't even coincide on the $U_i$'s (going from a B-sheaf to a sheaf uses the gluing axiom in an essential manner) | |
| Jun 14, 2018 at 9:22 | comment | added | BrianT | Max: ok, thanks a lot, it’s clearer now. About the open cover thing, suppose that the open cover is a basis, and that we know what the presheaf looks like above this basis. Can’t we sheafifise « above the basis » to get a B-sheaf (I think this is the terminology for sheaves over a basis), and then reconstruct the whole sheafification of the original sheaf from it ? | |
| Jun 14, 2018 at 8:45 | comment | added | Maxime Ramzi | (cont) But that's not the case in $\mathbf{Sh}(X)$: the previous formula does not hold. However, there is still a notion of coproduct in $\mathbf{Sh}(X)$, and sheafification transforms the previous "naive" coproduct in $\mathbf{Psh}(X)$ into the "weird" coproduct in $\mathbf{Sh}(X)$: one needs to be careful with what $\bigoplus$ means. In other words, the coproduct of sheaves,as computed in $\mathbf{Psh}(X)$, that is, pointwise, is not the coproduct of sheaves as computed in $\mathbf{Sh}(X)$ (for this to make sense you have to know what "coproduct" means in a category) | |
| Jun 14, 2018 at 8:42 | comment | added | Maxime Ramzi | Well you can't recover $F$ from its values on a basis because $F$ is only a presheaf (if it were a sheaf, we could recover its values from those on a basis); and I think (here I'm not so sure but almost) that we cannot recover the sheafification either. For my edit, the point is that coproducts in the category $\mathbf{Sh}(X)$ might look very different from those in $\mathbf{Psh}(X)$. In the latter (which is just a functor category) colimits (in particular coproducts) are computed pointwise, that is here $(\bigoplus_l F^l)(U) = \bigoplus_l F^l(U)$ (cont) | |
| Jun 14, 2018 at 8:26 | comment | added | BrianT | Max : thanks for your answer. Does this help if the open cover is a basis for the topology ? Also, could I didn’t understand very well your edit. Could you explain it ? Thanks a lot again | |
| Jun 14, 2018 at 7:41 | comment | added | Maxime Ramzi | BrianT : you cannot sheafify a "restriction to an open cover". Sheafifying $F$ uses the whole $F$. I gave you the example where the open cover is $X$ for instance and there are no global sections : you cannot recover $F$ from that. | |
| Jun 14, 2018 at 7:38 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Jun 14, 2018 at 7:35 | comment | added | Maxime Ramzi | @Malkoun : I am so sorry; I made a huge mistake... Indeed; let me edit my answer and try to answer all the last points made here | |
| Jun 14, 2018 at 7:01 | comment | added | BrianT | That’s weird. It seemed that the argument above was correct. Also, I really need to understand this this about open covers... | |
| Jun 14, 2018 at 6:52 | comment | added | Malkoun | We need to fix this discussion! The direct sum of sheaves, as a presheaf, is not a sheaf in general. There are some posts such as this one math.stackexchange.com/questions/193816/…, which contain counterexamples. | |
| Jun 13, 2018 at 22:18 | comment | added | BrianT | Would it help if the open cover was a basis for the topology of X ? | |
| Jun 13, 2018 at 22:10 | comment | added | BrianT | @Max : I may be tired too, so correct me if I’m wrong. I think that in order to answer my last question, one needs to be able to do the following : given the restriction of a presheaf to an open cover, sheafifising this restriction to get a sheaf on this open cover; then recovering the whole sheaf by extending (somehow) to any open set, and finally prove that the sheaf so obtained is the sheafification of the original presheaf. Do you think this is possible ? | |
| Jun 13, 2018 at 21:58 | comment | added | Malkoun | @Max, I see. Thank you for explaining this carefully. I have also learned today about the "co-continuity of left-adjoint functors". I have not seen the proof, but it should not be too difficult, I hope. Nice discussion BrianT and Max. | |
| Jun 13, 2018 at 21:50 | comment | added | Maxime Ramzi | @Malkoun : your second-to-last comment was however relevant ! Consider indeed a direct sum of sheaves $F=\bigoplus_l F^l$. Then its sheafification is $\bigoplus_l L(F^l)\simeq \bigoplus F^l$ (the $F^l$ are sheaves) and so it's $F$ : $F$ is a sheaf. | |
| Jun 13, 2018 at 21:47 | comment | added | Malkoun | I deleted my last two comments. I am tired, so forgive me for my "concrete nonsense". | |
| Jun 13, 2018 at 21:39 | comment | added | Maxime Ramzi | @Malkoun : your last statement is not correct ! Consider the example I gave just above; where $F(X) = F(X)\oplus F(X)$ but $F$ need not be isomorphic to $F\oplus F$ | |
| Jun 13, 2018 at 21:38 | comment | added | Maxime Ramzi | To answer your last question : indeed my answer doesn't take that into question (I was thinking it was a different question, i.e. in general is the sheafification of a direct sum the direct sum of sheafifications ). In your general situation where you compare two presheaves on some open cover, I don't know that much can be said... Suppose for instance $I$ has one element and $U_i =X$ and suppose there are no nontrivial global sections; then you can't say anything about the sheafification .. | |
| Jun 13, 2018 at 21:19 | comment | added | BrianT | There is also one last thing: I said in my question that the pre-sheaf was a direct sum a priori only on a fixed open cover (which is not necessarily a basis of topology). How do I get the result for the sheafifications on any open sets ? | |
| Jun 13, 2018 at 21:02 | comment | added | BrianT | Last question, is it possible tu apply the same argument to show that a direct sum of sheaves is a sheaf ? | |
| Jun 13, 2018 at 20:53 | comment | added | BrianT | Thanks a lot Max and Malkoun, I’ll try to learn the basics to understand your argument which seems to be almost straightforward as soon as you know the definitions and concepts. | |
| Jun 13, 2018 at 19:24 | comment | added | Maxime Ramzi | @Malkoun unfortunately I don't personnally know any such references, but looking around a bit on MO gives the following questions (and answers !) : mathoverflow.net/questions/216215/… and mathoverflow.net/questions/212548/… | |
| Jun 13, 2018 at 18:57 | comment | added | Malkoun | I guess part of me likes to keep things very concrete and as explicit as possible, but this makes what I write quite difficult to read, and I get bogged down in detail. I do appreciate though the power of looking at maps rather than objects, well exploited by Grothendieck. I do appreciate the power of abstract nonsense, but I tend not to think that way. Do you have any good reference for category theory for geometers, with an emphasis on its connections with algebraic geometry (as in "Categories for the Working Geometer"?). | |
| Jun 13, 2018 at 18:48 | comment | added | Maxime Ramzi | @Malkoun : once the right notions are in place, you can let category theory guide you gently ;-) (for question 2.; neither argument I'm using is complicated once you know the definitions of the words used) | |
| Jun 13, 2018 at 18:44 | comment | added | Malkoun | nice. I need to learn more catergory theory, it helps make the arguments cleaner. My answer for question 2 was a bit clumsy. I am glad you confirmed, via a more "high-tech" point of view. | |
| Jun 13, 2018 at 18:41 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |