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I heard many times that quasi-coherent sheaves of $\mathcal O_X$-modules are morally the same thing as the sheaves of sections of a bundle $V\to X$ over $X$. We think of a ring $A$ as of the ring of functions of the imagined space $\newcommand{\Spec}{\operatorname{Spec}}\Spec A$, and of an $A$-module $M$ as of a sheaf sections of some vector bundle with locally varying dimensions over $\Spec A$. I like to find references which make this concrete and discuss some of those bundles $V\to X$ in detail, preferably in the functor of points approach to algebraic geometry. Here are examples of the kind of results I am interested in:

  • In this answer Martin Brandenburg defines a pre-vector bundle on a scheme $X$ to be a morphism of schemes $V\to X$ together with a $\mathcal O(T)$-module structure on $\text{Hom}_X(T,V)$ for each scheme $T\to X$ over $X$ which varies naturally with restriction maps. A morphism of pre-vector bundles is a map over $X$ which respects the extra structure. Every pre-vetor bundle gives me a sheaf of $\mathcal O_X$-modules through taking sections. Which sheaves of $\mathcal O_X$-modules arise this way? Is there a construction in the other direction? Is there an adjunction? Where can I read more?

  • It is shown in many algebraic geometry books that locally free rank $r$ sheaves of $\mathcal O_X$-modules are in equivalence with ank $r$ vector bundles on the scheme $X$. Here is an example. It is also an exercise in Harthshorne.

  • One can associate two $X$-schemes $\underline{\Spec}_X(\operatorname{Sym}\mathcal E)\to X$ and $\underline{\Spec}_X(\operatorname{Sym}\mathcal E^\vee)\to X$ to a quasi-coherent sheave $\mathcal E$ of modules on $X$. The sheaf of sections of the first one is naturally isomorphic to the dual $\mathcal E^\vee$, and consequently the sheaf of sections of the second one is naturally isomorphic to $(\mathcal E^\vee)^\vee$. Is there a way to construct a bundle $V(\mathcal E)\to X$ such that its sheaf of sections is $\mathcal E$ for any quasi-coherent sheaf $\mathcal E$? The two constructions are discussed in this question, but I like to learn more.

  • Here is an example of something I like: Let $D = \Spec \mathbb Z[\varepsilon]/(\varepsilon^2)$ be the space of dual numbers. Then $\mathbb Z[\varepsilon]/(\varepsilon^2)\to \mathbb Z$ (sending $\varepsilon$ to zero) yields a map $D\to \mathbf 1$ and thus a map $X^D\to X$. This is the tangent bundle in synthetic differential geometry, and it turns out that it is also the tangent bundle in algebraic geometry. See here and here. Which other important quasi-coherent sheaves in algebraic geometry come from a bundle $V\to X$? When we increase the space we have from the category of schemes to the category of sheaves on the big Zariski site, can we make interesting bundle constructions $V\to X$ which may not exist in $\operatorname{Sch}$?

  • There is a nice chapter in the Algebraic Geometry book by Görtz and Wedhorn which shows that the category of quasi-coherent sheaves on a scheme is contravariantly equivalent to a category of quasi-coherent bundles on $X$. But their category of quasi-coherent bundles is made just so that it works out. The definition is not geometrically motivated in the book. Also the pseudo-invers is not the sheaf of sections construction, which is dissatisfying. (Chapter 11 in Görtz & Wedhorn's Algebraic Geometry I: Schemes)

  • I am aware that a quasi-coherent sheaf on a scheme $X$ is equivalent to a collection of modules, one for each generalised point $r:\Spec R \to X$ which vary pseudo-functorially. This is in line with the intuition that a quasi-coherent sheaf is a vector space attached to each (field-valued) point of $X$. But I am specifically interested in a global approach $V\to X$ from which I can extract the fibres by pullbacks.

  • I am familiar with the internal language of the big Zariski topos (as discussed in the second chapter of Ingo Blechschmidt's PhD thesis. If there is a characterisation of those $X$ schemes $V\to X$ which are in some sense locally non-trivial bundles of vector spaces in the internal language, then I am more than happy to learn it!

  • Moerdijk and Reyes define vector bundles on page 195 of their book Models of smooth infinitesimal Analysis. The definition works in any smooth topos, of which the gros Zariski topos Sh(Aff,Zar) is, according to the nLab, probably an example. What does their definition mean in the context of algebraic geometry? Can we characterize the sheaves of section which come from vector bundles in the sense of Moerdijk and Reyes?

Most algebraic textbooks which I find just introduce the theory of sheaves of $\mathcal O_X$-modules without a lot of geometric explanation. Where can I find books/texts/papers which discuss bundles $V\to X$ of schemes and their connections to quasi-coherent sheaves in detail?

Edit. I want to provide evidence for my extraordinary claim that $QCoh(X)$ embeds in two ways fully faithfully into a category of $\mathbb A_X$ modules.

The first construcion is one that I did not mention in the previous version of my question, because I did not know about it. It works as follows. You pick a scheme $X$ and view as an object in one of the big topoi via the functor of points approach. For simplicitly, let me take the big Zariski topos $Zar$. There is a local geometric morphism $\pi: Zar/X \to Sh(X)$ from the big topos of $X$ to the little. Its pushforward part $\pi_\ast$ takes a space over $X$ to its sheaf of section. It has a left adjoint $\pi^{-1}$ and one defines a fully faithful covariant functor $\pi^\ast:Mod(\mathcal O_X)\to Mod_{Zar/X}(\mathbb A_X)$ by setting $\pi^\ast\mathcal F = \pi^{-1}\mathcal F\otimes_{\pi^{-1}\mathcal O_X}\mathbb A_X$. A detailed construction and a proof that this functor is fully faithful can be found at the beginning of part 2 of Ingo Blechschmidt's PhD thesis. In fact, $\pi_\ast \pi^\ast = id$.

The second construction is already mentioned above. It is contravariant and works only for quasicoherent modules, but it always produces an $\mathbb A_X$-module which is a scheme.

Given an quasicoherent $\mathcal O_X$-module $\mathcal F$, we let $V(\mathcal F)$ be the $X$-scheme $Spec_X(Sym(\mathcal F))\to X$, where we now use the relative spec construction. The $\mathcal O_X$-algebra $Sym(\mathcal F)$ is $\mathbb N$-graded, and hence we get an $(\mathbb A_X,\cdot)$-monoid action on $V(\mathcal F)$. This is the first part of what we need to turn $V(\mathcal F)$ into an $\mathbb A_X$-module. What is left is to define the addition map. For that we switch to a local chart affine chart $U$ of $X$. We need a morphism $$ +: Spec(\,Sym(\mathcal F(U)))\times_U Spec(Sym(\mathcal F(U)))\to Spec(Sym(\mathcal F(U))) $$ relative $U$. This is the same thing as an $\mathcal O(U)$ algebra map $$Sym(\mathcal F(U))\to Sym(\mathcal F(U)) \otimes_{\mathcal O(U)} Sym(\mathcal F(U))$$ which in turn is the same thing as an $\mathcal O(U)$-module map $$\mathcal F(U) \to Sym(\mathcal F(U))\otimes_{\mathcal O(U)} Sym(\mathcal F(U))$$ We take the map which sends $f$ to $f\otimes 1 + 1\otimes f$ and we are done.

We have a functor $V:QCoh(\mathcal O_X)^{op} \to Mod_{Sch/X}(\mathbb A_X)$. There is a construction $L$ in the opposite direction which takes an $\mathbb A_X$-module to its sheaf of linear functions. We have that $LV = id$, and this shows that $V$ is fully faithful.

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    $\begingroup$ If you downvote my question, then it would be nice if you tell me what is wrong with it. I spend a lot of time and effort to search for myself before I have written it. $\endgroup$
    – Nico
    Oct 8, 2022 at 12:05
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    $\begingroup$ I think you are asking too much for what is supposed to be just an analogy. There are no infinite-dimensional vector bundles but there are certainly locally free sheaves of infinite rank. Even for coherent sheaves I do not think there is a general way of realising them as (generalised) vector bundles. $\endgroup$
    – Zhen Lin
    Oct 8, 2022 at 13:14
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    $\begingroup$ @ZhenLin How is my functor the dualization functor? It is true that $\pi_\ast V(\mathcal F)= \mathcal F^\vee$, but this is not what I am doing is it? Also maybe try to be a bit friendlier. you write unhelpful comments all over this site at it annoys the hell out of me honestly. $\endgroup$
    – Nico
    Mar 3, 2023 at 10:44
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    $\begingroup$ I am being honest and direct: what you say contradicts my intuition. And 6 others agree with me. I think the onus is on you to explain why we were mistaken. $\endgroup$
    – Zhen Lin
    Mar 3, 2023 at 11:14
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    $\begingroup$ I have no doubt that the symmetric algebra construction is faithful. I am less sure about the properties of the relative spectrum but I can believe it has properties analogous to the global spectrum. Fullness is what you need to show. And my biggest doubts are regarding the contravariant functor from "vector bundles" back to sheaves. $\endgroup$
    – Zhen Lin
    Mar 3, 2023 at 14:20

2 Answers 2

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This is not an answer. I just want to add a proof that the functor $V$ which I describe in my question is fully faithful, since Zhen Lin in the comments believes that this is not the case.

Step 1. I will first prove that $V$ is faithful, since this is simpler. Since the forgetful functor $Mod(\mathbb A_X)\to Sch/X$ is faithful, it is enough to show that $V$ is faithful as a functor $V: QCoh(X)^{op}\to Sch/X$. Here it is well-known (and written down in EGA) that $V$ is well-defined. There is also a contravariant functor $\mathcal A:(Sch/X)^{op}\to Alg(\mathcal O_X)$ which takes a scheme over $X$ to its $\mathcal O_X$-algebra of functions. This functor is described in EGA, and it is also shown in EGA that $\mathcal A \operatorname{Spec}_X=\operatorname{Id}$ where $\operatorname{Spec}_X$ denotes the relative spectrum construcion. The functor $\mathcal A$ and the relative spectrum construction provide an equivalence between the category of quasicoherent $\mathcal O_X$-modules and the category of affine $X$-schemes.

We now see that $$\mathcal AV = \mathcal A \operatorname{Spec}_X \operatorname{Sym} = \operatorname{Sym}$$ So to show that $V$ is faithful it is enough to check that $\operatorname{Sym}: QCoh(X) \to qcAlg(\mathcal O_X)$ is faithful. But this is the case, since $\operatorname{Sym}$ factors through the category of $\mathbb N$-graded quasicoherent $\mathcal O_X$-algebras, and taking the degree 1 piece defines a functor $(-)_1: \mathbb NgrqcAlg(\mathcal O_X)\to QCoh(X)$ which satisfies $(-)_1 \operatorname{Sym} = \operatorname{Id}$.

Step 2. Next I will show that $V$ yields a fully faithful contravariant functor into the category of $X$-schemes with a fiberwise $\mathbb G_m$-action. I work my way up to $\mathbb A_X$-modules step by step, because the the results in this step are also well-known and I can point to references.

In the case of an affine base scheme $X = Spec(k)$, where $k$ is any ring, there is a well-known contravariant equivalence between the category of $\mathbb Z$-graded $k$-algebras and algebra morphisms which preserve the grading and the category of affine $k$-schemes with a $(\mathbb G_m)_k$-group action and scheme morphisms which are equivariant.

This glues and gives us a contravariant equivalence between the category of quascoherent $\mathcal O_X$-algebras with a $\mathbb Z$-grading and the category of affine $X$-schemes with an action by $(\mathbb G_m)_X$. In one direction we have the relative spec construction, and in the other direction we have $\mathcal A$. If $Y\to X$ has an action by $(\mathbb G_m)_X$ then the function on $Y$ can be graded by homogeneous degree, and if $\mathcal B$ is a $\mathbb Z$-graded algebra, then we can use it to define a $(\mathbb G_m)_X$-group action on $\operatorname{Spec}_X(\mathcal B)$.

Let me denote the category category of $X$-schemes with a fiberwise group action by $\mathbb G_m$ and equivariant maps between them by $\mathbb G_m(Sch/X)$. What the discussion above shows is that we can promote $\operatorname{Spec}_X$ to a fully faithful functor $(\mathbb ZgrqcAlg(\mathcal O_X))^{op} \to G_m(Sch/X)$. Since $\operatorname{Sym}$ is a fully faithful functor into $\mathbb ZgrqcAlg(\mathcal O_X)$ (taking the degree 1 part defines a retraction of $\operatorname{Sym}$), and since $V = \operatorname{Spec}_X\operatorname{Sym}$ we now see that $V$ defines a fully faithful functor

$$V: QCoh(X)^{op} \to \mathbb G_m(Sch/X)$$

Step 3. Since each $\operatorname{Sym}(\mathcal F)$ is $\mathbb N$-graded (has nothing in the negative degrees), we can upgrade the fiberwise $\mathbb G_m$-action to a fiberwise $(\mathbb A,\cdot)$-monoid action. This already the first part of what I need to turn $V(\mathcal F)$ into an $\mathbb A_X$-module. Let me describe how the action is defined. Given a $\mathbb N$-graded quasicoherent $\mathcal O_X$-algebra $\mathcal B$, I need to define a morphism

$$\mathbb A\times \operatorname{Spec}_X(\mathcal B) \to \operatorname{Spec}_X(\mathcal B)$$

above $X$. I may assume without loss of generality that $X = Spec(k)$ is affine, and that $B$ is a $k$-algebra, we may later glue the results to a propert action on the total space. So I need a morphism $$\mathbb A \times \operatorname{Spec}(B) \to \operatorname{Spec} (B) $$ above $Spec(k)$, which is the same thing as a $k$-algebra map of the format $$B\to k[x]\otimes_k B$$ The $\mathbb N$-grading of $B$ tells me how to define the map. It sends $f\in B_n$ to $x^n\otimes f$. The construction is exactly analogue to the one in the previous step, only that the map here could not have been defined if there are functions in negative degree.

Everything else is also completely analogous to the previous case. Let me denote the category of $X$-schemes with a fiberwise monoid action by $(\mathbb A,\cdot)$ and the equivariant maps between such by $\mathbb A(Sch/X)$. We see that we can promote $\operatorname{Spec}_X$ to a fully faithfull embedding $(\mathbb NgrqcAlg(\mathcal O_X))^{op}\to \mathbb A(Sch/X)$. Since $V = \operatorname{Spec_X}\operatorname{Sym}$ we see that we can promote $V$ to a fully faithful functor $$V:QCoh(X)^{op} \to \mathbb A(Sch/X)$$

Step 4. An object in $\mathbb A(Sch/X)$ is almost an $\mathbb A_X$-module, the only thing that is missing is the addition map. The only missing step is to notice that each $V(\mathcal F)$ can be equipped with an addition map (this is described above in my question), and that a morphism $V(\mathcal F) \to V(\mathcal G)$ of $X$-schemes which preserves the $\mathbb A$-action (i.e. scalar multiplication) automatically also preserves addition. This is the case. In fact a stronger statement is true. If $W$ is any $\mathbb A_X$-module in $Sch/X$ and $\phi:W\to V(\mathcal F)$ is any function of $X$-schemes which preserves the $(\mathbb A,\cdot)$-action, then $\phi$ automatically also preserves addition.

I will add a proof of the final step when no one has an objection to the first three, since this is already really long.

References.

  • For step 1 I can point to corollary 12.2 in Görtz and Wedhorn's book Algebraic Geometry 1. They define the relative spectrum and $\mathcal A$ and show that they yield a contravariant equivalence between affine $X$-schemes and quasicoherent $\mathcal O_X$-algebras.

  • For step 2 I can point tp chapter I part 2 of the book Cox rings by Arzhantsev, Derenthal, Hausen and Laface. It discusses the contravariant equivalence between graded algebras and categories of schemes with a group action.

As a final bit of evidence I want to mention that I have spoken with a professional algebraic geometer on twitter, and they have told me that step 2 is correct. :D

Step 5. Since everything seems to be more or less okay so far, I will now do the last step :) This is indeed the most difficult one, and it is the part where I most likely have made a mistake if I have made one, so I will try to be extra careful.

I will now use an ingredient of synthetic differential geometry. Let $\mathcal S$ be any topos with an internal ring object $R$ in it. An internal $R$-module $V$ in $\mathcal S$ is said to be euclidean if it is microlinear and satisfies the KL-axiom

$$\mathcal S\models \forall t: V^D. \exists ! v:V.\, \forall d:D.\, t(d) = t(0) + d v$$

Such modules satisfy a very special property, namely that any morphism $\phi: W\to V$ from another $R$-module $W$ which is homogeneous (that is satisfies $\phi(\lambda w) = \lambda \phi(w)$ in the internal language) is automatically additive.

The topos $(Zar,\mathbb A)$ and each of its slices are famously a model of the generalized KL-axiom, and as such the $\mathbb A_X$-module $\mathbb A_X$ itself is euclidean. This has the following consequence.

Lemma A. If $W$ is an $\mathbb A_X$-module (it does not even have to be a scheme), and if $\phi: W\to \mathbb A_X$ is homogeneous, then it is automatically linear.

Proof. Lavendhomme, Basic concepts of synthetic differential geometry, 1.2.4 proposition 10. Alternatively Proposition 1.5 in chapter V of Moerdijk and Reye's Models for smooth infinitesimal analysis. $\square$

The same fact is also mentioned in the sixth paragraph of this answer by Martin Brandenburg. I am not aware of a short direct proof, the only one I could offer would be a direct translation of the internal argument.

But with the lemma at hand, I can now do the missing steps.

Lemma B. If an $\mathbb A_X$-module $V$ satisfies the property of lemma A, then the addition map of $V$ is already determined by its fiberwise $\mathbb A$-action. By this I mean that if $+'$ is any other addition map such that the space $V$ together with its fiberwise scalar action is an $\mathbb A_X$-module, then $+'$ is equal to the addition map of $V$.

Proof. Let me denote $V$ with the second addition map by $V'$. The identity of the underlying spaces is clearly homogeneous, since we haven't touched the scalar action. By the lemma A property of $V$ the identity map $V'\to V$ is also linear. It follows that the addition maps on both $V$ and $V'$ must agree. $\square$

The point of the lemma is of course that for such modules the addition map is not really an extra structure but a property, and once we have shown that the $V(\mathcal F)$ satisfy the lemma A property, we will see that step 3 was already enough. We already know that we can turn the $V(\mathcal F)$ into $\mathbb A_X$-modules (I have explained how in my question above).

Lemma C. The $\mathbb A_X$-modules $V(\mathcal F)$ satisfy the lemma A property. There is at most one way to equip them with an addition map, and a morphism into them which is homogeneous is automatically linear.

Proof. We assume that we have an $\mathbb A_X$-module $W$ and a homogeneous map $\phi:W\to V(\mathcal F)$. We like to show that $$\phi(x) + \phi(y) = \phi(x+y)$$I am using a little bit of the internal language here to avoid diagrams. By the universal property of $V(\mathcal F)$ it is enough to show that both scheme morphisms pull-back functions in the same way. By assumption $\phi$ is homogeneous, so that the associated pullback maps of $\mathcal O_X$-algebras preserve the gradings. By the universal property of $\operatorname{Sym}(\mathcal F)$ it is thus enough to show that $\phi(x) + \phi(y)$ and $\phi(x+y)$ pull the 1-homogeneous functions on $V(\mathcal F)$ back in the same way. That is we need to show that for each open subspace $U$ of $X$ and for each 1-homogeneous function $f:V(\mathcal F|_U)\to \mathbb A_U$ we have that $$f(\phi|_U(x) + \phi|_U(y)) = f(\phi|_U(x + y))$$ (I am again using the internal language, but what I mean is the associated commuting diagram). Now we use the result from SDG! The function $f$ is automatically linear, and so is $f\circ \phi|_U$. The result follows immediately. $\square$

Now that we know that, we just need to patch everything together. Each $V(\mathcal F)$ can be equipped with an addition map in one and only one way, and a morphism between $V(\mathcal F)\to V(\mathcal G)$ which is homogeneous is automatically linear. This shows that the fully faithful functor $$V: QCoh(X)^{op} \to \mathbb A(Sch/X)$$ of step 4 becomes a fully faithful functor $$V:QCoh(X)^{op} \to Mod_{Sch/X}(\mathbb A_X)$$ as claimed. We also have a nice way to compute $\mathcal LV(\mathcal F)$. By definition $\mathcal LV(\mathcal F)$ is the sheaf of linear functions on $V(\mathcal F)$. But we have just shown that the linear functions on $V(\mathcal F)$ are precisely the homogeneous once, and hence $$\mathcal LV = (-)_1 \mathcal A V = (-)_1 \mathcal A \operatorname{Spec}_X \operatorname{Sym} = \operatorname{Id}$$ The functor $\mathcal L$ is a retraction of $V$.

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To address your question "Is there a way to construct a bundle $V(\mathcal{E})\rightarrow X$ such that its sheaf of sections is $\mathcal{E}$ for any quasi-coherent sheaf $\mathcal{E}$?" I would first like to point out that we typically save the words "vector bundle" for the case that $\mathcal{E}$ is locally free of finite rank, in which case our answer is already in the affirmative. (As then, Zariski-locally on $X$, $Spec_X(\mathcal{E}^\vee)\rightarrow X$ is of the form $\mathbb{A}^n_U\cong Spec_X(\mathcal{E}^\vee|_U) \cong U\times_XSpec_X(\mathcal{E}^\vee)\rightarrow U$, and all vector bundles over $X$ are of this form.)

In general though, one idea could be to consider the functor $Mor_{X}(Spec_X(\mathcal{E}),\mathbb{A}^1_X)$ (see Stacks 0BL0). For any $X$-scheme $Y\rightarrow X,$

$$Mor_{X}\big(Spec_X(\mathcal{E}),\mathbb{A}^1_X\big)(Y)\,=\,\text{Hom}_{Sch/X}\big(Y\times_XSpec_X(\mathcal{E}),\mathbb{A}^1_X\big),$$

which is exactly the global sections of the structure sheaf on $Y\times_XSpec_X(\mathcal{E})$. Then when $Y=U\hookrightarrow X$ is an open subscheme, this is exactly $\Gamma(U,\mathcal{E}|_U)=\mathcal{E}(U)$. However, the only time that Stacks claims this is representable by a scheme is when $Spec_X(\mathcal{E})\rightarrow X$ is "finite locally free" (so, an affine morphism which is "locally free") which is exactly when $\mathcal{E}$ is locally free of finite rank. There are some conditions imposed on the structure morphism $\mathbb{A}^1_X\rightarrow X$ as well, but I thought this would be enough to address this part of your question.

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  • $\begingroup$ I've taken the liberty of adding a link to the section of the Stacks Project you referenced. FYI, it's preferable to reference the Stacks Project by tags - these are guaranteed to never change, whereas chapter/section numbers can change depending on edits to the text. $\endgroup$
    – KReiser
    Jan 29 at 22:52
  • $\begingroup$ @KReiser much appreciated! $\endgroup$ Jan 29 at 23:04

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