# 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.

• 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. Sep 9, 2012 at 12:54

• In Dualizing sheaves, differentials and residues on algebraic varieties, Lipman works over a perfect field $k$. Is there some other source (alternative to chap VII section 3 of Hartshorne's Residues and Duality) treating Serre's duality over an arbitrary field $k$? (I have to say, I don't know how perfection of $k$ is used in Lipman's book, I haven't read it.) Aug 16, 2023 at 11:59