= Solution
Use the finite map $f:C\to\mathbb P^1$ from the preceding solution. The inverse images of the two standard affine charts and of their intersection are affine, because $f$ is finite. The <vanishing of quasi-coherent cohomology on an affine scheme> therefore lets their two-term <Čech cochain complex> compute $H^i(C,E)$. It has terms only in degrees zero and one, proving
$$
\boxed{H^i(C,E)=0\qquad(i\geq2).}
$$
This actually applies to every <quasi-coherent sheaf> on $C$.
For a <smooth projective curve> with $H^0(C,\mathcal O_C)=k$, define its <genus of a smooth projective curve> by
$$
\boxed{g=\dim_kH^1(C,\mathcal O_C).}
$$
The preceding <Serre duality> also gives $g=\dim_kH^0(C,\Omega^1_C)$. The condition on constants holds in particular for a geometrically connected smooth projective curve. It makes $g$ equal to the <arithmetic genus> $1-\chi(\mathcal O_C)$. We state the theorem with this usual convention and then give the formula valid without that condition.
For a <divisor on an algebraic curve> $D$ and a <canonical divisor> $K$ with $\mathcal O(K)\cong\Omega^1_C$, the <Riemann-Roch theorem> is
$$
\boxed{\ell(D)-\ell(K-D)=\deg_kD+1-g,\qquad\ell(D)=h^0(C,\mathcal O(D)).}
$$
The degree over an arbitrary <field> is weighted by residue degrees: $\deg_kD=\sum_pn_p[k(p):k]$ when $D=\sum_pn_pp$. Equivalently, for a <line bundle> $L$,
$$
h^0(L)-h^0(\Omega^1_C\otimes L^{-1})=\deg_kL+1-g.
$$
We now prove it via <Riemann-Roch via elementary modifications>.
For any closed point $p$ and any <line bundle> $L$, the exact sequence
$$
0\longrightarrow L(-p)\longrightarrow L\longrightarrow L|_p\longrightarrow0
$$
has a <skyscraper sheaf> as its quotient, with $k$-dimension $[k(p):k]$. This follows locally from a uniformizer of the <discrete valuation ring> at $p$: a local line-bundle generator modulo that uniformizer spans one copy of $k(p)$. The quotient has zero higher cohomology, as it is the direct image from a finite affine scheme. The <long exact sequence in sheaf cohomology> and the proved higher vanishing give
$$
\chi(L)-\chi(L(-p))=[k(p):k],\qquad\chi(L)=h^0(L)-h^1(L).
$$
Every <line bundle> has a nonzero <rational section of a line bundle>, whose local orders give a <divisor on an algebraic curve> $D$ with $L\cong\mathcal O(D)$. Iterating the displayed identity, adding or subtracting closed points with their multiplicities, gives
$$
\chi(\mathcal O(D))=\deg_kD+\chi(\mathcal O_C).
$$
For the constants convention above, $\chi(\mathcal O_C)=1-g$. The <dualizing sheaf on a smooth projective curve> gives $h^1(\mathcal O(D))=h^0(\mathcal O(K-D))$, which converts this Euler-characteristic identity into the stated <Riemann-Roch theorem>. This proves the formula over arbitrary $k$, without assuming the closed points are $k$-rational.
For <vector bundles>, the corresponding formula is
$$
\boxed{\chi(E)=\deg_k(\det E)+\operatorname{rank}(E)(1-g).}
$$
To see it, saturate a rational rank-one subspace to a <line subbundle> and induct on rank. Both <Euler characteristic of a coherent sheaf> and the degree of the determinant are additive in the resulting short exact sequence. The line-bundle formula just proved starts the induction.
Taking $L=\Omega^1_C$ and using duality gives $h^0(\Omega^1_C)=g$, $h^1(\Omega^1_C)=1$, and hence $\deg_k\Omega^1_C=2g-2$. In particular a <line bundle> of degree greater than $2g-2$ has zero first cohomology: its dual twisted by $\Omega^1_C$ has negative degree, and a negative-degree line bundle has no nonzero section, since any section would have an <effective divisor> of zeros.
If the term “curve over $k$” is used without requiring $H^0(\mathcal O_C)=k$, the unconditional formula we proved is
$$
\boxed{\chi(L)=\deg_kL+\chi(\mathcal O_C).}
$$
Writing $g=h^1(\mathcal O_C)$ then replaces $1-g$ by $h^0(\mathcal O_C)-g$. For example, $\mathbb P^1_{k'}$ viewed as a $k$-curve for a finite separable extension $k'/k$ has $h^0(\mathcal O_C)=[k':k]$, so a literal constant term $1$ would be incorrect. Alternatively the <arithmetic genus> $p_a=1-\chi(\mathcal O_C)$ gives the formula with constant $1-p_a$. The constants hypothesis for the usual genus is recorded in https://stacks.math.columbia.edu/tag/0BY6[Stacks Project, genus of a curve].
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