I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri.
I quote from the paper:
Let $W_1$ and $W_2$ be two vector bundles of rank $n$ on the compact Riemann surface $X$. A (holomorphic) homomorphism $f:W_1\to W_2$ is said to be of maximal rank if the canonical extension $\bigwedge^nf:\bigwedge^nW_1\to\bigwedge^nW_2$ is a non-zero homomorphism. If $f:W_1\to W_2$ is a homomorphism of maximal rank we have $d(W_1)\le d(W_2)$, and if $d(W_1)=d(W_2)$, $f$ is an isomorphism. (These statements follow from the corresponding statements for line bundles.)
Let $V$ and $W$ be two vector bundles on $X$, not necessarily of the same rank. Let $f:V\to W$ be a non-zero homomorphism. Since the structure sheaf $\mathbf O_x$ is a sheaf of principal ideal domains, we see that $f$ has the following canonical factorisation $$\begin{array}\\ 0&\to&V_1&\to&V&\overset\eta\to&V_2&\to&0\\ &&&&&&\downarrow\small g\\ 0&\gets&W_2&\gets&W&\underset i\gets&W_1&\gets&0 \end{array}$$ where $V_1,V_2,W_1,W_2$ are vector bundles, each row is exact, $f=i\circ g\circ\eta$ and $g$ is of maximal rank. We call $W_1$ the subbundle of $W$ generated by the image of $f$.
Can someone please explain how does any non-zero homomorphism of vector bundles can be factored through a maximal rank homomorphisms? It will be helpful if someone provides with an simple to read reference.
First, take $V_2=f(V)$. Since we are over a non-singular curve, $V_2$ is a vector bundle and we have the top exact sequence where $V_1$ is the kernel of $f$. Let $W_1$ be the set of all elements in $W$ which go to a torsion element in $W/V_2$. Then we have the bottom exact sequence, where $W_2=W/W_1$. Also, $V_2\subset W_1$. Both have the same rank and the inclusion (as sheaves) says it is of maximal rank.