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started oo-vector bundle on my personal web, following my latest remarks in the thread here on deformation theory.
I do not understand your remark about free abelianization. The smallest abelian category containing the category of finite rank locally free sheaves of O-modules (aka algebraic vector bundles) is the category of coherent sheaves, rather than quasicoherent. The fibered category of quasicoherent sheaves appears rather as the category of qcoh sections of the fibered category of O-modules.
Another thing is that many functors are not defined at the level of coherent sheaves, thus to look at cohomology, injective resolutions etc. one needs to look coherent sheaves in a bigger category of qcoh sheaves.
Oh, so I got this worng. Thansk for pointing it out.
Hm, I think Lurie's theorem says that the oo-category of modules over a simplicial ring is equivalane to the stabilization of the overcategory . This means, I think, that the oo-stack of oo-quasicoherent sheaves as used by Ben-Zvi et al is nothing but the assignment .
This is the main statement that I am after.
But I guess I was too careless with associating this free stabilization with the abelianization that takes one from the ordinary Cat of vector bundles to quasicoherent sheaves.
So could you say this again: can one say in which sense the ordinary category of quasicoherent sheaves is a universal envelope of the category of vector bundles? Is there a good way?
I also think that it is other way around: finite rank locally free sheaves are just very special objects in Qcoh. It is like in affine situation finite rank free modules among all modules. But the fibered category of quasicoherent sheaves of O-modules may be obtained by a procedure from the fibered category of all O-modules. I would be also careful when generalizing from the affine case. For example it is an open problem wheather the category of qcoh sheaves on an arbitrary scheme has enough injectives.
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<p>Thanks a lot for all the comments!</p>
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Vect should start from Ring^op.
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<p>I don't think it should. It should be a contrvariant functor on spaces, hence a covariant functor on rings.</p>
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finite rank locally free sheaves are just very special objects in Qcoh.
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<p>Sure, that's why I made a standout box saying that I use somewhat non-standard terminology. It's just that I find "oo-category of oo-quasi-coherent sheaves", denoted <img src="/extensions/vLaTeX/cache/latex_dc11e2fea52fdc295ffe86cf97326f0f.png" title=" QC(X)" style="vertical-align: -20%;" class="tex" alt=" QC(X)"/> not suggestive enough for such an important basic concept. I would like to write <img src="/extensions/vLaTeX/cache/latex_471faff06932f1061a4f9b6d0e05b11f.png" title="Vect(X)" style="vertical-align: -20%;" class="tex" alt="Vect(X)"/> instead. Maybe it's a bad idea to change notation this way, but I find "quasi-coherent" really a terrible term. It doesn't convey at all what it is about, and that it is about something very fundamental.</p>
<p>So my entry is really to be read as labeled "oo-quasi-coherent sheaves", only that I idosyncratically say "oo-vector bundle" instead. Do you see what I mean? :-)</p>
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Urs: but that's exactly the same thing!
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<p>Yes, exactly. Now, I had thought there was a nice discussion of this in our entry on <a href="https://ncatlab.org/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a>, but checking now I see that there isn't. This needs to be included, now. Thanks for pushing me!</p>
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Well, effectively the statement is in the subsection As sheaves on Aff/X. But the punchline should be highlighted much more...
Renaming quasicoherent modules-sheaves-bundles into vector modules-sheaves'bundles is to my opinion bad. Coherence and quasicoherence are very intuitive and good terms. You see in smooth setup you can have bump functions, they appear when you go local, they are LOCAL surprises, not predicted from nearby fibers. On the other hand, coherence and quasicoherence have an essential ingredient that locally the localization is a procedure, not a passage to unknown completion with unknown new bumps. It is predictable result of a functor, which is the localization functor. All fibers over pointswork together in a coherent fashion, no democratic jumps around. Coherent dictature is the law!
Moreover, the maps between the fibers of the corresponding fibered category (here fiber is in a different sense than above) are isomorphisms, which could be considered as coherences also in the sense of the category theory. In fact, if we take a flat topology and look at the cover corresponding to the principal bundle over the base space, then the equivariant descent gives exactly the cocycle for the equivariance. Both are special cases of the same phenomenon of strictification of a fibered category.
I do not know why do you insist on a stabilization at the place where you discuss quasicoherence ? I mean abelianess is obtained before, at the level of O-modules. So you first have to get somehow by abstract nonsense (from your codomain fibration) the analogue of a category of O-modules. The second step is just the strictification to get from the fibered category of O-modules the fib category of qcoh modules. I can suspect something like abelianization or stabilization in the first step. The second is purely of quasicoherator nature (the functor which is outlines in Orlov's article by the way and before at places like EGA). I keep telling you this for last two years but it never develops into conversation, as you forget that there is first step and want to make both steps in one, and then I can not help you as I do not see them as one.
An important thing to have in mind is that the isomorphism with the localization is true only along affine morphisms. In the treatment of affine schemes this difference is NOT seen as EVERY morphisms among affine schemes is affine (this is a fundamental property of affiness in any known setup), and affine schemes are just affine morphisms over Spec(Z), hence the conclusion. On the other hand, if you look at nonaffine schemes, then you have to look at which morphisms are affine to have the localization property.
<p>Thanks again for all the comments. I went offline yesterday before I could reply to them.</p>
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I keep telling you this for last two years but it never develops into conversation, as you forget that there is first step and want to make both steps in one, and then I can not help you as I do not see them as one.
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<p>My apologies, Zoran, for being dense. I suppose I never understood what you were telling me.</p>
<p>So let's sort this out in detail for plain QC sheaves first, before we continue talking about the oo-categorification further.</p>
<p>I just made an attempt to add most of what I understand about QC sheaves to the entry <a href="https://ncatlab.org/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a>. Please see the announcement of the additions I made at <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=608&page=1#Item_1">quasicoherent sheaf (forum)</a>. Let me know what you think.</p>
<p>Also Domenico, please have a look at the entry on QC sheaves now. It tries to make very explicit the way QC sheaves may be thought of from the <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a> as simply homs into the the stack of modules.</p>
I will change the notation. I mean calling f* is fit to inverse image functor, not the map of rings. Map in the category of rings is just opposite to the map in the category of spectra. So I would either call f map of rings and Spec f map of spaces, or call f map of rings and f circ or f op the map of spectra.
Thanks, Zoran and Domenico, for fixing things!
The strictification indirect way: take any category over a category. The category opposite to the category of it sections is the category of modules in it. Take any section and from it make a Grothendieck construction. Then take a section of the latter, what you obtain is a quasicoherent section. Quasicoherent modules are objects of the opposite category. They can be also expressed via cartesian sections from a trivial fibration.
this morning I have worked on the entry and basically rewritten it, taking into account all the discussion we had, and also polishing this and that. I'd be very grateful if you could have another look
There are various immedie things to be done from here on. I indicate one in the new section on Associated oo-vector bundles.
I have forwarded this discussion to the blog, here
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<p>right, associating vector bundles is just composition of cocycles, which is just composition of morphisms in the oo-topos, yes.</p>
<p>Concerning the other aspect you mention: it is true that with the tangent oo-category approach we don't get <em>just</em> the classical theory. But we get something into which the classical theory embeds.</p>
<p>When we do this over <img src="/extensions/vLaTeX/cache/latex_34d0a67079d60aa8f0cd6ea8ad81abfa.png" title=" C = SRing^{op} " style="vertical-align: -20%;" class="tex" alt=" C = SRing^{op} "/> then, as the example 8.6 in "Stable oo-Categories" shows, <img src="/extensions/vLaTeX/cache/latex_03cf21e0dd38b8f3d0170d22802c53f4.png" title=" \infty Mod(R) " style="vertical-align: -20%;" class="tex" alt=" \infty Mod(R) "/> over an ordinary ring <img src="/extensions/vLaTeX/cache/latex_07473a8bf5ab2af6f9d299667342c3db.png" title=" R" style="vertical-align: -20%;" class="tex" alt=" R"/> is the oo-category whose homotopy category is the derived category of <img src="/extensions/vLaTeX/cache/latex_07473a8bf5ab2af6f9d299667342c3db.png" title=" R" style="vertical-align: -20%;" class="tex" alt=" R"/> -modules, yes. But ordinary modules still sit inside there.</p>
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here one can come back to one of the original issues between urs and zoran: what is Qcoh(X) to Vect(X)? we know that it is not abelianization, since Coh(X) is enough.
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<p>right, we need to say this precisely, eventually.</p>
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do coherent sheaves (not quasi-) form a bifibration? I.e. are they closed under push and pull?
they're not, right? already due to the finiteness condition.
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<p>Okay. Do we know if <img src="/extensions/vLaTeX/cache/latex_bab8fa3b10abe063ced5002f58a8b2a4.png" title=" QCOh(X)" style="vertical-align: -20%;" class="tex" alt=" QCOh(X)"/> is <em>exactly</em> what we get when we start with coherent sheaves and then complete under push-forward?</p>
<p>(I have to say I'd be interested, but I don't really care so much. :-) The general nonsense def of QCoh(X) is so good, that I don't see a pressing need to find another universal characterization.)</p>
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I've the doubt we may be unable to distinguish between two stably equivalent modules. but that's my unexperience in the field; I'll try to think more on this.
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<p>Right, yes. I am not sure. Here, too, I don't really care too much. we know the right picture. What we are asking here is to which extent the wrong picture is preserved by the right picture. :-) Would be good to nail it down, but I am not losing sleep over this.</p>
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started a subsection on flat oo-vector bundles and D-modules
There is a wording ordinary quasicoherent sheaves in the entry. As if D-modules were not ordinary quasicoherent sheaves, Full name is quasicoherent sheaves of O-modules in the first case and in the secondc case are the quasicoherent sheaves of D-modules. They are both quasicoherent sheaves in most ordinary sense, the difference is just weather of O-modules or of D-modules. The fact that by abuse of language geometers do not say qcoh sheaves of D-modules but only D-modules and not quasicoherent sheaves of O-modules but only qcoh sheaves or only qcoh modules is a different thing,
One should see if the qcoh 2-sheaves in the sense recently introduced by Toen in the arxiv preprint fits in this framework.
by "ordinary" I mean "1-categorical" as opposed to the oo-version. I have now tried to clarify this.
By the way, what's a good reference that we could cite at D-module for the equivalence of D-modules with qcoh sheaves on the deRham space?
I don't know, this is another point of view, equating D-modules with qcoh O-modules for another space, which I was not talking about. I put the reference of Beilinson-Bernstein on proof of Jantzen conjecture which is the main classical reference on D-affinity and algebraic D-geometry. I added some nlab links to D-module and the elementary textbook from Coutinho. I should say that I like your treatment of the interpretation via de Rham space.
For usual vector bundles, I put some pain into producing a reference section in vector bundle with main monographs having lots of material on basic of the vector bundles.
Wait a second there is a strange section titled “Virtual vector bundles” in vector bundle – but instead of virtual vector bundles (like in Karoubi’s book: allowing vector bundles of different dimension/rank on different connected components of the base and also allowing negatives of vector bundles), there is a discussion on $Z_2$ gradings which is related but not exactly what I recall from Karoubi (the pairs mean Grothendieck construction so one has negatives, but not the case of only locally constant rank) and there is also strange remark about “vectorial vector bundles” (??) there which I can not relate to virtual v.b.
the notion of vectorial bundle is used by Gomi to model twisted K-theory using geometric objects. Have a look at the entry. These cocycles are not exactly the same as those of Karoubi, but their equivalence classes do model the same K-theory.
Oh I remember looking at the paper, but not the expression. I should have followed the link.
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