= Solution
Order the index set of the <open cover> $\mathcal U=(U_a)$. The <Čech cochain groups> are
$$
\check C^p(\mathcal U,\mathcal F)=\prod_{a_0<\cdots<a_p}\mathcal F(U_{a_0}\cap\cdots\cap U_{a_p}),
$$
and their differential is the alternating sum of restrictions:
$$
(\delta c)_{a_0\ldots a_{p+1}}=\sum_{j=0}^{p+1}(-1)^j c_{a_0\ldots\widehat a_j\ldots a_{p+1}}|_{U_{a_0}\cap\cdots\cap U_{a_{p+1}}}.
$$
Terms obtained by deleting two indices cancel in pairs, so $\delta^2=0$. The <Čech cohomology> is $\check H^p=\ker\delta/\operatorname{im}\delta$, with zero incoming differential in degree zero. Its zeroth group is the group of <global sections> by <sheaf> gluing.
The <acyclic cover theorem> applies if every nonempty finite intersection is acyclic for $\mathcal F$. For a <quasi-coherent sheaf> on a variety it is sufficient that every such intersection is affine; on a separated variety, any affine <open cover> has this property. On $\mathbb P^n$, take the $n+1$ standard charts $U_i=D_+(X_i)$. Every intersection is a principal open in an affine chart, hence affine. The Čech complex has no terms above degree $n$, proving
$$
\boxed{H^i(\mathbb P^n,\mathcal F)=0\quad(i>n).}
$$
This is the <cohomological dimension bound from an affine cover> and requires only quasi-coherence, not finite generation.
For the remaining projective calculations assume $n\ge1$. There is a necessary zero-dimensional exception to the negative-twist assertion: $\mathbb P^0$ is a point, every twist is trivial there, and $H^0(\mathbb P^0,\mathcal O(m))\cong k$ even for $m<0$.
Put $S=k[X_0,\ldots,X_n]$ with its usual grading. The <twisting sheaf on projective space> is the <sheaf> associated with the shifted graded <module> $S(m)$; on an intersection $U_I=\bigcap_{i\in I}U_i$ its sections are
$$
\mathcal O(m)(U_I)=(S_{\prod_{i\in I}X_i})_m.
$$
On $U_i$ a generator is $e_i=X_i^m$, with transition $e_j=(X_j/X_i)^m e_i$. These are regular units on overlaps and satisfy the cocycle relation, so they glue an <invertible sheaf> for every integer $m$, including negative $m$.
A <global section> is a compatible family of these homogeneous fractions, hence a single element of degree $m$ in $\bigcap_iS_{X_i}\subseteq\operatorname{Frac}S$. For $n\ge1$, $X_0$ and $X_1$ are relatively prime in the <unique factorization domain> $S$, so $S_{X_0}\cap S_{X_1}=S$; therefore the full intersection is $S$. It follows that
$$
\boxed{H^0(\mathbb P^n,\mathcal O(m))\cong S_m=\begin{cases}
\text{homogeneous polynomials of degree }m,&m\ge0,\\
0,&m<0.
\end{cases}}
$$
For $m\ge0$ its dimension is $\binom{n+m}{n}$. The proof uses all-chart compatibility; regularity on one chart alone would permit poles on its complement.
If some $d_i=0$, the ideal $(X_i^{d_i})$ contains $1$, so the requested containment is immediate. Otherwise every $d_i\ge1$. A <monomial> $X_0^{a_0}\cdots X_n^{a_n}$ of degree $\sum_id_i$ must have $a_i\ge d_i$ for some $i$: if not, its total degree would be at most $\sum_i(d_i-1)<\sum_id_i$. Thus every degree-$\sum_id_i$ <monomial>, and hence every <homogeneous polynomial> of that degree, belongs to $(X_0^{d_0},\ldots,X_n^{d_n})$. This is <monomial containment in an ideal of coordinate powers>.
For $n>0$, a top-degree Čech cochain for $\mathcal O$ is a degree-zero Laurent <polynomial> on the full intersection. Write it with a common denominator as
$$
\frac{P}{X_0^{d_0}\cdots X_n^{d_n}},\qquad \deg P=\sum_i d_i.
$$
The containment just proved gives $P=\sum_iQ_iX_i^{d_i}$, with homogeneous $Q_i$ of degree $\sum_jd_j-d_i$. Consequently
$$
\frac P{\prod_jX_j^{d_j}}=\sum_i\frac{Q_i}{\prod_{j\ne i}X_j^{d_j}}.
$$
The $i$th term is regular on the intersection omitting $U_i$, and has degree zero. Give it the sign $(-1)^i$ in that component of the preceding Čech cochain. Its coboundary is the original fraction. Every top cochain is thus a coboundary, and
$$
\boxed{H^n(\mathbb P^n,\mathcal O)=0\quad(n>0).}
$$
This is <top Čech cohomology from missing-denominator monomials>.
To obtain the negative-twist bound by induction on dimension, let $H\cong\mathbb P^{n-1}$ be a hyperplane, with inclusion $j$. Its equation gives the <hyperplane exact sequence for twisting sheaves>
$$
0\longrightarrow\mathcal O(m-1)\longrightarrow\mathcal O(m)\longrightarrow j_*\mathcal O_H(m)\longrightarrow0.
$$
The associated <long exact sequence in sheaf cohomology> and <sheaf cohomology under a closed inclusion> give
$$
H^{n-1}(H,\mathcal O_H(m))\longrightarrow H^n(\mathbb P^n,\mathcal O(m-1))\longrightarrow H^n(\mathbb P^n,\mathcal O(m))\longrightarrow0.
$$
For $n=1$, restriction $H^0(\mathbb P^1,\mathcal O)\to H^0(H,\mathcal O_H)$ is the surjection $k\to k$. Exactness and the already proved $H^1(\mathbb P^1,\mathcal O)=0$ give $H^1(\mathbb P^1,\mathcal O(-1))=0$. The displayed surjections then give the same vanishing for every $m\ge0$. This handles the point hyperplane without making a false negative-twist claim on $\mathbb P^0$.
For $n\ge2$, the induction hypothesis in dimension $n-1$ gives $H^{n-1}(H,\mathcal O_H(m))=0$ whenever $m\ge-(n-1)$. Therefore consecutive top-degree groups are isomorphic for $m\ge1-n$. Starting with $H^n(\mathbb P^n,\mathcal O)=0$ and stepping down through $m=0,-1,\ldots,1-n$ reaches $H^n(\mathbb P^n,\mathcal O(-n))=0$. Stepping upward proves all positive twists too. We conclude
$$
\boxed{H^n(\mathbb P^n,\mathcal O(-r))=0\quad\text{for all integers }r\le n,\ n>0.}
$$
This is <top-twist vanishing by hyperplane induction>.
Finally, on $U_0$ write $x_a=X_a/X_0$ and $\omega_0=dx_1\wedge\cdots\wedge dx_n$. On $U_i$, let $\omega_i$ be the wedge of $d(X_a/X_i)$ for $a\ne i$, taken in increasing index order. For $i>0$, differentiating $X_0/X_i=x_i^{-1}$ and $X_a/X_i=x_a/x_i$ gives
$$
\boxed{\omega_i=(-1)^i x_i^{-n-1}\omega_0.}
$$
In the wedge, all terms involving a second copy of $dx_i$ disappear; the surviving powers are $x_i^{-2}$ from $d(x_i^{-1})$ and $x_i^{-1}$ from the other $n-1$ differentials. Moving $dx_i$ into its original position produces the stated sign. Rescale each local generator by $\eta_i=(-1)^i\omega_i$. Then $\eta_j=(X_j/X_i)^{-n-1}\eta_i$ on every overlap, exactly the transition of the twist with $m=-n-1$. This <canonical-form transition on projective space> proves the <canonical bundle of projective space>:
$$
\boxed{\Omega^n_{\mathbb P^n}\cong\mathcal O_{\mathbb P^n}(-n-1).}
$$
For line bundles the <Serre duality> pairing becomes
$$
H^i(\mathbb P^n,\mathcal O(m))^*\cong H^{n-i}(\mathbb P^n,\mathcal O(-m-n-1)).
$$
In particular $H^n(\mathcal O(-r))$ is dual to $H^0(\mathcal O(r-n-1))$, which is zero for $r\le n$, precisely the independently obtained bound. The case $r=0$ recovers $H^n(\mathcal O)=0$, and $H^n(\mathcal O(-n-1))\cong k$ matches $H^0(\mathcal O)\cong k$. Thus the calculated transition functions, <global sections> and top-degree vanishing agree with <Serre duality>.
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