$\def\VB{\mathsf{VB}}
\def\sO{\mathcal{O}}
\def\Mod{\mathsf{Mod}}
\def\LFMod{\mathsf{LFMod}}$For a ringed space $(X,\sO_X)$ and a ring $R$, denote $\Mod(\sO_X)$ and $\Mod(R)$ to the categories of $\sO_X$-modules and of $R$-modules. Let $M$ be a smooth manifold and denote $\sO_M$ to the sheaf of smooth functions on $M$. Denote $\VB(M)$ to the category of smooth vector bundles over $M$ with vector bundle homomorphisms. For a vector bundle $\pi:E\to M$ over $M$, denote $\Sigma_E$ to the sheaf of sections of $\pi$. There is a functor
\begin{align*}
\Sigma:\VB(M)&\to\Mod(\sO_M)\\
E&\mapsto\Sigma_E\\
F:E\to E'&\mapsto F_*:\Sigma_E\to\Sigma_{E'},
\end{align*}
where $F_*$ is induced by postcomposition by $F$. (Actually, postcomposition induces a morphism of hom-sheaves $\mathcal{H}om_{\VB(M)}(E,E')\to\mathcal{H}om_{\sO_M}(\Sigma_E,\Sigma_{E'})$). Working in a local frame, one can see that $\Sigma$ is faithful by seeing that $\Gamma\circ\Sigma$ is faithful, where $\Gamma:\Mod(\sO_M)\to\Mod(\underbrace{\Gamma(M,\sO_M)}_{C^\infty(M)})$ denotes the global sections functor. Less trivial is that the functor $\Sigma$ is also full. For this, one first checks that the functor $\Gamma\circ \Sigma$ is full: this is Lemma 10.29 of Lee's Introduction to Smooth Manifolds. Leveraging Lee's proof, one can show after fullness of $\Sigma$: Given vector bundles $E$ and $E'$ over $M$ and an $\sO_M$-linear map $\varphi:\Sigma_E\to\Sigma_{E'}$, denote $F^U:E|_U\to E'|_U$ to the unique vector bundle homomorphism such that $\Gamma\circ\Sigma(F^U)=\Gamma(F^U_*)=\varphi_U$. Using the explicit definition of $F^U$ given in Lee's proof of Lemma 10.29, one can check that $F^U|_{U\cap V}=F^V|_{U\cap V}:E|_{U\cap V}\to E'|_{U\cap V}$. Therefore, the $F^U$'s glue to a vector bundle homomorphism $F:E\to E'$ such that $F|_U=F^U$. Thus, since $F_*|_U=(F|_U)_*=F_*^U$, one has $F_{*,U}=\Gamma(F^U_*)=\varphi_U$, i.e., $F_*=\varphi$.
This shows that $\Sigma$ is a fully faithful functor and hence an equivalence of categories into its essential image. It is clear that the image of $\Sigma$ is contained in the full subcategory of locally free $\sO_M$-modules. What I am interested is on finding a detailed proof that locally free $\sO_M$-modules are in fact the essential image of $\Sigma$. The only reference I am aware of is Ramanan's Global Calculus. He discusses this essential surjectivity in Chapter 2, between definitions 2.8 and 2.9. However, I find the argument sketched on his discussion a little bit unsatisfactory.
So my questions are:
-
Do you know any reference which proves the essential surjectivity of $\Sigma:\VB(M)\to\LFMod(\sO_M)$ with more detail than Ramanan does?
-
Do you know yourself a more detailed proof of this fact than Ramanan's one?
I was trying to combine the Vector Bundle Chart Lemma of Lee's book (Lemma 10.6 of the 2nd edition) with Ramanan sketch, but I wasn't sure how to actually do it.
Best Answer
$\def\VB{\mathsf{VB}} \def\sO{\mathcal{O}} \def\Mod{\mathsf{Mod}} \def\LFMod{\mathsf{LFMod}} \def\frm{\mathfrak{m}} \def\bbR{\mathbb{R}} \def\ev{\operatorname{ev}} \def\Ker{\operatorname{Ker}} \def\grm{\operatorname{grm}} \def\sE{\mathcal{E}} \def\op{\oplus} \def\bbK{\mathbb{K}} \def\sF{\mathcal{F}} \def\sG{\mathcal{G}}$The following proof is for real vector bundles over real smooth manifolds. However, I think that mutatis mutandis it works as well for complex vector bundles over real smooth manifolds (and maybe over complex manifolds too). But I haven't checked the details, so in the following everything is real in case of doubt. Anyhow, I will write $\bbK$ instead of $\bbR$ along the proof to suggest the possibility of generalization. (I would be grateful if someone points out a place in the literature where they do the complex generalization.)
Let $M$ be a smooth manifold and let $\sO_M$ be the sheaf of smooth functions on $M$. For each $p\in M$, denote $\frm_p\subset\sO_{M,p}$ to the (unique) maximal ideal of germs of smooth functions vanishing at $p$. The known result that evaluation at a point induces an isomorphism of vector spaces $\sO_{M,p}/\frm_p\cong\bbK$ generalizes to the following result:
Proof. Fix $p\in M$. For an open neighborhood $U\subset M$ of $p$, the evaluation maps \begin{align*} \Sigma_E(U)&\to E_p\\ \sigma&\mapsto\sigma(p) \end{align*} make up a cocone over $E_p$. Hence, evaluation at $p$ induces a well-defined map $\ev_p:\Sigma_{E,p}\to E_p$. This map is easily seen to be onto by working at a local frame. It then suffices to see that $\Ker(\ev_p:\Sigma_{E,p}\to E_p)=\frm_p\Sigma_{E,p}$. The containment to the left is trivial. Conversely, let $\sigma\in \Sigma_{E,p}$ be in the kernel of $\ev_p$. Picking a local frame $(\sigma^i)$ around $p$, we get smooth functions $f_i$ defined locally around $p$ with $\sigma=\grm_p(f_i\sigma^i)$, where $\grm_p(-)$ denotes the germ at $p$. Then $0=\ev_p(\sigma)=f_i(p)\sigma^i(p)$. Hence $f_i(p)=0$ for all $i$. Thus $\sigma=\grm_p(f_i\sigma^i)=\grm_p(f_i)\grm_p(\sigma^i)\in\frm_p\Sigma_{E,p}$.
For the naturality, note that given a vector bundle homomorphism $F:E\to E'$, from the map on stalks of $F_*:\Sigma_E\to\Sigma_{E'}$ at $p$, we get an induced linear map $\overline{F}_{*,p}:\Sigma_{E,p}/\frm_p\Sigma_{E,p}\to\Sigma_{E',p}/\frm_p\Sigma_{E',p}$. On the other hand, $F$ induces a linear map $F_p:E_p\to E_p'$. Since the map $\overline{F}_{*,p}$ is given by postcomposition by $F$, we have $\ev_p\circ\overline{F}_{*,p}=F_p\circ\ev_p$. That is, the isomorphism is natural.
The proof of the last claim is immediate. Such a described section always exist by working in a local frame. $\square$
Proof. Let $v\in E_p$. By the proof of Lemma 10.29 of Lee's book, we have $F(v)=\varphi_M(\tilde{v})(p)$, where $\tilde{v}$ is a global section of $\pi$ with value $v$ at $p$. So \begin{align*} F_p(v)&=\varphi_M(\tilde{v})(p)\\ &=\ev_p(\varphi_M(\tilde{v}))\\ &=\ev_p(\grm_p(\varphi_M(\tilde{v})))\\ &=\ev_p(\varphi_p(\grm_p(\tilde{v})))\\ &=\ev_p(\varphi_p(\grm_p(\tilde{v}))+\frm_p\Sigma_{E',p})\\ &=\ev_p\circ\overline{\varphi}_p(\grm_p(\tilde{v})+\frm_p\Sigma_{E,p}). \end{align*}
This expresses commutativity of the diagram if we traverse it as in $ \begin{matrix} \bullet&\rightarrow&\bullet\\ \downarrow&&\uparrow\\ \bullet&\rightarrow&\bullet \end{matrix} $. $\qquad\square$
We are ready to show essential surjectiveness of the functor $\Sigma:\mathsf{VB}(M)\to\mathsf{LFMod}(\sO_M)$.
Proof. The previous results impel us to define the vector space $E_p= \sE_p/\frm_p\sE_p$, for each $p\in M$. The dimension of this vector space is $r$, for
$$ \frac{\sE_p}{\frm_p\sE_p}\cong \frac{\sO_{M,p}^{\op r}}{\frm_p\sO_{M,p}^{\op r}} =\frac{\sO_{M,p}^{\op r}}{(\frm_p\sO_{M,p})^{\op r}} =\frac{\sO_{M,p}^{\op r}}{\frm_p^{\op r}} \cong\left( \frac{\sO_{M,p}}{\frm_p} \right)^{\op r} \cong\bbK^r. $$ We want to apply the vector bundle chart lemma (Lemma 10.6 of Lee's book) to the vector spaces $E_p$. For this, we will use the cover of $M$ of open subsets $U\subset M$ for which $\sE|_U$ is free of rank $r$.
Pick open subsets $U,V\subset M$ for which there are isomorphisms $\varphi:\sE|_U\cong\sO_U^{\op r}$ and $\psi:\sE|_V\cong\sO_V^{\op r}$. Consider the following diagram in the category of vector spaces: $$ \require{AMScd} \begin{CD} E_p @>>> \{p\}\times\bbK^r \\ @| @AA{\cong}A \\ \sE_p/\frm_p\sE_p @>>> \sO_{M,p}^{\op r}/\frm_p^{\op r} \end{CD}$$ where the isomorphism on the right is canonical and given by the component-wise evaluation at $p$. For each $p\in U$ (resp., for each $p\in V$) we define the map $\Phi_p$ (resp., the map $\Psi_p$) to be the unique map $E_p\to\{p\}\times\bbK^r$ for which the previous diagram commutes if the bottom map is $\overline{\varphi}_p$ (resp., $\overline{\psi}_p$).
Define $E=\bigsqcup_{p\in M}E_p$ and $\pi:E\to M$ to be the natural projection. Define also a pair of maps over $M$ \begin{align*} \Phi&:\pi^{-1}(U)\to U\times\bbK^r \\ \Psi&:\pi^{-1}(V)\to V\times\bbK^r \end{align*} given by $\Phi|_{E_p}=\Phi_p$ for $p\in U$ and $\Psi|_{E_p}=\Psi_p$ for $p\in V$. To show that $(E,\pi)$ is a vector bundle over $M$, there is one thing left to verify from the statement of the Vector Bundle Chart Lemma. Namely, that the composite $ \Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r} $ is a bundle homomorphism.
Let $p\in U\cap V$ and consider the following commutative diagram: $$ \require{AMScd} \begin{CD} (U\cap V)\times\bbK^r @>{\Phi^{-1}}>> \pi^{-1}(U\cap V) @>{\Psi}>> (U\cap V)\times\bbK^r \\ @AAA @AAA @AAA \\ \{p\}\times\bbK^r @>{\Phi^{-1}_p}>> E_p @>{\Psi_p}>> \{p\}\times\bbK^r \\ @A{\cong}AA @| @AA{\cong}A \\ \sO_{M,p}^{\op r}/\frm_p^{\op r} @>{\overline{\varphi}^{-1}_p}>> \sE_p/\frm_p\sE_p @>{\overline{\psi}_p}>> \sO_{M,p}^{\op r}/\frm_p^{\op r} \end{CD} $$ By functoriality of Observation 3, the bottom composite equals $(\overline{\psi\circ\varphi^{-1}})_p$. By Lemma 2, we have that $\Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r}$ is a bundle homomorphism. Namely, it is the unique automorphism of the trivial $r$-vector bundle over $U\cap V$ such that the induced morphism on the sheaf of sections equals $(\psi\circ\varphi^{-1})|_{U\cap V}$. (Recall that the sheaf of smooth sections of the trivial bundle $W\times \bbK^r\to W$ is canonically isomorphic to $\Sigma_{W\times \bbK^r}\cong\sO_{W}^{\op r}$.) In particular, $\Psi\circ\Phi^{-1}|_{(U\cap V)\times\bbK^r}$ is smooth and fiber-wise linear.
Hence, by the Vector Bundle Chart Lemma, $(E,\pi)$ is a vector bundle and $\Phi$ and $\Psi$ are local trivializations.
It remains to show that $\sE\cong\Sigma_E$. Let $U\subset M$ be an open subset such that there is an isomorphism $\varphi:\sE|_U\cong\sO_U^{\op r}$. As before, denote $\Phi:E|_U\to U\times\bbK^r$ to the associated trivialization of $(E,\pi)$. We have an isomorphism of $\sO_U$-sheaves
$$ \sE|_U \xrightarrow{\varphi} \sO_U^{\op r} \cong \Sigma_{U\times\bbK^r} \xrightarrow{\Phi_*^{-1}|_U} \Sigma_E|_U. $$ Abusing notation by identifying $\sO_U^{\op r} \cong \Sigma_{U\times\bbK^r}$, denote $\Phi_*^{-1}|_U\circ\varphi$ to this composite isomorphism. Let's compute its value. Let $s\in\sE(U)$. Then $$ (\Phi_{*,U}^{-1}\circ\varphi_U)(s) =\Phi_{*,U}^{-1}(\varphi_U(s)) =\Phi^{-1}\circ(\varphi_U(s)). $$ For $p\in U$, we have \begin{align*} (\Phi_{*,U}^{-1}\circ\varphi_U)(s)(p) &=[\Phi^{-1}\circ(\varphi_U(s))](p)\\ &=\Phi^{-1}(\varphi_U(s)(p))\\ &=\Phi_p^{-1}(\varphi_U(s)(p))\\ &=\overline{\varphi}_p^{\,-1}(\grm_p(\varphi_U(s))+\frm_p^{\op r})\\ &=\varphi_p^{-1}(\grm_p(\varphi_U(s)))+\frm_p\sE_p\\ &=\grm_p(\varphi_U^{-1}\varphi_Us)+\frm_p\sE_p\\ &=\grm_ps+\frm_p\sE_p. \end{align*} This computation shows that $\Phi^{-1}_{*}\circ\varphi$ “does not depend” on the choice of isomorphism $\varphi:\sE_U\cong\sO_{U}^{\op r}$. More precisely: if $\psi:\sE_V\cong\sO_{V}^{\op r}$ is another isomorphism for another open $V\subset M$, the computation we just did proves that $(\Phi^{-1}_{*}\circ\varphi)|_{U\cap V}=(\Psi^{-1}_{*}\circ\psi)|_{U\cap V}$. Therefore, the map $\sE\to\Sigma_E$ which on sections at any open $U\subset M$ is given by \begin{align*} \sE(U)&\to\Sigma_E(U)\\ s&\mapsto U\to E\\ &\quad\;\;\; p\mapsto\grm_p s+\frm_p\sE_p \end{align*} is well-defined (i.e., $p\mapsto\grm_p s+\frm_p\sE_p$ is smooth), is $\sO_M$-linear and is an isomorphism, since locally it has the form $\Phi^{-1}_{*}\circ\varphi$. (We are using that for $\sF,\sG\in\Mod(\sO_M)$, the assingment $U\subset M\mapsto\operatorname{Hom}_{\sO_U}(\sF|_U,\sG|_U)$ is a sheaf and that “being an isomorphism” is a local condition for morphisms of sheaves.) $\quad\square$
EDIT (2/2/23): The previous proof is done by exploiting the natural s.e.s.
$$ 0\to \frm_p\Sigma_{E,p}\to\Sigma_{E,p}\to E_p\to 0 $$ that exists for any smooth vector bundle $E$. On the other hand, I just learned that there is also another natural s.e.s. $$ 0\to\mu_p\Gamma(E)\to\Gamma(E)\to E_p\to 0 $$ where $\Gamma(E):=\Gamma(E,\Sigma_E)$ and $\mu_p:=\{f\in C^\infty(M)\mid f(p)=0\}$ is the maximal ideal of $C^\infty(M)$ of functions that vanish at $p$.
Thus, we get a natural isomorphism $$ \tag{1}\label{1} \Gamma(E)/\mu_p\Gamma(E)\cong E_p. $$
I assume that the proof of the essential surjectiveness I gave here can be rewritten, mutatis mutandis, in terms of the isomorphism \eqref{1} instead of the one from Lemma 1 (e.g., one should rewrite the proof of Proposition 4 by defining $E_p=\Gamma(M,\sE)/\mu_p\Gamma(M,\sE)$ at the beginning). I think the main advantage of \eqref{1} is that it allows one to only work with cosets of sections, instead of with cosets of germs of sections, as we did above; as a result, the proof would use less cumbersome notation.
EDIT (2/4/23): An interesting corollary I've just thought of: this equivalence of categories is really between multicategories. Smooth vector bundles over a smooth manifold $M$ and sheaves of modules over a ringed space $(X,\sO_X)$ can be regarded as multicategories with the multilinear maps as multimorphisms. In the former case, they are the fiber-wise multilinear smooth maps $E_1\oplus\cdots \oplus E_n\to E$ over $M$. In the latter case, they are the morphisms of sheaves $\mathcal{M}_1\oplus\cdots\oplus\mathcal{M}_n\to\mathcal{N}$ that are section-wise $\sO_X$-multilinear. The functor $\Sigma:\VB(M)\to\Mod(\sO_M)$ of categories promotes to a multifunctor of multicategories by observing that $\Sigma_{E\oplus E'}\cong\Sigma_E\oplus\Sigma_{E'}$ (verify that the anti-parallel maps $\Sigma_{E\oplus E'}\rightleftarrows\Sigma_E\oplus\Sigma_{E'}$ obtained from the universal property of the (co)product of sheaves are mutually inverse). In particular, it follows that $\Gamma(E_1\oplus\cdots\oplus E_n)\cong\Gamma(E_1)\oplus\cdots\oplus\Gamma(E_n)$ as $C^\infty(M)$-modules. On the one hand, essential surjectivity of the multifunctor $\Sigma$ is the same as essential surjectivity as a functor. On the other hand, the proof that $\Sigma$ is a fully faithful multifunctor is similar to the proof in the functor case, except that one now works with the generalization of Lemma 10.29 of Lee's Introduction to Smooth Manifolds to the multilinear case.