0
$\begingroup$

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 $k(s)$?

$\endgroup$
4
  • $\begingroup$ Is $k(s)$ separable over $k$? If so, then the residue should be an element of $k$. $\endgroup$ Jun 25, 2015 at 14:12
  • $\begingroup$ Yes, in the case I'm interested in $k$ has characteristic zero. Could you please explain why? Thanks! $\endgroup$
    – zyx
    Jun 25, 2015 at 14:18
  • $\begingroup$ 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. $\endgroup$ Jun 25, 2015 at 14:42
  • $\begingroup$ 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. $\endgroup$ Jun 25, 2015 at 15:24

1 Answer 1

2
$\begingroup$

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 $\mathbf{Q}[t]/(t^2 + 1)$.

$\endgroup$
6
  • $\begingroup$ Could you give more details about how you computed it? $\endgroup$
    – zyx
    Jun 25, 2015 at 14:27
  • $\begingroup$ I used the definition of the residue. $\endgroup$ Jun 25, 2015 at 14:31
  • $\begingroup$ I agree with your computation. But I am confused about how this jibes with the Residue Theorem as stated, for instance, on p. 15, Proposition 6 of Serre's "Algebraic Groups and Class Fields". To extend the Residue Theorem over non-closed fields, presumably at some point we need to take traces. $\endgroup$ Jun 25, 2015 at 14:54
  • $\begingroup$ 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. $\endgroup$ Jun 25, 2015 at 15:22
  • 1
    $\begingroup$ If you apply the trace to the example given in the answer, you find 0, which is compatible with the residue theorem but not very interesting. $\endgroup$
    – abx
    Jun 25, 2015 at 16:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.