# Serre duality and Hirzebruch-Riemann-Roch in the non-projective case

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?

(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.

(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).

(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.

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For a purely "non-projective" proof (ie not referring to projective space) see M. Nori's paper "The Hirzebruch-Riemann-Roch theorem", Michigan Math. J. 48 (2000). In positive characteristic, there is a non-projective proof based on the Frobenius; see the paper "On the Adams-Riemann-Roch theorem in positive characteristic", Math. Z. 270 (2012) by R. Pink and myself. – Damian Rössler Sep 9 '12 at 12:54

Yes, sure.

1. See Theorem 15.2 (at least in my edition) of Fulton's Intersection Theory for Grothendieck-Riemann-Roch for a proper map of smooth varieties. Now take the target to be a point to obtain HRR.

2. 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 Residues and Duality. Now specialize as above to get Serre duality. (It just occurred to me that Lipman's Dualizing sheaves, differentials and residues on algebraic varieties is probably a more reasonable reference for this.)

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Perfect! Thanks! – Piotr Achinger Sep 8 '12 at 20:43
Note that Fulton's (-Baum-MacPherson-...) proof uses Chow's lemma to cover the general proper case and is thus still based on the projective case (this is related to (3) in the question). – Damian Rössler Sep 9 '12 at 13:04