# rationality of residues of differentials

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 $k(s)$?

• Is $k(s)$ separable over $k$? If so, then the residue should be an element of $k$. Jun 25, 2015 at 14:12
• Yes, in the case I'm interested in $k$ has characteristic zero. Could you please explain why? Thanks!
– zyx
Jun 25, 2015 at 14:18
• I believe Count Dracula is correct. I was assuming that the residue would be the trace of the residue obtained from a geometric point after base change. But that seems to be wrong. Jun 25, 2015 at 14:42
• I think the story is more complicated than it seems at first. Tate defines residues 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. Jun 25, 2015 at 15:24

The residue of $\text{d}t/(t^2 + 1)$ at the point $(t^2 + 1)$ of $\mathbf{A}^1_\mathbf{Q}$ is the class of $-t/2$ in $\mathbf{Q}[t]/(t^2 + 1)$.
• Hey ... wait a minute! According to Tate's "Residues of Differentials on Curves", the residue is always defined to be an element of $k$. I guess that Tate builds the trace into his definition of the residue. Jun 25, 2015 at 15:22