= Solution
For a <smooth projective curve> $C$, let $g=h^1(C,\mathcal O_C)$ be its <geometric genus>, let $K_C$ be a <canonical divisor>, and write $\ell(D)=h^0(C,\mathcal O_C(D))$ for the <dimension of a vector space> of the <Riemann-Roch space>. The <Riemann-Roch theorem> says
$$
\boxed{\ell(D)-\ell(K_C-D)=\deg D+1-g}
$$
for every <divisor on an algebraic curve> $D$ on $C$.
Here is the <cohomological proof of Riemann-Roch for curves>. At a <closed point> $P$, the <local ring> is a <discrete valuation ring>. A local uniformizer shows that increasing the allowed pole order by one gives the <short exact sequence> of <sheaves>
$$
0\longrightarrow\mathcal O_C(D-P)\longrightarrow\mathcal O_C(D)
\longrightarrow k(P)\longrightarrow0.
$$
The last term is a length-one <skyscraper sheaf>; its precise identification with $k$ is noncanonical, but its <sheaf cohomology> has $h^0=1$ and $h^1=0$. All the relevant groups of <sheaf cohomology> are finite-dimensional because $C$ is a <projective variety>, and coherent <sheaf cohomology> above degree one vanishes on a <algebraic curve>. The <long exact sequence in sheaf cohomology> therefore gives
$$
\chi(\mathcal O_C(D))=\chi(\mathcal O_C(D-P))+1,
\qquad \chi(\mathcal F)=h^0(C,\mathcal F)-h^1(C,\mathcal F).
$$
Since $k$ is an <algebraically closed field>, every <closed point> has degree one. Iterating this identity for both positive and negative coefficients of $D$ proves
$$
\chi(\mathcal O_C(D))=\deg D+\chi(\mathcal O_C).
$$
Every <global regular function> on a <projective variety> which is an <irreducible variety> is constant; hence $h^0(C,\mathcal O_C)=1$ and $\chi(\mathcal O_C)=1-g$.
Finally apply <Serre duality> for a <smooth projective curve>:
$$
H^1(C,\mathcal O_C(D))^*
\cong H^0(C,\omega_C\otimes\mathcal O_C(-D))
=H^0(C,\mathcal O_C(K_C-D)).
$$
Taking <dimensions of a vector space> and inserting this in the <Euler characteristic of a coherent sheaf> identity proves the displayed <Riemann-Roch theorem>. The substantial inputs from <sheaf cohomology> are finiteness and vanishing for coherent <sheaves> on a <projective curve>, and <Serre duality>; the change of <Euler characteristic of a coherent sheaf> is proved directly by the point exact sequence. In particular the argument handles negative as well as effective <divisors on an algebraic curve>.
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