A hyperplane in projective space is effective Cartier with O(H) = O(1)

Given: a field $k$, an integer $n\ge 1$, the polynomial ring $S=k[x_0,\dots ,x_n]$ graded by total degree with $\mathrm{deg}{x}_i=1$, the scheme $X=\mathrm{Proj}S$ with its twisting sheaf ${\mathcal{O}}_X(1)$, and the closed subscheme $H=\{x_0=0\}=V_{+}(x_0)$ of the projective space ${\mathbb{P}}_k^n$.

The Axiom of Choice is the statement that every family of nonempty sets has a choice function (The Axiom of Choice).

For a commutative ring $A$ and $n\ge 0$ there is a canonical isomorphism of $\mathrm{Spec}A$-schemes $\mathrm{Proj}A[x_0,\dots ,x_n]\cong {\mathbb{P}}_A^n$ carrying the standard chart $D_{+}(x_i)$ to the standard chart $U_i$ and the coordinate $x_{\mathrm{\ell }}\mathrm{/}{x}_i$ to $x_{\mathrm{\ell }}^{(i)}$; it is natural in $A$, and for $n=0$ both sides are $\mathrm{Spec}A$. Its proof assumes the Axiom of Choice, inherited from the affine-scheme construction (Projective space is Proj of a polynomial ring).

For a field $k$ and $n\ge 0$, with $S=k[x_0,\dots ,x_n]$ standard graded, the chart of $\mathrm{Proj}S$ at $x_i$ is $D_{+}(x_i)=\mathrm{Spec}k[x_{\mathrm{\ell }}\mathrm{/}{x}_i:\mathrm{\ell }\mathrm{\ne }i]$ in the coordinates $x_{\mathrm{\ell }}^{(i)}=x_{\mathrm{\ell }}\mathrm{/}{x}_i$ of the standard charts $U_i$ of ${\mathbb{P}}_k^n$, and the overlap identifications are the transition formulas $x_a^{(i)}\mapsto x_a^{(j)}\mathrm{/}{x}_i^{(j)}$ of those charts; the identification $\mathrm{Proj}S={\mathbb{P}}_k^n$ is the canonical one (Projective space is Proj of a polynomial ring, Relative projective space from standard charts).

Let $k$ be a field, $n\ge 0$ and $0\mathrm{\ne }F\in k[x_0,\dots ,x_n]$ homogeneous of degree $e\ge 1$. The theorem on closed subschemes cut out by homogeneous ideals identifies $V_{+}(F)=\mathrm{Proj}(k[x_0,\dots ,x_n]\mathrm{/}(F))$ with a closed subscheme of ${\mathbb{P}}_k^n$ and gives its standard-chart ring as $B_{(x_i)}\mathrm{/}(F{)}_{(x_i)}$, where $B=k[x_0,\dots ,x_n]$; in the standard chart $B_{(x_i)}=k[x_a\mathrm{/}{x}_i:a\mathrm{\ne }i]$, and the degree-zero localized ideal is $(F\mathrm{/}{x}_i^e)$. Thus the chart ring is $k[x_a\mathrm{/}{x}_i:a\mathrm{\ne }i]\mathrm{/}(F\mathrm{/}{x}_i^e)$, so these charts cover $V_{+}(F)$ (Closed subschemes of projective space and saturated ideals).

If $S$ is a commutative nonnegatively graded ring generated as an $S_0$-algebra by $S_1$ and $X=\mathrm{Proj}S$, then every twisting sheaf ${\mathcal{O}}_X(n)$ is invertible and the multiplication maps ${\mathcal{O}}_X(m)\otimes {\mathcal{O}}_X(n)\to {\mathcal{O}}_X(m+n)$ are isomorphisms; the proof assumes the Axiom of Choice, inherited from the Proj sheaf construction (Invertible twists for degree-one generated rings).

${\mathcal{O}}_X(n)=\widetilde{S(n)}$ is the associated sheaf of the shifted graded module $S(n)$, so that for a homogeneous $f\in S_{+}$ of positive degree one has $\mathrm{\Gamma }(D_{+}(f),{\mathcal{O}}_X(n))=S(n{)}_{(f)}$, the degree-zero part of the homogeneous localisation $S(n)[f^{-1}]$, and ${\mathcal{O}}_X(0)={\mathcal{O}}_X$; no invertibility is asserted by the definition itself (Twisting sheaf on Proj).

There is a scheme $\mathrm{Proj}S$ whose charts $D_{+}(f)\cong \mathrm{Spec}{S}_{(f)}$ for homogeneous $f\in S_{+}$ form an affine open cover, compatibly with the standard-open basis (Proj carries a scheme structure, Standard opens of Proj).

The standard opens satisfy $D_{+}(f)=\{\mathfrak{p}\in \mathrm{Proj}S:f\mathrm{\notin }\mathfrak{p}\}$ and $D_{+}(f)\cap D_{+}(g)=D_{+}(fg)$ for homogeneous $f,g\in S_{+}$ (Standard opens of Proj).

Sections of a sheaf on the members of an open cover that agree on overlaps glue to a unique global section (A sheaf on a topological space).

A section of ${\mathcal{O}}_X$ over $U$ is regular when multiplication by each of its germs is injective on the corresponding local ring; the regular sections form the multiplicative set $S_X(U)$ used to build the sheaf ${\mathcal{K}}_X$ of meromorphic functions (Sheaf total quotient rings).

For a field $k$ and $r\ge 0$ the polynomial ring $k[x_1,\dots ,x_r]$ is a unique factorisation domain, hence an integral domain (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes).

Let $X$ be a scheme, $\mathcal{L}$ an invertible ${\mathcal{O}}_X$-module and $s\in \mathrm{\Gamma }(X,\mathcal{L})$ a regular global section, with generators $e_i$ of $\mathcal{L}$ on an open cover $\{U_i\}$ and coefficients $f_i$ defined by $s{\mathrm{\mid }}_{U_i}=f_i{e}_i$. Then the coefficients are regular sections of ${\mathcal{O}}_{U_i}$, their ratios are units on overlaps, they glue to an effective Cartier divisor $D$ with local-equation datum $\{(U_i,f_i)\}$, there is a canonical isomorphism $\phi :{\mathcal{O}}_X(D)\to \mathcal{L}$ with $\phi (1_D)=s$, and the ideal sheaf of the vanishing subscheme $Z_D$ satisfies $I_D{\mathrm{\mid }}_{U_i}=f_i{\mathcal{O}}_{U_i}$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor).

An effective Cartier divisor on $X$ is a Cartier divisor admitting a local-equation datum $\{(U_i,f_i)\}$ with $f_i\in {\mathcal{O}}_X(U_i)$ regular; such a divisor has a vanishing subscheme whose ideal sheaf is locally $(f_i)$, and the unit equation $1$ gives the empty effective divisor (Effective cartier divisor).