Hypersurface cohomology sequence

The morphism of sheaves. By [F3] and [F2] the degree-zero graded map $a\mapsto fa$ induces a morphism $\varphi =\widetilde{(\cdot f)}:\mathcal{O}(-d)=\widetilde{B(-d)}\to \tilde{B}=\mathcal{O}$ whose component on the chart $D_{+}(x_i)$ is the localised map $B(-d{)}_{(x_i)}\to B_{(x_i)}$, $a\mathrm{/}{x}_i^k\mapsto fa\mathrm{/}{x}_i^k$.

Chartwise form and injectivity. Transporting the source of the component along the isomorphism of [F3], multiplication by $x_i^d$, exhibits $\varphi {\mathrm{\mid }}_{D_{+}(x_i)}$ as multiplication by $f_i:=f\mathrm{/}{x}_i^d$ on $B_{(x_i)}$; hence $\varphi {\mathrm{\mid }}_{D_{+}(x_i)}$ is injective if and only if $f_i$ is a nonzerodivisor of $B_{(x_i)}$, and by the locality of exactness [F6] $\varphi$ is injective if and only if this holds for every $i$, which is the stated hypothesis. In the field case $B_{(x_i)}=k[x_j\mathrm{/}{x}_i:j\mathrm{\ne }i]$ is a polynomial ring over a field, hence a domain, and $f_i\mathrm{\ne }0$ because $f\mathrm{\ne }0$ and localisation of a domain at a nonzero element is injective; so the hypothesis is automatic then.

The cokernel on a chart. On the affine chart $D_{+}(x_i)=\mathrm{Spec}{B}_{(x_i)}$ the module sequence $0\to B_{(x_i)}\stackrel{\cdot f_i}{\to }{B}_{(x_i)}\to B_{(x_i)}\mathrm{/}(f_i)\to 0$ is exact, so by [F5] the restriction of $0\to \mathcal{O}(-d)\stackrel{\varphi }{\to }\mathcal{O}$ to the chart is short exact with cokernel the associated sheaf of $B_{(x_i)}\mathrm{/}(f_i)$; its last map is the quotient map.

The structure map on a chart. By [F4] the closed immersion restricts over $D_{+}(x_i)$ to the canonical closed immersion with coordinate ring $B_{(x_i)}\mathrm{/}(f_i)$, so the component of $i^{\mathrm{♯}}$ on the chart is the quotient map $B_{(x_i)}\to B_{(x_i)}\mathrm{/}(f_i)$ with kernel the principal ideal $(f_i)$; hence $(i_{\ast }{\mathcal{O}}_X){\mathrm{\mid }}_{D_{+}(x_i)}$ is the associated sheaf of $B_{(x_i)}\mathrm{/}(f_i)$ and $\ker (i^{\mathrm{♯}}{\mathrm{\mid }}_{D_{+}(x_i)})=\mathrm{im}(\varphi {\mathrm{\mid }}_{D_{+}(x_i)})$ inside $\mathcal{O}{\mathrm{\mid }}_{D_{+}(x_i)}$.

The short exact sequence. On each chart of the cover $D_{+}(x_i)$ the maps $\varphi$ and $i^{\mathrm{♯}}$ satisfy $\ker i^{\mathrm{♯}}=\mathrm{im}\varphi$ by [step 3.1], and $i^{\mathrm{♯}}$ is surjective on each chart by [F4]; by the locality of exactness [F6] the sequence $0\to \mathcal{O}(-d)\stackrel{\varphi }{\to }\mathcal{O}\stackrel{i^{\mathrm{♯}}}{\to }{i}_{\ast }{\mathcal{O}}_X\to 0$ is short exact. This is the displayed short exact sequence.

The long exact sequence. Applying [F7] to the short exact sequence of [step 4.1] gives the long exact sequence with connecting maps ${\mathrm{\partial }}^q:H^q({\mathbb{P}}_A^n,i_{\ast }{\mathcal{O}}_X)\to H^{q+1}({\mathbb{P}}_A^n,\mathcal{O}(-d))$, whose other terms are $H^q({\mathbb{P}}_A^n,\mathcal{O}(-d))$ and $H^q({\mathbb{P}}_A^n,\mathcal{O})$.

The cohomology of the hypersurface. Since ${\mathcal{O}}_X$ is quasi-coherent by [F8], the closed-immersion isomorphism gives $H^q(X,{\mathcal{O}}_X)\cong H^q({\mathbb{P}}_A^n,i_{\ast }{\mathcal{O}}_X)$ for every $q\ge 0$; composing with [step 5.1] replaces the third term of the long exact sequence by $H^q(X,{\mathcal{O}}_X)$ and identifies ${\mathrm{\partial }}^q$ with a map $H^q(X,{\mathcal{O}}_X)\to H^{q+1}({\mathbb{P}}_A^n,\mathcal{O}(-d))$.

Substitution of the explicit groups. By [F9], $H^q({\mathbb{P}}_A^n,\mathcal{O})=0$ and $H^{q+1}({\mathbb{P}}_A^n,\mathcal{O})=0$ for every $q\ge 1$, so exactness of the sequence of [step 6.1] at $H^q({\mathbb{P}}_A^n,i_{\ast }{\mathcal{O}}_X)$ and $H^{q+1}({\mathbb{P}}_A^n,\mathcal{O}(-d))$ makes ${\mathrm{\partial }}^q$ an isomorphism for $q\ge 1$; in degree zero the same sequence reads $0\to H^0({\mathbb{P}}_A^n,\mathcal{O}(-d))\to A\to H^0(X,{\mathcal{O}}_X)\to H^1({\mathbb{P}}_A^n,\mathcal{O}(-d))\to 0$ because $H^0({\mathbb{P}}_A^n,\mathcal{O})\cong A$ and $H^1({\mathbb{P}}_A^n,\mathcal{O})=0$ by [F9]. Moreover $H^0({\mathbb{P}}_A^n,\mathcal{O}(-d))=0$ for $n\ge 1$ and $d>0$ by [F9], while for $n=0$ it is $\cong A$, and $H^1({\mathbb{P}}_A^n,\mathcal{O}(-d))=0$ for $n\mathrm{\ne }1$ since then $1$ is neither $0$ nor $n$.

Vanishing and the top group. For $q\ge 1$, [F9] gives $H^{q+1}({\mathbb{P}}_A^n,\mathcal{O}(-d))=0$ unless $q+1\in \{0,n\}$, and $q+1\ge 2>0$, so $H^q(X,{\mathcal{O}}_X)=0$ for every $q\ge 1$ with $q\mathrm{\ne }n-1$, and for $n\ge 2$ the remaining positive-degree group is $H^{n-1}(X,{\mathcal{O}}_X)\cong H^n({\mathbb{P}}_A^n,\mathcal{O}(-d))$, the free $A$-module on the $(n+1)$-tuples of negative integers with sum $-d$ by [F9]. For $A=k$ a field these tuples correspond bijectively to the compositions of $d$ into $n+1$ positive parts via $g_i=-e_i$, and [F10] counts them by $\left(\genfrac{}{}{0px}{}{d-1}{n}\right)$, which is $0$ when $d-1<n$, i.e. for $d\le n$; hence for $n\ge 2$ one has ${\dim }_k{H}^{n-1}(X,{\mathcal{O}}_X)=\left(\genfrac{}{}{0px}{}{d-1}{n}\right)$ and this group vanishes for $d\le n$. When $n=1$, every $q\ge 1$ group vanishes by the same connecting-map argument, while degree zero has the exact sequence $0\to A\to H^0(X,{\mathcal{O}}_X)\to H^1({\mathbb{P}}_A^1,\mathcal{O}(-d))\to 0$ from step 7.1. For $A=k$ a field, the last group has dimension $d-1$ by [F9, F10] (including $d=1$, when it is zero), so ${\dim }_k{H}^0(X,{\mathcal{O}}_X)=d$.

Boundaries and choice accounting. If $A=0$ then ${\mathbb{P}}_A^n=\mathrm{\varnothing }$, $X=\mathrm{\varnothing }$ and every sheaf and group above is zero, so the sequence and all isomorphisms hold. If $A=k$ is a field and $n=0$, then $f=c{x}_0^d$ with $c\mathrm{\ne }0$ and $X=\mathrm{Spec}(k\mathrm{/}(c))=\mathrm{\varnothing }$: the chart sequence is $0\to k\stackrel{c}{\to }k\to 0\to 0$, the long exact sequence reads $0\to k\stackrel{\cong }{\to }k\to H^0(X,{\mathcal{O}}_X)=0\to H^1({\mathbb{P}}^0,\mathcal{O}(-d))=0$, and $H^q(X,{\mathcal{O}}_X)=0=H^{q+1}({\mathbb{P}}^0,\mathcal{O}(-d))$ for all $q\ge 1$. For general $A$ and $n=0$ the chart ring is $A$ and $X=\mathrm{Spec}(A\mathrm{/}(c))$ for the scalar $c$ with $f=c{x}_0^d$, so $H^0(X,{\mathcal{O}}_X)\cong A\mathrm{/}(c)$ and the degree-zero sequence is exact by construction of the cokernel; the case $f=x_0^d$ has $c=1$ and $X=\mathrm{Spec}(A\mathrm{/}(1))=\mathrm{\varnothing }$. Over a field and $n\ge 2$, $d\le n$ gives $\left(\genfrac{}{}{0px}{}{d-1}{n}\right)=0$. Nonreduced examples are covered, e.g. $f=x_1^d$ over a field for $n\ge 1$; when $n=1$ its degree-zero section space has dimension $d$ by step 8.1, and when $n\ge 2$ the top positive-degree group is computed by the stated isomorphism. The Axiom of Choice is consumed exactly through the suppliers [F7] and [F8] and the twist computation [F9]; no resolution, chart or trivialisation is chosen in this proof.