Let $C$ be a smooth curve over a field $k$, $\overline{C}$ the smooth compactification and $S=\overline{C} \setminus C$. We think of $S$ as a reduced divisor defined over $k$. Take the sheaf of logarithmic differentials $\Omega^1(\log S)$ and a non-zero rational section $\eta$. Then there is a notion of residue of $\eta$ at points in $S$.

*Question*: Let $s$ be a point in $S$. Is $\mathrm{Res}_s(\eta)$ an element of $k$ or of the residue field $k(s)$?

definesresidues to be elements in $k$, even if the residue field extension is not separable. Presumably for any other definition of residue, if you apply the trace, you get back to Tate's definition. $\endgroup$