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Taking Hom category is Yoneda strictification, and as I pointed out many time (and emphasised in Rosenberg-Kontsevich), this is taking quasicoherent sections of a fibered category of O-modules. What you describe (Kan extension) is a standard way to extend a stack to the category of presheaves, though I may not understand some aspects of your approach. For example, I do not see why do you insist on "bifibration" if the only thing used in such constructions is a fibered category structure (I also find it confusing to me talking "classifying" in this context for a pseudofunctor f*). See the section below in quasicoherent sheaf which quotes
The fibered category of O X-modules can be replaced by a more general fibered category. Then the category of quasicoherent modules in this fibered category is the category opposite to the category of cartesian sections of ...(I do not see who to cut and paste from nlab to see the special symbols!!). This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks (dvi, ps).
If one considers a ring as a one object additive category, instead of Yoneda one can consider a similar functor explained in MacLane-Moerdijk book in the treatment of the tensor products in the chapter on geometric morphism (one of the exercises). In infinitedimensional context (ind-schemes) there are two different definition of qcoh sheaves, one agrees with this one, while another is more sensible for infinitedimensional schemes.
The category of R-modules is the category of all sections of Mod \to Ring over R.
Well but this is NOT the category of O-modules. This input is already strictified. When you do Kan extension you are simultaneously doing extension to a bigger category and strictification. The starting category over a category does not need even to be fibered, this follows in general.
Please do not use term "classifies" for what is traditionally called cleavage, it confuses me, each time (as classifying has so many other meanings which I do use). I am not saying that there is an error, but I think you chose a complicated way using all these end formulas to say a special case of something what is much simpler in fibered category setup.
Look at page 1 and page 4 of the Kontsevich-Rosenberg paper Also look at 1.1.5 how one gets the special case inducing qcoh modules one the category of presheaves, which corresponds to your case. Quasicoheretor which I mentioned yeterday is the right adjoint from 1.1.3.
o you are tinking of some fibration such that its "quasicoherent sections" are O-modules.
Urs, no. Sheaves of O-modules are not necessarily quasicoherent. They can have unpredictable bumps, as I explained before. Quasicoherent sheaves on schemes is just a short for "quasicoherent sheaves of O-modules". Now there is a forgetful functor from quasicoherent sheaves of O-modules to just O-modules which has a right adjoint, the quasicoheretor functor, which is well explained in Orlov's paper listed in the nlab article.
A cleavage of a fibered category is A CHOICE of inverse image pseudofunctor, given a fibered category, it is not unique and we need axiom of choice to find it. This is definitely not Grothendieck construction, the Grothendieck construction is the inverse: getting from a pseudofunctor to a fibered category. One can start with a category over a category, but such that it is not fibered. From that one can make a pseudofunctor as well, and then perform the Grothendieck construction, the composition is the big Lim (= bicategorical colim) of the original category of a category as a functor. This is a strictification equal to taking opposite of the category of qcoh sections. A third related construction is Street's first strictification which starts with a pseudofunctor but produces strict functor (thus one has a splitting instead of just a cleavage).
Yet another warning: the idea section starts with a notion of qcoh sheaf on a locally ringed space. Later at some point somebody wrote that this can be interpreted via sheaf theortical nonsense, on presheaves of sets etc. This is true ONLY if the locally ringed space is locally affine and the topology is of rather special type, generated by affine maps, otherwise we can not view it as a sheaf on the affines in a subcanoncial topology.
In infinite-categorical situation one could take again qcoh sections for any left fibration of quasicategories what is more general than taking homs in special kind of objects which you call QC.
<div>
<blockquote>
Yet another warning: the idea section starts with a notion of qcoh sheaf on a locally ringed space. Later at some point somebody wrote that this can be interpreted via sheaf theortical nonsense, on presheaves of sets etc. This is true ONLY if the locally ringed space is locally affine and the topology is of rather special type,
</blockquote>
<p>So the statement in the section</p>
<p><a href="http://ncatlab.org/nlab/show/quasicoherent%20sheaf#AsSheaves">As sheaves over Aff/X</a></p>
<p>needs to state more conditions on <img src="/extensions/vLaTeX/cache/latex_04d0eb37f084cb02c47fa2a271035fef.png" title="X" style="vertical-align: -20%;" class="tex" alt="X"/> and on the topology?</p>
<p>Zoran, could you just go ahead and put into the entry precisely the statements that you say are needed. Somehow we keep talking past each other and maybe it's quicker if you write down the right version than trying to tell me to write down what you think needs to be written. :-)</p>
<p>After you've done it I look at it.</p>
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No, I won't do much now. I left more extensive changes for some time in future. I am already wasting too much time for nlab, and now it is much harder when there are so many additions to the entry. But if you read just 2/3 pages of Orlov and Konstevich/Rosenberg we would save all the effort we had in discussion. On Aff/X the thing is OK if Aff is what it is and the topology is Zariski. Kontsevich-Rosenberg ad other references are more careful about the topology. Note that the usual quasicoherent presheaves are automatically shevaes in Zariski case. As we talked when we talked about descent, quasicoherent presheaves KNOW what is the underlying maximal topology in a sense.
Okay. I should make clear that I haven't found the time and energy to look into the classical theory of quaasicoherent sheaves on schemes much. I know you provided the references, and I thank you for that, but I am already doing too many things at once.
All I was focusing on here is the statement that the definition "as sheaves on Aff/X" as stated currently in the entry, characterizing pre(!)sheaves on Aff/X with a certain property of their restriction morphisms is equivalent -- purely by abstract nonsense -- to the definition in terms of morphisms of presheaves .
I mentioned that this construction is a right Kan extension only for the record, it doesn't really matter much so far. Also that the definition by the hom of stacks exists naturally is clear. The point under discussion is how exactly this does indeed capture what is described by other means in the literature.
but, wait. take paragraph 1.1.5 in Kontsevich Roserberg.
They say that their definition of the cat of quasicoherent sheaves on a presheaf X on a site E with respect to fibration is the cat of cartesian functors over .
Isn't that the same as the cat of morphisms , with the pseudofunctor corresponding to under the Grothendieck construction?
Mike? Todd?
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<p>I wrote:</p>
<blockquote>
Isn't that the same
</blockquote>
<p>Sure it is.</p>
<p>Somebody please give me a sanity check:</p>
<p>Paragraph 1.1.5 in <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=2333">Kontsevich-Rosenberg</a> is nothing, I think, but the category of pseudofunctors that I am talking about -- but described under the equivalence given by the Grothendieck construction in terms of fibered categories:</p>
<ul>
<li><p>their <img src="/extensions/vLaTeX/cache/latex_62ee0dd9f4d9bb034c4b5a0e3fe8e324.png" title="E" style="vertical-align: -20%;" class="tex" alt="E"/> is <img src="/extensions/vLaTeX/cache/latex_b111c2e8748afb8b7665b27bea395a99.png" title=" E = Ring^{op} " style="vertical-align: -20%;" class="tex" alt=" E = Ring^{op} "/> for us</p></li>
<li><p>so their <img src="/extensions/vLaTeX/cache/latex_89eabda702ae2421acfbfda451243ea5.png" title=" E/X " style="vertical-align: -20%;" class="tex" alt=" E/X "/> is nothing but the <a href="https://ncatlab.org/nlab/show/category+of+elements">category of elements</a> of the presheaf <img src="/extensions/vLaTeX/cache/latex_affd89a8cd39b956ec4d2f2a51464690.png" title=" X : Ring \to Set" style="vertical-align: -20%;" class="tex" alt=" X : Ring \to Set"/>, hence the image under the <a href="https://ncatlab.org/nlab/show/Grothendieck+construction">Grothendieck construction</a> of <img src="/extensions/vLaTeX/cache/latex_5b33586fe138ebca0c30bd0e3643f6b1.png" title=" X" style="vertical-align: -20%;" class="tex" alt=" X"/></p></li>
<li><p>their <img src="/extensions/vLaTeX/cache/latex_3881474e0d8aa3b215bf05ddb6b75b5c.png" title="F \to E" style="vertical-align: -20%;" class="tex" alt="F \to E"/> is our <img src="/extensions/vLaTeX/cache/latex_2c6cb3c76d33dbbc62e82ff8c6d62089.png" title=" Mod \to Ring" style="vertical-align: -20%;" class="tex" alt=" Mod \to Ring"/>. By definition, what I wrote as <img src="/extensions/vLaTeX/cache/latex_bc5fe60f38700778e321eb1e23186169.png" title=" QC : Ring \to Cat" style="vertical-align: -20%;" class="tex" alt=" QC : Ring \to Cat"/> is the pseudofunctor corresponding to this under the Grothendieck construction</p></li>
<li><p>they say the cat of quasicoherent sheaves on <img src="/extensions/vLaTeX/cache/latex_5b33586fe138ebca0c30bd0e3643f6b1.png" title=" X" style="vertical-align: -20%;" class="tex" alt=" X"/> is the category of cartesian morphisms from <img src="/extensions/vLaTeX/cache/latex_64f37e308576f3303a4923fc1ee38893.png" title=" X/E" style="vertical-align: -20%;" class="tex" alt=" X/E"/> to <img src="/extensions/vLaTeX/cache/latex_5ec3dcd4fa96727f6008946ebbeebfa2.png" title="F" style="vertical-align: -20%;" class="tex" alt="F"/> over <img src="/extensions/vLaTeX/cache/latex_62ee0dd9f4d9bb034c4b5a0e3fe8e324.png" title="E" style="vertical-align: -20%;" class="tex" alt="E"/>. But it's the very statement of the equivalence induced by the Grothendieck construction that says that this is the same as the functor category <img src="/extensions/vLaTeX/cache/latex_124bf0084d7f39958907ac76b4941d38.png" title=" X \to QC" style="vertical-align: -20%;" class="tex" alt=" X \to QC"/>: the nLab sort of says this a bit at <a href="https://ncatlab.org/nlab/show/Grothendieck+fibration">Grothendieck fibration</a> (maybe this could be made more explicit and highlighted more).</p></li>
</ul>
<p>I will have to go offline in a minute. But think about this. It looks to me that Rosenberg-Kontsevich make exactly my point, just viewed from the dual perspective of fibered categories.</p>
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Your analysis of Kontsevich-Rosenberg is surely correct. The simpler procedure of taking just (the opposite to) the category of cartesian sections as quasicoherent modules is fully general, not applying only to presheaves. In your case (the 1.1.5) Yoneda strictification (CARTESIAN hom into a cleavage) gives an equivalent result, both the Yoneda and coherence are taken together in one step by limiting hom to cartesian hom in Yoneda, what makes it strictified. I find it not only more general, but also easier to think and perform separately, because taking cartesian section as opposed to all sections is about strictification/coherence and Yoneda is about extending to presheaves. Each thing has its own meaning and the steps or choices are manifest; furthermore it is general and finally, it does not involve considering enriched/weighted limits, what is for most practical mathematician hardly comprehensible (I myself read lines with weighted co/limits only when forced).
Ne entry cleavage.
Domenico: One should also point out that the morphisms of pseudofunctors (if the pseudofunctor aprpoach is taken instead of invariant fibered category approach) involve 2-cells which satisfy prescribed coherences (currently not listed in the entry, see the lecture notes on gerbes by Moerdijk for that approach). The managing of coherences instead of invariant property approach via cartesian functors is why Grothendieck and Gabriel left pseudofunctors and chose fibered categories. Try also to see how the definitions work in some other well-known cases (rather than just the (pre)sheaves of sets and simplicial sets). One good example are configuration schemes.
Urs: at your personal page it still says that the abelianization of Vect/X is the category of qcoh sheaves of modules, the correct is just coherent, as it is smaller and already abelian.
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<blockquote>
Yoneda strictification (CARTESIAN hom into a cleavage)
</blockquote>
<p>Hm, sorry, I'm still not sure what you mean by this.</p>
<p>The equivalence of bicategories given by the Grothendieck construction is between</p>
<ul>
<li><p>the bicatgeory of pseudofunctors <img src="/extensions/vLaTeX/cache/latex_34bf529d998c2b522f58576022ebfe4a.png" title=" E^{op} \to Cat" style="vertical-align: -20%;" class="tex" alt=" E^{op} \to Cat"/>, pseudonatural transformations and modifications of these</p></li>
<li><p>and the bicategory of Grothendieck fibrations <img src="/extensions/vLaTeX/cache/latex_3881474e0d8aa3b215bf05ddb6b75b5c.png" title="F \to E" style="vertical-align: -20%;" class="tex" alt="F \to E"/>, cartesian morphisms between these over <img src="/extensions/vLaTeX/cache/latex_62ee0dd9f4d9bb034c4b5a0e3fe8e324.png" title="E" style="vertical-align: -20%;" class="tex" alt="E"/> and transformations between these.</p></li>
</ul>
<p>Isn't it? The fact that we have cartesian morphisms between the fibered categories is precisely what models that this comes from a transformations between the corresponding pseudofunctors.</p>
</div>
Correct Urs, but in category over a category formalism the distinction is important. Your Yoneda hom can be taken in that category, rather than in the bicategory of pseudofunctors. In orbifold theory for example in 1990s people talked about wrong morphisms of orbifolds as they did not take care of cartesian property. Once they realized that they started calling such correct morphisms strong or good morphisms of orbifolds. Of course, the people working in stack language had never had that problem. But if we allow to consider categories over categories, not only stacks, then one can easily slip and not taking care of it. The two languages are precisely equivalent as you quoted.
In a pseudofunctor approach one does not need to think about non-cartesian morphism, it is rather important to keep the 2-cells involving morphisms between pseudofunctors coherent and a part of the data (not just existence).
Urs: are you still finding word quasicoherent unintuitive ?
I added a new section As cartesian morphisms of fibrations.
I guess we agree on what I wrote there now. But have a look.
I like it.
Okay, good. I am glad we clarified this. And I am glad that Kontsevich-Rosenberg agree with us! :-)
This is a really powerful nice abstract nonsense definition. Combined with Lurie's stuff on tangent (infinity,1)-category this gives a amazingly fundamental way of thinking about all things related to modules and vector bundles.
This is why I don't like to call this "quasicoherent sheaf", yes. It feels like calling something as fundamental as, say, the Yoneda lemma, a "quasi-something". I think that this appears as quasi-somethingelse is a historical coincidence. This stuff here is somewhere at the very bottom of the structure of math. It's fundamental and should have a more suggestive name, as such.
That's my feeling anyway, But I understand that the term "quasi--coherent" is widely accepted and does describe some aspects of this.
But indeed in practical everyday mathematics coherent sheaves are usually more often used than quasicoherent. But from the fundamental point of view, quasicoherent sheaves work correctly in more general situations. Coherent sheaves are good for locally noetherian schemes and for complex analytic manifolds for example, but not for nonnoetherian schemes: but quasicoherent work always. I was asking about the intuition about "coherent" part of the name (which is even better seen when looking at local sections in locally ringed space point of view). Quasi may be kind of accident. But saying vector bundle can get much harsher critique from many points of view (for example if you want to discuss infinitedimensional vector bundles in algebraic geometry properly, you need to get out of quasicoherent category as shown by Drinfeld!!). I agree that the Lurie's quasicategory tangent bundle is amazing discovery and treated in such subtle and natural detail that it is breathtaking. I am glad that you find the quasicoherent sheaves fundamental.
I have created a new section
in quasicoherent sheaf having a direct variant of the definition (which I learned a year and half ago from the Beilinson-Drinfeld paper on Hitchin fibration, cited somewhere on the page) which is of course equivalent (quite obviously), but its explicit character makes less need to think of Grothendieck constructions, Kan extensions etc. although they are facing you directly into your eyes. I spelled unusually precisely the version of coherence for that approach which is usually not spelled out.
I hope you like it.
Some expansion on terminology at cleavage.
<div>
<p>Zoran wrote:</p>
<blockquote>
I have created a new section: Direct definition for presheaves of sets on Aff
</blockquote>
<p>Nice, thanks. That's in fact better/more detailed than the previous section. It's good that you mention the cocycle condition. You see, this comes out out the pseudofunctor description: it's the "pseudonaturality prism" equation on triangles for a pseudonatural transformation of pseudofunctors. When I typed my proof, I was wondering why Goerss didn't mention it! Now I am glad to see that other authors to mention it. :-) So I added one more item to my discussion of the pseudonatural transformations and said that also the coycle which you mention comes out.</p>
<p>I think we should eventually try to merge the new section that you added with the previous one. They are really about the same thing. Yours is more detailed, but there is also now some repetition of the statement of the context.</p>
<p>Also: we need to try to harmonize and sort out what is a fibration and what is an opfibration.</p>
<p>Let's see, we have the pseudofunctor <img src="/extensions/vLaTeX/cache/latex_d40c607adfb83ae73204cc3c3a4eb706.png" title=" QC : CRings \to Cat " style="vertical-align: -20%;" class="tex" alt=" QC : CRings \to Cat "/> given by
<img src="/extensions/vLaTeX/cache/latex_7d363aa2a281ecaa1c1f6dc18eea6f83.png" title=" QC : (R \stackrel{f}{\to} S) \mapsto (R Mod \stackrel{-\otimes_f S}{\to} S Mod) " style="vertical-align: -20%;" class="tex" alt=" QC : (R \stackrel{f}{\to} S) \mapsto (R Mod \stackrel{-\otimes_f S}{\to} S Mod) "/>.</p>
<p>By Grothendieck constructions this yields an _op_fibration</p>
<p><img src="/extensions/vLaTeX/cache/latex_932e6f1e6f86908e98006ae3f69551a3.png" title=" Mod \to CRing " style="vertical-align: -20%;" class="tex" alt=" Mod \to CRing "/>,</p>
<p>right? Hence a fibration <img src="/extensions/vLaTeX/cache/latex_47e3c0c9d12568a71ff5e01402433bf5.png" title=" Mod^{op} \to CRing^{op} " style="vertical-align: -20%;" class="tex" alt=" Mod^{op} \to CRing^{op} "/> .</p>
<p>Of course since <img src="/extensions/vLaTeX/cache/latex_e6eac473632253fcf93de52f50ce87e6.png" title="Mod to CRing" style="vertical-align: -20%;" class="tex" alt="Mod to CRing"/> is a bifibration, there is also a fibration <img src="/extensions/vLaTeX/cache/latex_7ea7809b7b68b91e36807a2397874873.png" title=" Mod \to CRing^{op}" style="vertical-align: -20%;" class="tex" alt=" Mod \to CRing^{op}"/>, but that's not the one we need here, as you emphasized above.</p>
<p>These op-things are easily mixed up. Could you give me a sanity check on the above reasoning? Once we agree on this, somenody should go through the entry and harmonize it. I think currently it is slightly inconsistent internally in notation.</p>
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I think that covariant pseudofunctor from $CRing$ is viewed as contravariant functor from $CRing^{op}$, both with target $Mod$ and not $Mod^{op}$, and one traditionally looks at the Grothendieck fibration for that contravariant functor. I do not want to change the point of view to cofibrations from $Mod^{op}$. Thus I do not want to interpret $f^*$ as direct image, it must be inverse image, and it does not matter weather downstairs I label morphism of rings or opposite, of affine spectra. Relabelling changes covariant to contravariant, but I still talk about inverse images. This is quite usual, having "covariant" fibrations.
I see, okay. Maybe we should make this explicit at the entry.
I edited coherent sheaf a bit, trying to prettify it.
I also mention quasicoherent sheaves there, now. And at quasicoherent sheaf I edited the first definition paragraph, especially where it refers to coherent sheaves.
I think there was an inaccuracy in the previous verision: it had said that if the and can be chosen finite, then the sheaf is coherent. But this is only true of the structure sheaf itself is coherent. Right? Otherwise one has to add the "finite type" condition. Please check my edits.
I wrote something at free module, to satisfy that link, but I am in a haste now.
I also saw now Toby's query box at what is now free construction > history. I made "free construction" instead redirect to free functor.
I quickly added in the link to free module in the examples section at free functor, but it's a bit rough there now. If nobody else does, I'll polish it later.
Let me speculate a bit: there is Giraud's reconstruction from given Grothendieck topos of a site which reproduced a topos; there is Rosenberg's reconstruction of a scheme, up to isomorphism, out of its category of quasicoherent sheaves. Is there a way to rediscover the essentials of the Rosenberg's reconstruction from knowing the Giraud's reconstruction and the general nonsense approach to quasicoherent sheaves ?
added to quasicoherent sheaf a brief note on the definition for structured infinity-toposes: here
following this, I have now added to differential cohesion a brief note here observing that (quasicoherent) sheaves of $\mathcal{O}$-modules are naturally axiomatized in differential cohesion.
[ there is an issue with left exactness, I may have to get back to this tomorrow, did leave a warning in the entry ]
What are some decent sufficient conditions for pullback of quasicoherent sheaves to have a left adjoint?
Ah, never mind, I got what I was looking for, in prop. 3.3.23 of DAG XII.
(I’ll be adding this now as an example at Wirthmüller context.)
It’s probably instructive to work out the affine case first. The Eilenberg-Watts theorem says a left adjoint (necessarily additive) $\mathbf{Mod}(A) \to \mathbf{Mod}(B)$ is isomorphic to a functor of the form $F \otimes_A (-)$, for some $(B, A)$-bimodule $F$, so a right adjoint $\mathbf{Mod}(B) \to \mathbf{Mod}(A)$ must be isomorphic to a functor of the form $Hom_B (F, -)$ for some $(B, A)$-bimodule $F$. Thus, if $B \to A$ is a commutative ring homomorphism such that $A \otimes_B {-}$ has a left adjoint, then $A \cong Hom_B (F, B)$ for some $(B, A)$-bimodule $F$; and since $A \otimes_B {-}$ is a left adjoint, $F$ must be finitely presented and projective as a $B$-module. So the same is true for $A$.
Now suppose $A$ is finitely generated and projective as a $B$-module. Then $A$ is a retract of a finitely generated free $B$-module, say $E$. Clearly, $E \otimes_B {-}$ preserves limits and filtered colimits, so it has a left adjoint (by the accessible adjoint functor theorem). Thus the necessary and sufficient condition is “$A$ is finitely generated and projective as a $B$-module”.
A similar argument should show that $f^* : \mathbf{Qcoh}(Y) \to \mathbf{Qcoh}(X)$ has a left adjoint when $f : X \to Y$ is a morphism such that $f^{-1} \mathcal{O}_X$ is locally free of finite rank as an $\mathcal{O}_Y$-module.
Thanks for the comment, that makes sense.
Somehow more attention in the literature is on (implicitly) Wirthmüller contexts for (holonomic) D-modules, e.g.
discussing stuff as listed here
Has anyone seen a derived geometry version of this?
At quasicoherent sheaf, there is the nice description of $QCoh(X)$ as sheaves on $Aff/X$ (with the Zariski topology) satisfying the condition “$N(b) \otimes_B A \cong N(a)$”. I believe that this condition in fact already implies the sheaf condition. Therefore $QCoh(X)$ can also simply by expressed as the (2-categorical) limit $\lim_{Spec A \to X} Mod(A)$; thus $QCoh : Sch^{op} \to Cat$ is the right Kan extension of $Aff^{op} \to Cat$ (sending $Spec A$ to $Mod(A)$) along the inclusion $Aff^{op} \to Sch^{op}$.
This is similar, but not identical, to the characterization as a right Kan extension along $Aff \to [CRing, Cat]$ mentioned in the article.
I’ll think about this some more and add it to the article if nobody objects.
It surely implies the sheaf condition – after all, every scheme has a cover by affine schemes, so the claim reduces to the theorem (which becomes a definition here…) that quasicoherent sheaves on affine schemes are the same as modules over rings.
Thanks for the confirmation! I don’t quite see how you reduce to affine schemes; my proof would go like this: Abusing notation and simply writing $N(A)$ for $N(a)$, where $a : Spec A \to X$ is some fixed morphism, we have to verify that
$N(A) \to \prod_i N(A[s_i^{-1}]) \rightrightarrows \prod_{i,j} N(A[s_i^{-1},s_j^{-1}]),$where $1 = \sum_i s_i \in A$, is an equalizer diagram. By assumption, this diagram is isomorphic to
$N(A) \to \prod_i N(A)[s_i^{-1}] \rightrightarrows \prod_{i,j} N(A)[s_i^{-1},s_j^{-1}].$That this diagram is an equalizer diagram (even for any module $N(A)$) is a basic fact of commutative algebra; for instance, it follows from the fact that $N(A)^\sim$ is a sheaf on $Spec A$.
Sorry, misunderstood question.
@Ingo
Yes, that’s what I meant. But I see that you meant to show that it is a sheaf on the gros Zariski site – I was thinking about the small Zariski site.
I added to quasicoherent sheaf a synthetic characterization of quasicoherence using the internal language of the big Zariski topos.
Briefly, a sheaf $N$ is quasicoherent if and only if the canonical map
$N \otimes_{\mathbb{A}^1} A \longrightarrow Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N)$is bijective for all finitely presented $\mathbb{A}^1$-algebras; this has a geometric interpretation outlined in the entry.
Also, for a quasicoherent $\mathbb{A}^1$-algebra $R$, the canonical map
$R \longrightarrow \prod_{\varphi : R \to \mathbb{A}^1} \mathbb{A}^1$is bijective, which codifies the usual intuition that an element of an algebra $R$ should be viewed as a function on the spectrum of $R$.
Let $A$ be a ring (commutative with identity). Then the category of $A$-modules is equivalent to the category of abelian group objects in the overcategory CommRings/$A$. Switching to the algebro-geometric picture, this is the same as saying that the category of quasi-coherent sheaves on Spec$A$ is equivalent to the category of abelian cogroup objects in the undercategory Spec$A$\AffSchemes.
Now let $X$ be a scheme. Is it true that the category of quasi-coherent sheaves on $X$ is equivalent to the category of abelian cogroup objects in the undercategory $X$\Schemes?
Re #45: It appears that both sides of your conjectural equivalence satisfy Zariski descent. So the claim follows from the affine case.
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