Morphism of vector bundles in algebraic geometric terms

Question

Hartshorne doesn't give a definition of morphism of vector bundles but only that of isomorphism between vector bundles. After looking at a few online references, I think I understand the following: Let $V\xrightarrow{f}X$ and $V'\xrightarrow{f'}X$ be two vector bundles of rank $n$ and $m$ respectively on a scheme $X$. Then a morphism of vector bundles is a morphism of $X$-schemes $\phi:V \longrightarrow V'$ such that the induced map $V|_{U_i}\cong\mathbb{A}_{U_i}^n \longrightarrow \mathbb{A}_{U_i}^m \cong V'|_{U_i}$ is linear for all $U_i$ where $\{U_i\}_{i \in I}$ is a cover of $X$ which provides local trivializations for both $V$ and $V'$.

I am not sure if my understanding is correct. Please correct me if I am wrong. I do know the differential geometry version of the definition but I want a definition of morphism of vector bundles corresponding to the way Hartshorne has defined vector bundles (II,Ex:5.18).

Thanks in advance!

Answer 1

It depends of the topology that you are using to define the bundle, vector bundle are not always locally trivial for the Zariski topology. It a reason which has motivates Grothendieck to define the Etale topology.

https://mathoverflow.net/questions/134267/are-principal-bundles-isotrivial

Answer 2

Let $X$ be a scheme and be $E,F$ finite rank locally trivial $\mathcal{O}_X$-modules. Assume for simplicity that $X:=Spec(A)$ and $E:=A^n,F=A^m$ are free $A$-modules of rank $n,m$. An $A$-linear map $\phi: E \rightarrow F$ induce an $A$-linear map $\phi^*: F^* \rightarrow E^*$ by $\phi^*(f):= f\circ \phi$. We get an induced map of $A$-algebras

$\phi^*: Sym_A(F^*) \rightarrow Sym_A(E^*)$.

Passing to schemes we get an induced map of "geometric vector bundles" (a morphism over $X$)

$\mathbb{V}(\phi): \mathbb{V}(E):=Spec(Sym_A(E^*)) \rightarrow \mathbb{V}(F):=Spec(Sym_A(F^*))$

and I believe the map $\mathbb{V}(\phi)$ is a "map of geometric vector bundles". It is a "linear map" in the sense that if you choose an element $f\in F^*$ it follows $\phi^*(f) \in E^*$. This construction is "functorial" in the sense that if $E,F$ are $A$-modules and $\phi$ is an arbitrary $A$-linear map, you may always define $\mathbb{V}(\phi)$. In Hartshorne Ex.II.5.18 some results on this construction are proved. It may be that with this definition it follows the category of finite rank locally trivial sheaves and morphism of $\mathcal{O}_X$-modules is equivalent to the category of geometric vector bundles and maps of geometric vector bundles, but this has to be checked. I guess you may find it in the litterature but at the moment I do not have a precise reference.

Example: If $k$ is an algebraically closed field of characteristic zero and if $G:=SL(V)$ is the special linear group on a finite dimensional $k$-vector space $V$ and $H \subseteq G$ a closed subgroup, you may always realize the "quotient" $G/H$ as a smooth quasi projective scheme of finite type over $k$. There is a canonical projection morphism $\pi: G \rightarrow G/H$ and $\pi$ is locally trivial in the etale topology in general. If $H$ is a parabolic subgroup of $G$, it follows $\pi$ is locally trivial in the Zariski topology with fiber $H$. In this case it follows the quotient $G/H$ is a smooth projective scheme of finite type over $k$. The map $\pi$ is a principal fiber bundle with fiber $H$, hence $\pi$ is not a "geometric vector bundle" in the above sense since $H$ is not an affine space.

Example: In the case when $H$ is a parabolic group it follows $H$ is classified. There is a flag of $k$-vector spaces $V_{\bullet} \subseteq V$ with the property that $H$ is the subgroup of $SL(V)$ fixing the flag. In this case the quotient $SL(V)/H$ is the flag variety $F(V_{\bullet})$ of $V$ of type $V_{\bullet}$. The $k$-rational points of $F(V_{\bullet})$ parametrize the set of flags in $V$ of type $V_{\bullet}$. In particular if $V_{\bullet}:=W \subseteq V$ where $W$ is a $d$-dimensional sub space, it follows $F(V_{\bullet})\cong \mathbb{G}(d,V)$ is the grassmannian of $d$-planes in $V$. Hence for the grasmannian here is a canonical projection map

$\pi: SL(V) \rightarrow \mathbb{G}(d,V)$

which is locally trivial in the Zariski topology with fiber $H$. Using $\pi$ me may associate to any linear representation $\rho: H \rightarrow GL_k(W)$ a vector bundle $\pi(\rho): E(\rho)\rightarrow \mathbb{G}(d,V)$ with a left $SL(V)$-action such that the map $\pi(\rho)$ is $SL(V)$-invariant. This sets up an equivalence of categories between the category of finite dimensional $k$-linear $H$-representations and the category of $SL(V)$-linearized vector bundles on the grassmannian. Because $\pi$ is locally trivial in the Zariski topology, it follows $\pi(\rho)$ is locally trivial in the Zariski topology. And from Hartshorne exercise II.5.18 there is a finite rank locally trivial sheaf $\mathcal{E}$ on $\mathbb{G}(d,V)$ with $E(\rho) \cong \mathbb{V}(\mathcal{E}^*)$.

Example. This construction relates the representation theory of $SL(V)$ to the geometry of the flag variety $F(V_{\bullet})$. Given any finite dimensional irreducible $SL(V)$-module $(W,\rho)$, there is a parabolic subgroup (not unique) $P \subseteq SL(V)$ and an invertible sheaf $\mathcal{L}(\rho)$ in $Pic(SL(V)/P)$ with an $SL(V)$-linearization and an isomorphism

$W \cong \operatorname{H}^0(SL(V)/P, \mathcal{L}(\rho))$

of $SL(V)$-modules - this is the Borel-Weil-Bott theorem.

Example. Let $k$ be a field of characteristic zero and let $V:=k\{e_0,e_1$} and $V^*:=k\{x_0,x_1\}.$ Let $l \subseteq V$ be a line and let $P \subseteq SL(V)$ be the parabolic subgroup fixing $l$. It follows the quotient $SL(V)/P$ is isomorphic to the projective line $\mathbb{P}^1_k$. For any integer $d \geq 1$ it follows the symmetric product $Sym^d(V^*)$ is an irreducible finite dimensional $SL(V)$-module, and these are all such modules. Let $\mathcal{O}(d)$ be the invertible sheaf corresponding to $d$. It follows $\mathcal{O}(d)$ has a canonical $SL(V)$-linearization and there is a canonical isomorphism ($d\geq 1$)

$Sym^d(V^*) \cong \operatorname{H}^0(SL(V)/P,\mathcal{O}(d))$

of $SL(V)$-modules. Hence all irreducible $SL(V)$-modules may be realized as global sections of invertible sheaves on the projective line.

In general all finite dimensional irreducible $SL(V)$-modules for any $V$ may be realized geometrically as global sections of certain invertible sheaves on the flag variety $F(V_{\bullet})$ for some flag $V_{\bullet}$. This is a much studied subject involving combinatorics, geometry and representations.

Question: "So what would be a correct definition of morphism of vector bundles in algebraic geometry terms in Zariski topology? – Sam"

Answer: The map

$\phi: f^{-1}(U) \cong \mathbb{A}^n_{U} \cong f'^{-1}(U) $

should be a map

$f:A[x_1,..,x_n] \rightarrow A[x_1,..,x_n]$

with $f(x_k)=\sum_i a_{ik}x_i$ with $a_{ik} \in A$ and $U:=Spec(A)$ a local trivialization of $V,V'$. To specify a map $f$ of $A$-algebras is equivalent to giving a set of $n$ polynomials $p_1(x_i),..,p_n(x_i)$. You define

$f(x_l):=p_l(x_i)$.

The ring homomorphism $f$ is "linear" iff $p_l(x_i)\in A[x_1,..,x_n]$ is a "linear polynomial" for $i=1,..,n$.