Suppose that $\mathbb{P}^r=\mathbb{P}^r_K$ is the projective space over a field $K$. Let $\mathcal{O}_{\mathbb{P}^r}(-1)^n\longrightarrow \mathcal{O}_{\mathbb{P}^r}^m$ be a morphism of vector bundles. How can I show that $$m\leq h^0(C)$$ where $C$ is the cokernel of the map and $h^0(-)=\dim_k H^0(-)$?
Edit
I realize that this seems not to hold, but then I can't understand why the following theorem doesn't fail. It's form Eisenbud's Geometry of Syzygies.
Corollary 5.9 is the following:
A linear presentation for a graded finitely generated module $M$ over a polynomial ring $R$ is an exact sequence of the form $$R(-1)^n\longrightarrow R^m\longrightarrow M\longrightarrow 0$$
I don't think this is true. For example, let $m = 1$; then there is a surjection $\mathcal{O}_{\mathbb{P}^r}(-1)^{r+1} \to \mathcal{O}_{\mathbb{P}^r}$, i.e. the cokernel $C$ is the zero sheaf.