TITLE: Is there a Riemann-Roch for smooth projective curves over an arbitrary field?
QUESTION [8 upvotes]: Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical
line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$. 
(1) When $k$ is algebraically closed then $\omega$ is a dualizing sheaf for $X$. From there it is
easy to prove Riemann-Roch for regular (holomorphic) line bundles $L$ over $X$: By this I mean
a precise formula which computes the Euler characteristic of $L$ in terms of the degree
of $L$ and the genus of $X$ (I think of both as being topological invariants).
(2) When $k$ is a finite field then one may consider the topological ring $\mathbf{A}_F$,
the ring of Adeles of $F$. Doing Fourrier analysis on this self-dual locally compact abelian group and doing a counting argument one may deduce Riemann-Roch.
Q1: Is it possible to generalize Riemann-Roch to other fields? What about real and $p$-adic numbers?
Q2: Is $\omega$ a dualizing sheaf when $k$ is finite? If not 
(I guess that in general one has to replace the notion of dualizing sheaf by some kind
of complex in a derived category)
Q3: Is there a way to prove simultaneously $(1)$ and $(2)$?
Q4: Is there some notion that would encompass simultaneuously $\mathbf{A}_F$ and $\omega$?

REPLY [25 votes]: Dear Hugo, the wonderful formalism of schemes allows us to have a Riemann-Roch theorem for a projective curve $X$  over an arbitrary field $k$, even without any assumption of smoothness.
It says, like in the good old times of Riemann surfaces, that for a Cartier divisor $D$ on $X$ we have
 $$\chi (\mathcal O_X(D))= deg(D)+ \chi (\mathcal O_X)$$ 
There is a dualizing sheaf $\omega$ and  Serre duality yields the formula 
$$  h^0(X,\mathcal O_X(D))-h^0(X,\omega \otimes\mathcal O_X(-D))=1-p_a(X)+deg D$$
where $p_a(X)=1-\chi(\mathcal O_x)$ is the so called arithmetic genus of the curve.
Everything is in our friend Qing Liu's fantastic book Algebraic Geometry and Arithmetic Curves but I bet he's too modest to give you this obvious answer!
Edit The first displayed formula is actually valid in  even greater generality: it holds for any projective curve $X$ (smooth or not) over an arbitrary artinian ring $k$. The proof is on page 164 of 
Altman, A.; Kleiman, S., Introduction to Grothendieck duality theory. Lecture
Notes in Mathematics No. 146, Springer-Verlag, Berlin-New York, 1970
Complement  As an answer to Hugo's question in his comment below, let me add that indeed, in the case of a smooth projective curve over a field $k$, the dualizing sheaf $\omega$ is nothing else than the canonical sheaf . More generally for a smooth projective variety of dimension $r$ over $k$, the dualizing sheaf is just the canonical sheaf $\omega=\Omega^r_{X/k}$.  This is (a special case of) Theorem I.4.6, page 14 in Altman-Kleiman's monograph.