Serre duality and Hirzebruch-Riemann-Roch in the non-projective case - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:23:48Z http://mathoverflow.net/feeds/question/106684 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106684/serre-duality-and-hirzebruch-riemann-roch-in-the-non-projective-case Serre duality and Hirzebruch-Riemann-Roch in the non-projective case Piotr Achinger 2012-09-08T19:39:21Z 2012-09-08T20:28:14Z <p>Serre duality and the Hirzebruch-Riemann-Roch formula are usually stated for $X$ a smooth projective algebraic variety. Do you know of a reference which proves these results for $X$ smooth and proper?</p> <p>(1) Since Hirzebruch-Riemann-Roch is true for compact complex manifolds, and Serre duality holds for compact Kähler manifolds, one could expect them to hold more generally. </p> <p>(2) As both results are traditionally proved by deducing the result from the case of $X=\mathbb{P}^n$ (either by embedding $X$ in $\mathbb{P}^n$ or finding a finite morphism $X\to \mathbb{P}^n$), it should be easy to adapt these proofs to the case when $X$ is $A_2$, that is, embeddable in a toric variety. ($A_2$ is in fact equivalent to the property that every two points admit a common affine open neighbourhood).</p> <p>(3) In characteristic $0$, we can connect $X$ to a smooth projective $X'$ by blow-ups and blow-downs with smooth centers. So I guess proving relative duality and HRR (that is, Grothendieck duality and Grothendieck-Riemann-Roch) for a blow-up in a smooth center should do the trick. I am more interested in the characteristic $p$ case though.</p> http://mathoverflow.net/questions/106684/serre-duality-and-hirzebruch-riemann-roch-in-the-non-projective-case/106687#106687 Answer by Donu Arapura for Serre duality and Hirzebruch-Riemann-Roch in the non-projective case Donu Arapura 2012-09-08T20:09:52Z 2012-09-08T20:28:14Z <p>Yes, sure.</p> <ol> <li><p>See Theorem 15.2 (at least in my edition) of Fulton's <em>Intersection Theory</em> for Grothendieck-Riemann-Roch for a proper map of smooth varieties. Now take the target to be a point to obtain HRR.</p></li> <li><p>This is certainly overkill, but you can find a proof of Grothendieck duality for proper maps (with finite Tor dimension) between noetherian schemes in chap VII section 3 of Hartshorne's <em>Residues and Duality</em>. Now specialize as above to get Serre duality. (It just occurred to me that Lipman's <em>Dualizing sheaves, differentials and residues on algebraic varieties</em> is probably a more reasonable reference for this.)</p></li> </ol>