Re #45: It appears that both sides of your conjectural equivalence satisfy Zariski descent. So the claim follows from the affine case.

]]>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?

]]>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 $R$.

]]>@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.

]]>Sorry, misunderstood question.

]]>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 $Spec A$.

]]>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.

]]>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.

]]>Thanks for the comment, that makes sense.

Somehow more attention in the literature is on (implicitly) Wirthmüller contexts for (holonomic) D-modules, e.g.

- Joseph Bernstein, around p. 18 of
*Algebraic theory of D-modules*(BernsteinDModule.pdf:file, ps, dvi)

discussing stuff as listed here

- Pavel Etingof,
*Formalism of six functors on all (coherent) D-modules*(pdf)

Has anyone seen a derived geometry version of this?

]]>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.

]]>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*.)

What are some decent sufficient conditions for pullback of quasicoherent sheaves to have a left adjoint?

]]>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 ]

]]>added to *quasicoherent sheaf* a brief note on the definition for structured infinity-toposes: here

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 ?

]]>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.

]]>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 see, okay. Maybe we should make this explicit at the entry.

]]>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.

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<p>Zoran wrote:</p>
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I have created a new section: Direct definition for presheaves of sets on Aff
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<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="https://nforum.ncatlab.org/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="https://nforum.ncatlab.org/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="https://nforum.ncatlab.org/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="https://nforum.ncatlab.org/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="https://nforum.ncatlab.org/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="https://nforum.ncatlab.org/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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consider the natural embedding CRing --> SCRing (C stands for commutative, S for simplicial). then we have the fibration SMod --> SCRing and we can hope it pulls back to Mod --> CRing. should this be so, we would be done: SMod --> SCRing is equivalent to TSCRing --> SCRing, and so Mod (and therefore Qcoh) would be completely recovered from CRing --> SCRing <-- TSCRing. ]]>

Some expansion on terminology at cleavage.

]]>I have created a new section

- Direct definition for presheaves of sets on Aff

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.

]]>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.

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