Why is $\frac{\partial\mathcal{L}}{\partial(\partial_\nu \bar{\psi})} = 0$, for the Dirac Lagrangian $\mathcal{L} = \bar{\psi}(i \gamma^\mu \partial_\mu - m)\psi$?
This comes up in deriving the Noether current for $\psi \rightarrow e^{i\alpha}\psi$ for example.
My confusion comes from the fact that we can write the following term in the Lagrangian $i\bar{\psi}\gamma^\mu\partial_\mu\psi = -i(\partial_\mu \bar{\psi})\gamma^\mu\psi$ by integrating by parts which makes it look like $\frac{\partial\mathcal{L}}{\partial(\partial_\nu \bar{\psi})} = -i \gamma^\mu \psi$. In fact, this is how we get the equations of motion for $\bar{\psi}$.