I would like some references for Serre duality for a proper noetherian scheme $X$ (over $\mathbb C$). I know that in this case there exists a dualizing sheaf. But can theorem 7.6, III, in Harshorne's "Algebraic Geometry" (Duality dor a Projective Scheme) be extended to the case of a proper noetherian Cohen-Macaulay scheme ? i.e are there natural functorial isomorphisms $${\rm Ext}^i({\cal F},\omega_X^0)\to H^{n-i}(X,{\cal F})^* \ ?$$ (where $n=\dim(X)$ and $\cal F$ is any coherent sheaf on $X$).
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3$\begingroup$ See Tags 0AU9 (existence of $\omega_X^\bullet$ on varieties) and 0AWT (dualising complex on CM lives in one degree). $\endgroup$– R. van Dobben de BruynCommented Nov 14, 2021 at 16:16
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3$\begingroup$ I think stacks.math.columbia.edu/tag/0FVZ is a more direct reference for what the OP is asking... $\endgroup$– Sándor KovácsCommented Nov 15, 2021 at 18:46
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$\begingroup$ Also, BTW, @Hephaistos, I think there is a dual sign missing from your displayed line, no? $\endgroup$– Sándor KovácsCommented Nov 15, 2021 at 18:46
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$\begingroup$ @R. van Dobben de Bruyn and @ Sándor Kovács. Thank you for the answers. The missing dual sign has been corrected. $\endgroup$– HephaistosCommented Nov 15, 2021 at 23:47
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$\begingroup$ @Sándor Kovács I assume (but it is not clear for me) that Lemma 48.27.5 implies that the isomorphisms are fully functorial, i.e. if we have an exact sequence $\Sigma$ of (quasi-)coherent sheaves then the isomorphisms provide an isomorphism between the long exact sequence of $\Sigma$ in cohomology and the dual of that with the Ext's (in other words the isomorphisms of Lemma 48.27.5 commute with $\delta$-operators). $\endgroup$– HephaistosCommented Nov 15, 2021 at 23:54
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1 Answer
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A basic reference for duality using just sheaves instead of derived categories is the paper:
Kleiman, Steven L.: Relative duality for quasicoherent sheaves. Compositio Math. 41 (1980), no. 1, 39–60.
http://www.numdam.org/item/?id=CM_1980__41_1_39_0
With this at hand you have a duality theorem for proper algebraic varieties in Lipman's blue book:
Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque No. 117 (1984)
http://www.numdam.org/item?id=AST_1984__117__1_0
See section 4 to begin with.
Finally, to get "higher duality" for a Cohen Macualay variety, this is discussed in the book
Kunz, Ernst: Residues and duality for projective algebraic varieties. University Lecture Series, 47. AMS, Providence, RI, 2008.
On page 110 (sec. 12) it is explained how to go form the case $i = 0$ to arbitrary $i \geq 0$. It is stated in the projective case, but the argument carries over the proper case.
answered Nov 17, 2021 at 11:16
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$\begingroup$ Thank you for these new references. $\endgroup$ Commented Nov 17, 2021 at 20:41