Timeline for Serre duality over a non-algebraically closed field
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2015 at 13:18 | vote | accept | CommunityBot | ||
| Aug 24, 2015 at 3:41 | answer | added | Karl Schwede | timeline score: 8 | |
| Aug 23, 2015 at 22:31 | comment | added | grghxy | The proof works verbatim for any projective Cohen-Macaulay scheme of pure dimension $n$ over any field. Look at the proof! The only place $k$ being algebraically closed is "used" is that the local ring at a closed point of $\mathbf{P}^N_k$ is regular, but every local ring at a closed point on an affine space over any field is regular (since $R[t]$ is a regular ring for any regular noetherian ring $R$, as a standard application of Serre's homological criterion for regularity: see Theorem 19.5 in Matsumura's book "Commutative Ring Theory"). | |
| Aug 23, 2015 at 19:07 | comment | added | Will Sawin | Need $X$ geometrically connected. In this case, I think you can prove it using flat base change, since passing to the algebraic closure is flat. | |
| Aug 23, 2015 at 18:41 | comment | added | user39380 | @KarlSchwede In case $k$ is not algebraically closed, is the morphism still an isomorphism? (I am not familiar with derived language, I am not sure how the duality concretely writes out?) | |
| Aug 23, 2015 at 18:25 | comment | added | Karl Schwede | See Grothendieck duality, then you get statements over much more general bases. | |
| Aug 23, 2015 at 18:19 | history | asked | user39380 | CC BY-SA 3.0 |