Cohomology of O(d) on projective space

Setup. Write $S=A[x_0,\dots ,x_n]={⨁}_{j\ge 0}{S}_j$ with the total-degree grading, $X={\mathbb{P}}_A^n\cong \mathrm{Proj}S$ and, for $i=0,\dots ,n$, $U_i=D_{+}(x_i)$; the index set $\{0<\cdots <n\}$ is linearly ordered, $D_{+}(x_i)$ is open, and the $U_i$ cover $X$ because a homogeneous prime $\mathfrak{p}\supseteq ̸S_{+}$ must miss some $x_i$, the $x_i$ generating $S_{+}$ by [F1] and [F8].

The members and their finite intersections are affine. For a nonempty finite subset $\sigma \subseteq \{0,\dots ,n\}$ put $g_{\sigma }={\prod }_{i\in \sigma }{x}_i$, a homogeneous element of degree $\mathrm{\mid }\sigma \mathrm{\mid }\ge 1$; iterating $D_{+}(f)\cap D_{+}(g)=D_{+}(fg)$ in [F1] gives ${\bigcap }_{i\in \sigma }{U}_i=D_{+}(g_{\sigma })$, and [F2] identifies this with $\mathrm{Spec}{S}_{(g_{\sigma })}$, an affine scheme.

The sheaves are quasi-coherent. By [F3] the twisting sheaf ${\mathcal{O}}_X(d)$ is quasi-coherent, and [F3] also gives that its restrictions to the open subschemes ${\bigcap }_{i\in \sigma }{U}_i$ are quasi-coherent.

Acyclicity and comparison. By 1.2 and 1.3, every nonempty finite intersection $W$ of members of the cover is affine and ${\mathcal{O}}_X(d){\mathrm{\mid }}_W$ is quasi-coherent, so $H^q(W,{\mathcal{O}}_X(d){\mathrm{\mid }}_W)=0$ for all $q>0$ by [F4]. Thus the ordered cover is acyclic for ${\mathcal{O}}_X(d)$ and [F5] gives isomorphisms for every $p\ge 0$.

Cohomology of the total Čech complex. By [F6] the ordered Čech complex is the direct sum of the complexes $K^{\bullet }(e)$. Formation of kernels and cokernels of families of linear maps commutes with direct sums by [F7], so for every $p\ge 0$, ${\check{H}}^p(\mathcal{U},{\mathcal{O}}_X(d))\cong {⨁}_{e\in {\mathbb{Z}}^{n+1},\sum e_i=d}{H}^p(K^{\bullet }(e))\mathrm{.}$

Degree-by-degree bookkeeping. By [F6] a summand $H^p(K^{\bullet }(e))$ can be nonzero only if $N(e)=\mathrm{\varnothing }$ and $p=0$, or $N(e)=\{0,\dots ,n\}$ and $p=n$; the remaining summands are contractible and contribute nothing in any degree. Hence, when $n\ge 1$, the degrees $0$ and $n$ do not overlap and ${\check{H}}^0(\mathcal{U},{\mathcal{O}}_X(d))={⨁}_{e\ge 0,\sum e_i=d}A,{\check{H}}^n(\mathcal{U},{\mathcal{O}}_X(d))={⨁}_{e<0,\sum e_i=d}A,$ while ${\check{H}}^p(\mathcal{U},{\mathcal{O}}_X(d))=0$ for $0<p<n$; here $e\ge 0$ means $e_i\ge 0$ for all $i$ and $e<0$ means $e_i<0$ for all $i$.

The degree-zero group. The exponent vectors $e\ge 0$ with ${\sum }_i{e}_i=d$ are exactly the exponent vectors of the monomials of total degree $d$ in $x_0,\dots ,x_n$, so by [F8] the first group of 1.6 is the free $A$-module on that set, namely $A[x_0,\dots ,x_n{]}_d$ when $d\ge 0$ and $0$ when $d<0$.

The top group. The exponent vectors $e<0$ with ${\sum }_i{e}_i=d$ are exactly the Laurent monomials of the statement. If $d\le -n-1$ then $e_0=d+n$, $e_1=\cdots =e_n=-1$ is such a vector, and for any such vector the substitution $f_i=-e_i-1\ge 0$ shows ${\sum }_i{f}_i=-d-(n+1)$, so these vectors form a finite set; if $d\ge -n$ then ${\sum }_i{e}_i\le -(n+1)$ is impossible, so the set is empty. Hence the second group of 1.6 is the free $A$-module on the stated finite set, which is nonzero exactly when the set is nonempty and $A\mathrm{\ne }0$, that is, exactly when $d\le -n-1$ and $A\mathrm{\ne }0$.

The case $n=0$. The cover has the single member $U_0=D_{+}(x_0)$ and ${\mathbb{P}}_A^0=\mathrm{Spec}A$ by [F2]. In the decomposition of 1.5 there is only the exponent vector $e=(d)$: if $d\ge 0$ then $N(e)=\mathrm{\varnothing }$ and [F6] gives ${\check{H}}^0=A$ with no higher term, while if $d<0$ then $N(e)=\{0\}=\{0,\dots ,n\}$ and [F6] gives the same in degree $0$. Hence ${\check{H}}^0(\mathcal{U},{\mathcal{O}}_X(d))=A$ and ${\check{H}}^p(\mathcal{U},{\mathcal{O}}_X(d))=0$ for $p>0$, so 1.4 gives $H^0(X,{\mathcal{O}}_X(d))\cong A$ and vanishing of all higher cohomology.

Conclusion for $n\ge 1$. Combining 1.4 with 1.6, 2.1 and 2.2 gives $H^0(X,{\mathcal{O}}_X(d))\cong A[x_0,\dots ,x_n{]}_d$ for $d\ge 0$, $H^0(X,{\mathcal{O}}_X(d))=0$ for $d<0$, $H^n(X,{\mathcal{O}}_X(d))$ free on the all-negative monomials of total degree $d$ (nonzero exactly for $d\le -n-1$ when $A\mathrm{\ne }0$), and $H^q(X,{\mathcal{O}}_X(d))=0$ for $0<q<n$; with 2.3 the case $n=0$ gives $H^0\cong A$ and no other nonzero group.

Boundaries and choice accounting. If $A=0$ then $S=0$, $\mathrm{Proj}S=\mathrm{\varnothing }={\mathbb{P}}_A^n$, all Čech terms vanish and 1.4-3.1 give zero groups, matching the descriptions $S_d=0$ and the free module on any set over the zero ring; the empty-scheme convention of [F1] and [F2] applies. The case $d=0$ is included: $H^0\cong A$ (for $n\ge 1$, the constant monomials) and the top group vanishes since $0\ge -n$; the endpoint $d=-n-1$ gives the top group $A\cdot (x_0\cdots x_n{)}^{-1}$, and $d=-1,\dots ,-n$ give $H^n=0$ while all intermediate groups vanish. The statement is not an equivalence and asserts no converse, so no iff case arises. The Axiom of Choice [F9] is consumed through the affine vanishing [F4] and the Leray comparison [F5]; the ordering of the cover, the monomial bases $x_{\sigma }^e$ and the substitution $f_i=-e_i-1$ are canonical, and no further selection is made.