I need help understanding the counterexample in Vakil's notes (section 13.1.9) that the category of vector bundles does not form an abelian category.
He considers the
"trivial line bundle on $\mathbb{R}$ (with coordinate $t$) to itself, corresponding to multiplying by $t$"
The problem, according to Vakil is that it $\textbf{jumps rank}$, and "if you try to define a kernel and cokernel you will get confused".
I tried to put this into precise terms, and I figured that the map he is referring to seems, according to me to be
$$ \mathbb{R} \times \mathbb{R} \xrightarrow{\pi} \mathbb{R} \times \mathbb{R},$$ taking an element $(x,t)$ to $(xt,t)$.
I am not sure what he means by "jumping rank", I am not even sure what is the rank of this map!
I know that the rank is obtained by taking a trivialization $\{ U_i, \phi_i\}$ and looking at the number of copies of $\mathbb{R}$ that appear on the homeomorphism.
$$\pi^{-1}(U_i) \xrightarrow{\phi} U_i \times \mathbb{R}^n $$
but I don't know how to find the rank of this example.
I only know that taking fibers away from the origin works well, but taking $\pi^{-1}(0,0)$ we get $\mathbb{R} \times \{ 0 \}.$
Can someone explain in detail what is the real problem with this example, and what does Vakil mean by "jumping rank" and "trouble defining kernels and cokernels"?