5.1. prerequisites

You should at least glance through the previous post to familiarize yourself with sheaves and the notion of manifolds as locally ringed spaces. For those who might already know a little about locally ringed spaces, let me note that I am actually only considering special types:

A firm command of basic commutative algebra is still required, including rings, algebras and their homomorphisms; modules and their homomorphisms; tensor products; and local rings.

A handy reference for details in sheaf theory is Tennison’s book cited in the References. My treatment of $G$-reductions and infinitesimal geometric structures is closely modeled on Conlon’s book on manifold theory, which is also a very nice reference. I am not assuming any prerequisite knowledge from either of these texts, however.

5.2. sheaves of modules

While we defined sheaves in the previous post, we did not define morphisms of sheaves. We do so now, restricting to the case of sheaves of sets:

There are multiple ways to transport sheaves from one topological space to another. One of the simplest is defined in:

If $X$ is a locally ringed space, we defined the stalks $\sco_{X,p}$ of the structure sheaves in the previous post. This definition was the obvious adaptation of the definition of the local rings $C^\infty_{M,p}$ over a manifold $M$. In both cases, the structure sheaves were sheaves of actual functions, and the germs in the stalks were certain equivalence classes of functions. (Remember, we are technically working with special types of locally ringed spaces, in which the structure sheaves are required to be sheaves of functions!)

For a general abstract sheaf $\cale$ on a topological space $X$, the sections in $\cale(U)$ may not be functions. Nevertheless, it is still possible to define stalks $\cale_p$ following essentially the same recipe.

We now give the first of the main definitions in this post:

Restricted sheaves and direct sums of sheaves will be vitally important when we talk about locally freeness. The first of these is defined in:

And now, direct sums:

The direct sum of sheaves of modules is built section-wise from direct sums of modules of sections. In particular, this is an example where a natural operation on modules induces a natural operation on sheaves of modules by applying the operation to modules of sections. This is the first time that we’re seeing this process in action, but we will see it again (repeatedly!) in the next post when we discuss many more operations on sheaves of modules.

I pointed out above that sections $s\in \cale(U)$ are not actual functions, they are simply elements in abstract modules. While that’s true, they actually induce, in a very natural way, certain functions whose codomains are made up of the stalks and the fibers of the module. While we defined stalks above, we haven’t met fibers yet; here’s their definition:

With the definition of fiber in hand, let me explain how a section $s\in \cale(U)$ may be conceptualized as an actual function in two different—but related—ways. Suppose that for each $p\in U$, we write $s(p)$ for the image of $s$ in the stalk $\cale_p$ under the localization map

\begin{equation}\label{loc-eqn} \cale(U) \to \cale_p \end{equation}

sending every section to its germ in the stalk. Then, if we take the disjoint union of the stalks, we get a function which, by an abuse of notation, we denote using the same symbol as the section:

\begin{equation}\label{etale-eqn} s: U \to \bigsqcup_{p\in U} \cale_p, \quad p \mapsto s(p). \end{equation}

In this way, the section $s$ becomes an actual function, but its codomain is a disjoint union of stalks which is (perhaps) a rather strange object the first time you meet it.

We can play the same game with fibers. Indeed, there is also the quotient map from the stalk to the fiber:

\begin{equation}\label{fib-eqn} \cale_p \to \cale_p/\mfm_p \cale_p = \cale_p \otimes_{\sco_{X,p}} \kappa(p) = E_p. \end{equation}

If we now write $s(p)$ for the image of $s$ in the fiber $E_p$ under the composition of \eqref{loc-eqn} and \eqref{fib-eqn}, then we have another function which, by a second abuse of notation, we denote using the same symbol:

\begin{equation}\label{vb-eqn} s: U \to \bigsqcup_{p\in U} E_p, \quad p \mapsto s(p). \end{equation}

Thus, the single section $s$ may be identified either with a function \eqref{etale-eqn} into a disjoint union of stalks, or with a function \eqref{vb-eqn} into a disjoint union of fibers.

It turns out that a pair of very beautiful theories may be developed based on these identifications of sections with actual functions. In the first case, when we think of functions of the form \eqref{etale-eqn}, we obtain the theory of étale spaces, which will prove to be very important in the next post when we discuss operations on sheaves of modules. In the second case, when we think of functions of the form \eqref{vb-eqn}, we obtain the theory of vector bundles, which is the subject of this post!

In fact, this idea of thinking of abstract sections $s\in \cale(U)$ as actual functions \eqref{vb-eqn} will be so important that we formalize it, along with several other notions, in:

We also need to observe that linear morphisms of sheaves of modules induce linear transformations on fibers:

5.3. locally free modules

The definition of freeness for regular modules adapts easily to the case of sheaves of modules. But for the latter there is also a notion of local freeness.

Transitioning from one trivializing open set $W_i$ to another $W_j$ is facilitated via the important functions defined in:

The significance of the prefix “$\GL(k)$-“ in $\GL(k)$-cocycle will be significant later when we talk about reduction of structure groups.

I will leave it to you to show that if $\eta$ is an isomorphism, then so are each of the fiber maps $\eta_p$. For the converse, we will use the well-known criterion that $\eta$ is an isomorphism if and only if each induced map $\cale_p \to \cale_p’$ on stalks is an isomorphism. This actually does not require local freeness, and a proof may be found in virtually any textbook that treats sheaves at an intermediate level; for example, Tennison’s book cited in the References contains a proof in Theorem 4.10. But, by local freeness, the stalks $\cale_p$ and $\cale_p’$ are free modules over the local ring $\sco_{X,p}$ of finite rank. Therefore, to prove the theorem, it will suffice to establish the following:

Again, it is clear that $\alpha$ being an isomorphism implies $\alpha \otimes_A k$ is one too. Conversely, if $\alpha \otimes_A k$ is an isomorphism, then its determinant is nonzero in the residue field $k$. Thus, the determinant of $\alpha$ is not in the maximal ideal $\mfm$, which means that it must be a unit. Hence, $\alpha$ is an isomorphism. This completes the proof of the theorem. Q.E.D.

5.4. vector bundles

Having defined the first main actor in this story, we now move on to the second.

In many references, the term vector bundle usually stands for locally trivial vector bundle (over a manifold), which we will define below. Also, though technically a vector bundle is a morphism $\pi:E \to X$, we shall often refer to the total space $E$ as if it were the vector bundle.

We now have two usages of the single symbol $E_p$ and the word fiber. It could represent the fiber

\begin{equation}\notag E_p = \cale_p \otimes_{\sco_{X,p}} \kappa(p) \end{equation}

of a sheaf of modules $\cale$ over $X$, or it could be the fiber

\begin{equation}\notag E_p = \pi^{-1}(p) \end{equation}

of a vector bundle $\pi:E \to X$. These two usages will be reconciled later, but for now, just be aware that you need to keep track of context in order to determine what I mean by fiber and the symbol $E_p$.

If $\pi:E \to X$ is a vector bundle, then the total space is clearly the disjoint union of the fibers:

\begin{equation}\label{disjoint-eqn} E = \bigsqcup_{p\in X} E_p. \end{equation}

Here, by writing the disjoint union, I do not mean the “external disjoint union” in which every element in $\bigsqcup_{p\in X} E_p$ is an ordered pair $(p,v)$ with $v\in E_p$; rather, I mean the “internal disjoint union,” which simply means $E = \bigcup_{p\in X} E_p$ and that the fibers are pairwise disjoint. Also, take care to notice that the decomposition \eqref{disjoint-eqn} is one of sets, and not necessarily one of topological spaces; that is, the topology on the total space $E$ is not required to be the disjoint union topology. From this point of view, a morphism $\phi:E\to E’$ of vector bundles over $X$ decomposes as a disjoint union of maps on fibers:

\begin{equation}\label{union-map} \phi = \bigsqcup_{p\in X} \phi_p : \bigsqcup_{p\in X} E_p \to \bigsqcup_{p\in X} E_p’, \quad \text{where $\phi_p: E_p \to E_p’$ is $\bbr$-linear}. \end{equation}

It’s another matter to go in the other direction, however. If one begins with a collection of $\bbr$-linear maps $\phi_p$ and forms the union \eqref{union-map}, then it is not guaranteed (in general) that one obtains a morphism of vector bundles.

Sometimes it is useful to compare vector bundles over different base spaces. To make sense of this, let $\alpha: X \to X’$ be a morphism of locally ringed spaces, and suppose $\pi:E \to X$ is a vector bundle over $X$ while $\pi’: E’ \to X’$ is a vector bundle over $X’$. Then a morphism from $\pi$ to $\pi’$ over $\alpha$ is a morphism $\phi: E\to E’$ on total spaces such that:

1. We have $\pi’ \circ \phi = \alpha \circ \pi$.

2. For each $p\in X$, the induced map on fibers \begin{equation}\notag \phi_p: E_p \to E’_{\alpha(p)}, \quad v\mapsto \phi(v), \end{equation} which is well-defined according to (1.), is an $\bbr$-linear map.

Notice then that the definition of a morphism as given in the box above is subsumed into this more general definition of morphisms of vector bundles over morphisms of locally ringed spaces. Indeed, a morphism of the first type is simply a morphism of second type over an identity map.

This second type of morphism fits in nicely with the next important definition, as you will see in the easy exercise immediately following:

The next important definition is the first step in associating to every vector bundle a sheaf of modules.

Remember that in passing from a sheaf of modules $\cale$ to its associated fibers, we simply changed the typeface from $\cale$ to $E_p$. We will often do the same for vector bundles: If $E$ is a vector bundle, then the module of sections $\Gamma(V,E)$ will be denoted $\cale(V)$, and we will refer to the $\sco_X$-module $V\mapsto \Gamma(V,E)$ by $\cale$.

For an $\sco_X$-module $\cale$ with fibers $E_p$, we already have several important evaluation maps. We now define analogs of these for vector bundles:

The evaluation maps \eqref{fiber-eval-eqn} will later be upgraded to isomorphisms for certain types of vector bundles (over manifolds). Thus, for these vector bundles, the vector bundle fibers $E_p$ and the fibers $\cale_p \otimes_{\sco_{X,p}} \kappa(p)$ of the modules of sections are the same thing, at least up to isomorphism.

Not only do we have a way to associate a sheaf of modules to every vector bundle, but this association is actually functorial:

Provided that $X$ is a manifold, we will see later that the functor $S$ is one half of an equivalence between certain full subcategories of $\Vect_X$ and $\Mod_X$.

5.5. locally trivial vector bundles and reduction of structure groups

Vector bundles in the previous section were defined over arbitrary locally ringed spaces. However, in order to define the analog for vector bundles of the locally free condition on sheaves of modules, we must specialize to manifolds.

I wrote this definition in a way that is exactly parallel to the definition of locally free sheaves of modules. Scroll back and forth between the definitions and compare them!

We now have transition functions and $\GL(k)$-cocyles for both locally free modules and locally trivial vector bundles. As we will see later, under the correspondence between such sheaves and vector bundles—which is the main objective of this post—these transitions functions and cocycles will coincide exactly.

I now want to explain the significance of the “$\GL(k)$-“ prefix in $\GL(k)$-cocycle.

Though technically a structure group is associated with a cocycle, when it will cause no confusion we shall often refer to a structure group of the vector bundle, leaving the representative cocycle implicit.

We have a fiber criterion for isomorphisms between locally free modules. Here’s the analog for locally trivial vector bundles:

I will let you prove that if $\psi$ is an isomorphism, then so too are each of the fiber maps $\psi_p$. Conversely, if each of the fiber maps $\psi_p$ are isomorphisms, then the inverse function

\begin{equation}\notag \psi^{-1} = \bigsqcup_{p\in M} \psi^{-1}_p : E’ \to E \end{equation}

exists as a set-theoretic function. We need to prove that it is smooth.

But smoothness is a local condition, so we can check smoothness of $\psi^{-1}$ on an open cover of its domain. In particular, let $W$ be an open subset of $M$ over which both $E$ and $E’$ trivialize, with trivializations

\begin{equation}\notag \phi: W\times \bbr^k \to E|_W \quad \text{and} \quad \phi’: W\times \bbr^k \to E’|_W. \end{equation}

We then have a commutative diagram

\begin{equation}\notag \begin{xy} \xymatrix{ E|_W \ar[r]^{\psi} & E’|_W \\ W \times \bbr^k \ar[u]^{ \phi} \ar[r]_{\color{\white}\tl{\psi}} & W \times \bbr^k \ar[u]_{\phi’} } \end{xy} \end{equation}

where $\tl{\psi}$ is a bijective smooth map of the form

\begin{equation}\notag (p,v) \mapsto (p,S(p)v) \end{equation}

for some function $S: W \to \GL(k)$. If we can prove that $S$ is smooth, then we’ll be done, for then the inverse

\begin{equation}\notag \tl{\psi}^{-1} : (p,v) \mapsto (p,S(p)^{-1}v) \end{equation}

is smooth since $S$ is smooth and so is inversion. But then $\psi^{-1} = \phi \circ \tl{\psi}^{-1} \circ (\phi’)^{-1}$, so $\psi^{-1}$ is indeed smooth.

To prove that $S$ is smooth, let $e_1,\ldots,e_k$ be the standard ordered basis of $\bbr^k$. Identifying operators in $\GL(k)$ with their standard matrices with respect to the $e_j$’s, we see that the $j$-th column of $S(p)$ is the vector $S(p)e_j$. But each of the the functions

\begin{equation}\notag W \to \bbr^k , \quad p \mapsto S(p)e_j, \quad j=1,\ldots,k, \end{equation}

are smooth, and hence each entry in the columns of $S(p)$ are smooth functions of $p$. Hence $S$ is smooth. Q.E.D.

5.6. infinitesimal geometric structures, examples

Reductions of structure groups of tangent bundles are of particular interest as they often correspond to familiar geometric structures on manifolds. We study these in more detail in this section, and then offer a class of very important examples of reductions that occur in physics.

First, let’s recall that the tangent bundle $T(M)$ over an $n$-dimensional manifold trivializes over open parametrizable subsets of the base space $M$. Indeed, suppose $\{W_i\}$ is an open cover of $M$ consisting of open parametrizable subsets, with diffeomorphisms

\begin{equation}\notag \psi_i: U_i \to W_i, \end{equation}

where each $U_i$ is an open subset of $\bbr^n$. Then the topology and smooth structure on $T(M)$ are the unique ones such that the global pushforwards

\begin{equation}\notag (\psi_i)_\ast : U_i \times \bbr^n \to T(M)|_{W_i}, \quad (t,v) \mapsto \left(\psi_i(t),(\psi_i)_{\ast,t}(v)\right), \end{equation}

are diffeomorphisms. The diffeomorphisms $\psi_i$ lift to isomorphisms of trivial vector bundles

\begin{equation}\notag \psi_i \times \id: U_i \times \bbr^k \to W_i \times \bbr^n, \quad (t,v)\mapsto \left( \psi_i(t),v\right), \end{equation}

over $\psi_i$, and then the trivialization

\begin{equation}\notag \phi_i: W_i \times \bbr^n \to T(M)|_{W_i} \end{equation}

is defined to be the composite

\begin{equation}\notag \phi_i = (\psi_i)_\ast \circ (\psi_i \times \id)^{-1} = (\psi_i)_\ast \circ (\psi_i^{-1} \times \id). \end{equation}

Putting everything together, we see that the fiber transition function

\begin{equation}\notag J_{ji} : W_i \cap W_j \to \GL(n) \end{equation}

is given by

\begin{equation}\label{jacob-eqn} J_{ji}(p) = (\psi_j^{-1} \circ \psi_i)_{\ast,t}, \quad p = \psi_i(t). \end{equation}

Now, the $\GL(n)$-cocycle $\{W_i,J_{ji}\}$ constructed in this way is a very special type of cocycle belonging to the tangent bundle $T(M)$; there may be other cocycles that do not manifest in this form. These special cocycles are given their own name in the first part of:

To help fix these ideas and concepts in our minds, let’s go through an example which is central to relativity theory. Let $n,k\geq 0$ be integers, not both zero, and consider the indefinite orthogonal group of signature $(n,k)$, denoted $\O(n,k)$, defined to be the subgroup of linear operators in $\GL(n+k)$ that preserve the scalar product

In particular, the class of indefinite orthogonal groups contains the regular orthogonal groups since $\O(n,0) = \O(n)$ for each $n\geq 1$.

Let $e_1,\ldots,e_{n+k}$ denote the standard ordered basis of $\bbr^{n+k}$. If we define $\eta \in \GL(n+k)$ by

\begin{equation}\notag \eta(e_j) = \begin{cases} e_j & : 1\leq j \leq n, \\ -e_j & : n < j \leq n+k, \end{cases} \end{equation}

then one easily sees that $\Lambda \in \O(n,k)$ if and only if

\begin{equation}\label{yup-eqn} \Lambda^t \eta \Lambda = \eta, \end{equation}

where $(-)^t$ denotes the transpose operation. Thus, $\O(n,k)$ is a real linear algebraic group and is, in particular, a Lie subgroup of $\GL(n+k)$ (since it is closed in the latter). If $k=0$, then $\eta$ is the identity operator and \eqref{yup-eqn} reduces to the fact that $\Lambda^{-1} = \Lambda^t$.

The lorentzian manifolds of signature $(1,3)$ provide the mathematical models of spacetimes in relativity theory.

Now, it turns out that an infinitesimal $\O(n,k)$-structure on a manifold $M$ is exactly what is needed in order to transport the scalar product \eqref{who-eqn} to the tangent spaces of $M$ in a coherent fashion. Indeed, suppose that $\{W_i,T_{ji}\}_{i,j\in I}$ is an $\O(n,k)$-cocycle on $T(M)$ with trivializations

\begin{equation}\notag \phi_i: W_i \times \bbr^{n+k} \to T(M)|_{W_i}, \end{equation}

and for each $p\in M$ choose an $i\in I$ such that $p\in W_i$. Then, define

where $\phi_i(p,\tl{v}_i) = v$ and $\phi_i(p,\tl{w}_i)= w$, and where the right-hand side of \eqref{who1-eqn} is the scalar product \eqref{who-eqn}. As you may then easily check, this definition of $\ang{v,w}_p$ does not depend on the particular choice of $W_i$ since the fiber transition functions $T_{ji}$ have their images in $\O(n,k)$.

Furthermore, the scalar product \eqref{who1-eqn} smoothly depends on $p$, in the following sense:

To see this, let

\begin{equation}\notag \theta: U \times \bbr^{n+k} \to T(M)|_U \end{equation}

be the associated local trivialization of the tangent bundle. Then

\begin{equation}\notag \bd_\mu|_p = (\theta \circ s_\mu)(p) \end{equation}

where $s_\mu: U \to U \times \bbr^{n+k}$ is the smooth map with $s_\mu(p)=(p,e_\mu)$ and where $e_\mu$ is the $\mu$-th standard basis vector of $\bbr^{n+k}$. To prove $g_{\mu\nu}$ is smooth, it will suffice to prove that the restriction

\begin{equation}\notag g_{\mu\nu} : U \cap W_i \to \bbr \end{equation}

is smooth, for each $i\in I$ such that $U\cap W_i \neq \emptyset$. But for each $p\in U\cap W_i$ we have

\begin{equation}\label{light-eqn} g_{\mu\nu}(p) = \ang{(\pi_2 \circ \phi_i^{-1} \circ \theta \circ s_\mu)(p),(\pi_2 \circ \phi_i^{-1}\circ \theta \circ s_\nu)(p)}, \end{equation}

where $\pi_2: (U\cap W_i) \times \bbr^{n+k} \to \bbr^{n+k}$ is the natural projection onto the second cartesian factor. But the right-hand side of \eqref{light-eqn} clearly smoothly depends on $p$, and hence so does the left-hand side.

The difference between flat and curved lorentzian manifolds is a central topic in the general theory of relativity. We will have to wait till a future post for a closer study of these gadgets.

The definition given above of the metric tensor is perhaps not the most mathematically elegant one, though it is the one that you use if you want to actually compute something. Later, we will see that the metric tensor is equivalently described by a certain type of $\sco_M$-linear morphism

\begin{equation}\notag g: \calx_M \otimes_{\sco_M} \calx_M \to \sco_M \end{equation}

where $\calx_M$ is the sheaf of vector bundles on $M$, i.e., the sheaf of sections of the tangent bundle $T(M)$. But notice that we are (apparently) tensoring $\calx_M$ against itself over the structure sheaf $\sco_M$; this tensor operation—which is a bit tricky to define—will be described in the next post of this series.

5.7. locally trivial vector bundles to locally free modules

Let’s return to the general theory. We have the functor

\begin{equation}\notag S: \Vect_M \to \Mod_M \end{equation}

which sends every vector bundle $\pi:E \to M$ over a manifold $M$ to its module of sections $\cale$; see the exercise above. My goal in this section is to prove that the image of the restricted functor

\begin{equation}\notag S: \ltVect_M \to \Mod_M \end{equation}

actually lands in the full subcategory $\lfMod_M$ of locally free modules. Here’s the theorem:

Given an open subset $V\subseteq W$ and a section $s\in \cale(V)$, suppose we consider the smooth maps

\begin{equation}\notag f_i: V \xrightarrow{s} E|_W \xrightarrow{\phi^{-1}} W \times \bbr^k \xrightarrow{\pi_i} \bbr, \quad i=1,\ldots,k, \end{equation}

where $\pi_i$ is the natural projection onto the $i$-th cartesian factor of $\bbr^k$. Then, for all $p\in V$, we have

\begin{equation}\notag s(p) = \phi(p,f_1(p),\ldots,f_k(p)). \end{equation}

Then, define

\begin{equation}\notag s = \Phi_V(f_1,\ldots,f_k). \end{equation}

I will let you check all the remaining details. Q.E.D.

We recall in the exercise above that the evaluation map $\ev_{E,p}$ is induced by the evaluation map $\dev_p:\cale_p \to E_p$ on the stalk whose kernel contains the submodule $\mfm_p \cale_p$. Thus, to prove the theorem, we need to show two things: (1) The kernel of $\dev_p$ is actually equal to the submodule $\mfm_p \cale_p$, and (2) the map $\dev_p$ is surjective.

So, we first prove $\ker{\dev_p} \subseteq \mfm_p \cale_p$. Let $s\in \cale_p$ be a section with $s(p)=0$. Choose an open neighborhood $W$ of $p$ over which $E$ trivializes, and by shrinking it if necessary, suppose that $s\in \cale(W)$. Then $\cale$ trivializes over $W$ as well via an $\sco_W$-linear map

\begin{equation}\notag \Phi: \sco_W^k \to \cale|_W \end{equation}

built from the trivialization

\begin{equation}\notag \phi: W \times \bbr^k \to E|_W \end{equation}

as described in the previous theorem. The map

\begin{equation}\notag \Phi_W: \sco_M(W)^k \to \cale(W) \end{equation}

is an $\sco_M(W)$-linear isomorphism; let $e_1,\ldots,e_k \in \cale(W)$ be the sections corresponding to the standard basis of the direct sum $\sco_M(W)^k$. Thus, we may write

\begin{equation}\label{mat-eqn} s = f_1 e_1 + \cdots +f_k e_k, \end{equation}

where $s = \Phi_W(f_1,\ldots,f_k)$. But $\phi_p(f_1(p),\ldots,f_k(p)) = s(p)=0$, and hence

\begin{equation}\notag f_1(p) = \cdots = f_k(p) = 0 \end{equation}

since the map on fibers $\phi_p:\bbr^k \to E_p$ is a linear isomorphism. Thus, $f_i \in \mfm_p$ for each $i=1,\ldots,k$, which shows $s\in \mfm_p\cale_p$ in view of \eqref{mat-eqn}.

All that remains is to prove that the evaluation map $\dev_p$ is surjective. Choose a vector $v\in E_p$ and a $k$-tuple $(u_1,\ldots,u_k)\in \bbr^k$ such that

\begin{equation}\notag \phi_p(u_1,\ldots,u_k) = v. \end{equation}

Choose functions $f_1,\ldots,f_k \in \sco_{M,p}$ such that $f_i(p) = u_i$ for each $i=1,\ldots,k$. We may choose an open neighborhood $V$ of $p$ small enough so that $V\subseteq W$ and $f_i \in \sco_M(V)$ for each $i=1,\ldots,k$. If we define

\begin{equation}\notag s = \Phi_V(f_1,\ldots,f_k), \end{equation}

then $\dev_p(s) = s(p) =v$. Q.E.D.

5.8. locally free modules to locally trivial vector bundles

Having described a method to associate a locally free sheaf of modules to every locally trivial vector bundle via the functor

\begin{equation}\notag S: \ltVect_{M} \to \lfMod_M, \end{equation}

we now describe a method to go the other direction and associate a locally trivial vector bundle to every locally free module. We will obtain a functor

\begin{equation}\notag V: \lfMod_M \to \ltVect_{M}, \end{equation}

and together $S$ and $V$ will be mutually inverse equivalences of categories.

So, let $\cale$ be a locally free sheaf of rank $k$ on a manifold $M$ with fibers

\begin{equation}\notag E_p = \cale_p \otimes_{\sco_{M,p}} \kappa(p). \end{equation}

The first step in associating a vector bundle to $\cale$ is to define the total space of the vector bundle:

\begin{equation}\notag E \defeq \bigsqcup_{p\in M} E_p. \end{equation}

Here, the disjoint union is the “external” one, so that an element in $E$ is a pair $(p,v)$ with $p\in M$ and $v\in E_p$. We then define the natural projection

\begin{equation}\notag \pi:E \to M, \quad (p,v) \mapsto p, \end{equation}

and we claim that $\pi$ is a locally trivial vector bundle of rank $k$. The proof this claim will use the Topology and Structure Sheaf Construction Lemmas proved in the appendix. In particular, I will use the notation in those lemmas, so you might want to at least glance at them quickly before continuing.

For the topology on the total space $E$, we need to cook up a cover $\{U_i\}_{i\in I}$ of $E$ and bijections

\begin{equation}\notag \phi_i: Y_i \to U_i, \end{equation}

where each $Y_i$ is a topological space. Here are the definitions:

We now need to check that these data have the four properties stated in the Topology Construction Lemma. The numbering of the properties here is the same as the numbering in that lemma.

We have thus constructed all the data needed to apply the Topology Construction Lemma to the total space $E$ to obtain a topology on it. Actually, we have also constructed the data needed to apply the Structure Sheaf Construction Lemma since we proved that the homeomorphisms \eqref{comp-eqn} are smooth, and thus $E$ is equipped with a structure sheaf $\sco_E$ endowing it with the structure of a manifold. We summarize the situation in a theorem:

The proof of the theorem follows immediately from the construction given above.

Now, we define a functor

\begin{equation}\label{vec-fun-eqn} V: \lfMod_M \to \ltVect_M \end{equation}

on objects by letting $V(\cale)$ be the vector bundle constructed in the theorem. For its action on morphisms:

5.9. the main equivalence

We have (finally!) reached the final section of this post, in which we prove that the functors

\begin{equation}\notag S: \ltVect_M \leftrightarrows \lfMod_M :V \end{equation}

are mutually inverse equivalences of categories. This will require us to construct the natural isomorphisms between the composite functors $S\circ V$ and $V\circ S$ and the respective identity functors on $\lfMod_M$ and $\ltVect_M$. The first step in this construction is given in the following, which uses the evaluation maps defined here and here.

(1.): We define the map \eqref{boo1-eqn} to have fiber maps \eqref{boo2-eqn}. So, we need only check that $\dev_E$ is smooth. For this, we use the two main theorems (here and here). From them, we know that if $E$ trivializes over an open set $W$ in $M$ with trivialization

\begin{equation}\notag \phi: W\times \bbr^k \to E|_W, \end{equation}

then:

1. The module of sections $\cale$ of $E$ trivializes over $W$ with a trivialization \begin{equation}\notag \Phi: \sco_W^k \to \cale|_W \end{equation}

2. The vector bundle $V(\cale)$ trivializes over $W$ with a trivialization \begin{equation}\notag \phi’: W \times \bbr^k \to V(\cale)|_W \end{equation}

3. The trivializations $\Phi$ and $\phi’$ fit into a composite commutative diagram \begin{equation}\notag \begin{xy} \xymatrix{ \bbr^k \ar[r]^-{ \phi_p’} & \cale_p \otimes \kappa(p) \ar[d]^{ = } \\ \kappa(p)^k \ar[r]^-{ \Phi_p} \ar[d]_{\ev_{\sco_M^k,p}} \ar[u]^{ \ev_{\sco_M^k,p}} & \cale_p \otimes \kappa(p) \ar[d]^{\ev_{E,p}} \\ \bbr^k \ar[r]_-{ \phi_p} & E_p} \end{xy} \end{equation}

Thus, we have a commutative diagram

\begin{equation}\notag \begin{xy} \xymatrix{ \bigsqcup_{p\in W} (\cale_p \otimes \kappa(p)) \ar[r]^-{ \dev_E} & \bigsqcup_{p\in W} E_p \\ W\times \bbr^k \ar[r]_-{=} \ar[u]^{\phi’} & W \times \bbr^k \ar[u]_{ \phi} } \end{xy} \end{equation}

from which it follows immediately that $\dev_E$ is smooth.

(2.): We define the map \eqref{you1-eqn} to have section maps \eqref{you2-eqn}. So, we first need to check that the map \eqref{you2-eqn} is well-defined, i.e., that the image of the adjoint evaluation map over $W$ actually lands in the module of sections $\Gamma(W,E)$. To do this, remember that the image of a section $s\in \cale(W)$ under the adjoint evaluation map is the function

\begin{equation}\label{blank-eqn} s: W \to \bigsqcup_{p\in W} E_p = E|_W, \quad s \mapsto (p,s(p)), \end{equation}

where $s(p)$ denotes the image of $s$ in the fiber $E_p = \cale_p \otimes_{\sco_{M,p}} \kappa(p)$. In particular, we are using the single symbol $s$ to denote the abstract section $s\in \cale(W)$, and also the function \eqref{blank-eqn}. Thus, to show that \eqref{you2-eqn} is well-defined, we need to show that the function \eqref{blank-eqn} is smooth.

Fixing a point $p\in W$ and shrinking $W$ if needed, we may assume that $\cale$ trivializes over $W$ with trivialization

\begin{equation}\notag \Phi: \sco_W^k \to \cale|_W. \end{equation}

Then, by construction, the vector bundle $E$ also trivializes over $W$ with trivialization

\begin{equation}\notag \phi: W \times \bbr^k \to E|_W, \end{equation}

such that $\Phi_W$ and $\phi$ coincide up to “evaluation over $W$”; see the main theorem. In particular, if we have

\begin{equation}\notag s = \Phi_W(f_1,\ldots,f_k) \end{equation}

for some $f_1,\ldots,f_k\in \sco_M(W)$, then

\begin{equation}\notag s(p) = \phi(p,f_1(p),\ldots,f_k(p)). \end{equation}

But this means that the composite of the diffeomorphism $\phi^{-1}$ with the map \eqref{blank-eqn} is given by

\begin{equation}\notag p \mapsto (p,f_1(p),\ldots,f_k(p)) \end{equation}

which clearly smoothly depends on $p$. Thus \eqref{blank-eqn} is smooth, so the image of \eqref{you2-eqn} does indeed land in $\Gamma(W,E)$. I will leave the rest of the proof up to you, to check that \eqref{you1-eqn} is indeed an $\sco_M$-linear morphism. Q.E.D.

We now have enough to state and prove the main theorem of this post:

I will quickly sketch the outline. First, note that \eqref{tv1-eqn} is an isomorphism of vector bundles since it is an isomorphism on fibers and both $V(\cale)$ and $E$ are locally trivial (see here). Then, to complete the proof of part (1.), one needs to check that $\dev$ is indeed a natural transformation, which I will leave for you.

Then, to prove that \eqref{tv2-eqn} is an isomorphism, we may similarly take advantage of the fact both $\cale$ and $S(E)$ are locally free and check that $\eta_\cale$ is an isomorphism on fibers (see here). But, as you may check, the fiber map

\begin{equation}\notag (\eta_\cale)_p : E_p \to S(E)_p \otimes_{\sco_{M,p}} \kappa(p) \end{equation}

is actually the inverse of the (isomorphic) evaluation map

\begin{equation}\notag \ev_{E,p}: S(E)_p \otimes_{\sco_{M,p}} \kappa(p) \to E_p. \end{equation}

Thus, \eqref{tv2-eqn} is indeed an isomorphism. Then, to complete the proof, one needs to check that $\eta$ is a natural transformation, which I will again leave to you. Q.E.D.

5.10. appendix, construction lemmas

Here, we prove two “construction” lemmas that we will use to obtain manifold structures on the vector bundles associated with locally free modules. The first lemma constructs topologies, while the second lemma constructs structure sheaves.

Suppose we expand the codomain of the $\phi_i$’s to

\begin{equation}\notag \phi_i:Y_i \to X. \end{equation}

Then the topology on $X$ is the finest topology for which each of the maps $\phi_i$ is continuous; thus, a subset $V\subseteq X$ is open if and only if $\phi_i^{-1}(V)\subseteq Y_i$ is open, for each $i\in I$. This topology is clearly the unique topology for which each $\phi_i$ is continuous. I will leave it to you to prove that this topology is second-countable and Hausdorff.

Thus, we need only prove that each $U_i$ is open and each $\phi_i$ is a homeomorphism onto $U_i$. To do this, it will be convenient to prove that the following two conditions on a subset $V\subseteq X$ are equivalent:

Of course, the condition in (1.) is nothing but the condition defining openness of $V$ in $X$. Note first that (1.) clearly implies (2.). For the converse, let $i \in I$ and note that

\begin{equation}\label{who10-eqn} \phi_i^{-1}(V) = \phi_i^{-1}(V\cap U_i) = \bigcup_{j\in J} \phi_i^{-1}(V\cap U_i \cap U_j). \end{equation}

But $\phi_j^{-1}(V\cap U_j)$ is assumed open in $Y_j$ for each $j\in J$, and therefore

\begin{equation}\notag \phi_j^{-1}(V\cap U_i \cap U_j)=\phi_j^{-1}(V\cap U_j) \cap \phi_j^{-1}(U_i\cap U_j) \subseteq \phi_j^{-1}(U_i \cap U_j) \end{equation}

is an open inclusion. Then $\phi_i^{-1}\circ \phi_j$ is a homeomorphism by hypothesis, and hence

\begin{equation}\notag \phi_i^{-1}(V\cap U_i \cap U_j)= (\phi_i^{-1}\circ \phi_j)\left(\phi_j^{-1}(V\cap U_i \cap U_j) \right) \subseteq \phi_i^{-1}(U_i\cap U_j) \end{equation}

is an open inclusion. However, $\phi_i^{-1}(U_i\cap U_j)$ is open in $Y_i$ by hypothesis, and hence each of the sets on the right-hand side of \eqref{who10-eqn} is open in $Y_i$. Thus, (2.) implies (1.).

In view of the equivalence of (1.) and (2.), it follows immediately that each $U_i$ is open. I will leave it to you to prove then that each $\phi_i$ is a homeomorphism, which would complete the proof. Q.E.D.

Suppose we expand the codomain of the $\phi_i$’s to

\begin{equation}\notag \phi_i:Y_i \to X. \end{equation}

Given a nonempty open set $V\subseteq X$, we define $\sco_X(V)$ to be the set of those functions $f:V \to \bbr$ such that $f\circ \phi_i \in \sco_{Y_i}(\phi_i^{-1}(V))$, for each $i\in I$. I will leave it to you to check that this definition yields a well-defined sheaf $\sco_X$ for which each $\phi_i$ is a morphism of locally ringed spaces, and that it is the unique sheaf on $X$ with this property.

Thus, we need only prove that each $\phi_i$ is actually an isomorphism of ringed spaces

\begin{equation}\notag \phi_i:(Y_i,\sco_{Y_i}) \to (U_i,\sco_{U_i}), \end{equation}

where $\sco_{U_i}$ is the restriction of $\sco_X$ to the open subset $U_i$. Letting $V\subseteq X$ be a nonempty open set, it will be convenient to prove that the following two conditions on a function $f:V \to \bbr$ are equivalent:

Notice that condition (1.) is exactly the defining condition for $f$ to be a function in $\sco_X(V)$. Obviously (1.) implies (2.), so we need only prove the converse. For that, let $i\in I$ and note that

\begin{equation}\label{union-eqn} \phi_i^{-1}(V) = \phi_i^{-1}(V\cap U_i) = \bigcup_{j\in J} \phi_i^{-1}(V\cap U_i \cap U_j). \end{equation}

But $f\circ \phi_j \in \sco_{Y_j}(\phi_j^{-1}(V\cap U_j))$ for each $j\in J$, and by restriction, we have

\begin{equation}\notag f\circ \phi_j \in \sco_{Y_j}(\phi_j^{-1}(V\cap U_i \cap U_j)) \end{equation}

for each $i\in I$ as well. But then

\begin{equation}\label{union1-eqn} f\circ \phi_i = f\circ \phi_j \circ (\phi_j^{-1} \circ \phi_i) \in \sco_{Y_i}(\phi_i^{-1}(V\cap U_i \cap U_j)) \end{equation}

since the composite $\phi_j^{-1} \circ \phi_i$ is a morphism. Thus, in view of \eqref{union-eqn} and \eqref{union1-eqn}, and the fact that $\sco_{Y_i}$ is a sheaf, we see that

\begin{equation}\notag f\circ \phi_i \in \sco_{Y_i}(\phi_i^{-1}(V)), \end{equation}

as desired. From the equivalence of (1.) and (2.), it follows immediately that each $\phi_i$ is an isomorphism. Q.E.D.

5.11. references

• L. Conlon. Differentiable manifolds. Reprint of the 2001 second edition. Mod. Birkhäuser Class. Birkhäuser Boston, Inc., Boston, MA, 2008.

• B. R. Tennison. Sheaf theory. London Mathematical Society Lecture Note Series, No. 20. Cambridge University Press, Cambridge, England-New York-Melbourne, 1975,