On finite separable morphisms

Question

Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective varieties over an algebraically closed field $k$. Let $d$ be the degree of $f$. We have an exact sequence $$0\longrightarrow\mathcal{O}_Y\longrightarrow f_\ast\mathcal{O}_X\longrightarrow \mathcal{F}\longrightarrow 0$$ When $\operatorname{char} k\nmid d$ the above exact sequence splits via $\frac{1}{d}Tr:f_\ast\mathcal{O}_X\longrightarrow \mathcal{O}_Y$, hence $H^0(Y,\mathcal{F})=0$. Does the latter equality hold also in the case $\operatorname{char} k\mid d$? I'm particularly interested in the case of curves.