= Solution
{c}
The <resolution principle for sheaf cohomology> states that an <exact sequence>
$$
0\longrightarrow\mathcal F\longrightarrow\mathcal A^0\longrightarrow\mathcal A^1\longrightarrow\cdots
$$
with $H^q(X,\mathcal A^j)=0$ for every $q>0$ computes $H^i(X,\mathcal F)$ as the cohomology of $\Gamma(X,\mathcal A^\bullet)$. In particular one may use a <flasque resolution>.
An <affine morphism> $\phi:X\to Y$ is one for which the inverse image of every affine open subset of $Y$ is affine. We use <vanishing of quasi-coherent cohomology on an affine scheme>: a <quasi-coherent sheaf> has zero higher <sheaf cohomology> on an <affine variety>. Take a <flasque resolution> $\mathcal F\to\mathcal I^\bullet$. Each <direct image sheaf> $\phi_*\mathcal I^j$ is flasque, because restriction on $V\subseteq U\subseteq Y$ is restriction on $\phi^{-1}V\subseteq\phi^{-1}U$. On an affine open $U\subseteq Y$, the augmented complex of sections is
$$
0\longrightarrow\Gamma(\phi^{-1}U,\mathcal F)\longrightarrow\Gamma(\phi^{-1}U,\mathcal I^0)\longrightarrow\Gamma(\phi^{-1}U,\mathcal I^1)\longrightarrow\cdots.
$$
Its positive-degree cohomology vanishes by the affine theorem, and its degree-zero kernel is $\Gamma(\phi^{-1}U,\mathcal F)$. Checking exactness on this affine basis shows that $\phi_*\mathcal I^\bullet$ is a <flasque resolution> of $\phi_*\mathcal F$. The global-section complexes are identical, proving <cohomology under an affine morphism>:
$$
\boxed{H^i(Y,\phi_*\mathcal F)\cong H^i(X,\mathcal F)\quad(i\geq0).}
$$
For <projective space>, put $S=k[x_0,\ldots,x_n]$ with its usual grading and $U_i=\{x_i\ne0\}$. Construct the <twisting sheaf on projective space> by setting its sections on $U_i$ to be the degree-$m$ part $(S_{x_i})_m$, viewed as a module over $(S_{x_i})_0$. It is free of rank one with generator $x_i^m$, even when $m$ is negative. On overlaps the generators differ by the unit $(x_i/x_j)^m$, so these modules glue to a <line bundle> $\mathcal O(m)$.
Assume first $n\geq1$. These local modules all embed in the graded fraction field, and agreement on overlaps identifies global sections with $\bigcap_i(S_{x_i})_m$. Since $S$ is a <unique factorization domain> and at least two distinct variables occur, a reduced fraction belonging to every $S_{x_i}$ has no nonconstant denominator. Thus the intersection is $S_m$. Counting monomials by <stars and bars> gives
$$
H^0(\mathbf P^n,\mathcal O(m))=S_m,\qquad
h^0_n(m)=\begin{cases}\binom{m+n}{n},&m\geq0,\\0,&m<0.\end{cases}
$$
Here $h^i_n(m)=\dim_kH^i(\mathbf P^n,\mathcal O(m))$. The printed formula needs the convention that a counting <binomial coefficient> is zero when its upper integer is smaller than its nonnegative lower integer; the generalized polynomial convention for negative upper integers would give incorrect dimensions. For $n=0$, every twist on the single point $\mathbf P^0$ is trivial, so $h^0_0(m)=1$ for every $m$ and its higher cohomology is zero. This case supplies the base of dimension induction; the displayed top-cohomology formula below is for $n\geq1$.
The standard $n+1$ affine opens have affine finite intersections. The <acyclic cover theorem> and affine vanishing identify <sheaf cohomology> with a <Čech cochain complex> ending in degree $n$. In particular $H^i(\mathbf P^n,\mathcal O(m))=0$ for $i>n$.
Let $H=\{x_n=0\}\cong\mathbf P^{n-1}$, with inclusion $i$. Multiplication by $x_n$ gives the <hyperplane exact sequence for twisting sheaves>
$$
0\longrightarrow\mathcal O_{\mathbf P^n}(m-1)\xrightarrow{\,x_n\,}\mathcal O_{\mathbf P^n}(m)\longrightarrow i_*\mathcal O_H(m)\longrightarrow0.
$$
Exactness is checked in the local trivializations, where a hyperplane equation is a non-zero-divisor and the quotient is its restriction to $H$. Since a <closed immersion> is an <affine morphism>, the first part identifies the cohomology of $i_*\mathcal O_H(m)$ with that on $H$.
For $n=1$, the <long exact sequence in sheaf cohomology> and $H\cong\mathbf P^0$ give
$$
h^1_1(m-1)-h^1_1(m)=1-\bigl(h^0_1(m)-h^0_1(m-1)\bigr)
=\begin{cases}0,&m\geq0,\\1,&m<0.\end{cases}
$$
Starting from the given vanishing for sufficiently large $m$ and descending gives $h^1_1(m)=0$ for $m\geq-1$ and $h^1_1(m)=-m-1$ for $m\leq-2$.
Now let $n\geq2$, and assume the formulas in dimension $n-1$. Restriction
$$
H^0(\mathbf P^n,\mathcal O(m))\longrightarrow H^0(H,\mathcal O_H(m))
$$
is surjective for every $m$: for $m\geq0$ it is polynomial restriction, and for $m<0$ both groups are zero since $\dim H\geq1$. The <long exact sequence in sheaf cohomology> therefore makes $H^1(\mathbf P^n,\mathcal O(m-1))\to H^1(\mathbf P^n,\mathcal O(m))$ injective. For $2\leq j<n$, dimension induction gives $H^{j-1}(H,\mathcal O_H(m))=0$, so the same sequence makes
$$
H^j(\mathbf P^n,\mathcal O(m-1))\longrightarrow H^j(\mathbf P^n,\mathcal O(m))
$$
injective. Composing these injections up to a sufficiently large twist, where the target is zero by the given hypothesis, proves $H^j(\mathbf P^n,\mathcal O(m))=0$ for every $m$ and $0<j<n$.
The remaining part of the <long exact sequence in sheaf cohomology> is consequently
$$
0\longrightarrow H^{n-1}(H,\mathcal O_H(m))\longrightarrow H^n(\mathbf P^n,\mathcal O(m-1))\longrightarrow H^n(\mathbf P^n,\mathcal O(m))\longrightarrow0,
$$
where the last zero uses the affine-cover dimension bound on $H$. Hence $h^n_n(m-1)=h^n_n(m)+h^{n-1}_{n-1}(m)$. Using dimension induction and eventual vanishing, downward summation yields
$$
h^n_n(m)=\sum_{j=m+1}^{-n}\binom{-j-1}{n-1}=\binom{-m-1}{n}\quad(m\leq-n-1),
$$
by telescoping Pascal's identity; for $m\geq-n$ it is zero. Thus, without invoking <Serre duality> in the induction,
$$
\boxed{
H^j(\mathbf P^n,\mathcal O(m))=0\ (j>0,\ j\ne n),\qquad
h^n_n(m)=\begin{cases}\binom{-m-1}{n},&m\leq-n-1,\\0,&m\geq-n.\end{cases}}
$$
Finally, the <canonical bundle of projective space> is $\omega_{\mathbf P^n}\cong\mathcal O(-n-1)$, obtained by taking determinants in the dual <Euler sequence>. <Serre duality> therefore predicts
$$
H^j(\mathbf P^n,\mathcal O(m))\cong H^{n-j}(\mathbf P^n,\mathcal O(-m-n-1))^*.
$$
It pairs the two computed extreme-degree dimensions and pairs zero intermediate groups with zero intermediate groups, exactly as required.
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