Residue pairing between H^0 and top cohomology of projective space

Fix $S=k[x_0,\dots ,x_n]$ and $U_i=D_{+}(x_i)$. For a nonempty subset $I$, the intersection $U_I=D_{+}({\prod }_{i\in I}{x}_i)$ is affine and its twist sections are $(S[({\prod }_{i\in I}{x}_i{)}^{-1}]{)}_m$ by [F5]. These are the Laurent monomials whose negative exponents occur only in $I$. The twists are quasi-coherent on these affines, so [F5] makes this an acyclic cover and identifies its Čech cohomology with sheaf cohomology. In top degree the quotient by Čech boundaries kills exactly the monomials having some nonnegative exponent: such a monomial already occurs on the intersection omitting that index. The remaining all-negative classes are the canonical basis from [F5], giving the basis in [F3]. For a homogeneous polynomial $s$ of degree $d$, the graded map $S(m)\to S(m+d)$ is multiplication by $s$; localizing shows that its map on every Čech term is ordinary Laurent multiplication.

Basis monomials. By [F3], for $n\ge 1$ and $d\ge 0$ the space $H^0({\mathbb{P}}_k^n,\mathcal{O}(d))$ has as a $k$-basis the monomials $x^{\underset{‾}{a}}=x_0^{a_0}\cdots x_n^{a_n}$ with $a_i\ge 0$ and ${\sum }_i{a}_i=d$, while $H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1-d))$ has as a $k$-basis the Laurent monomials $x^{\underset{‾}{e}}=x_0^{e_0}\cdots x_n^{e_n}$ with every $e_i<0$ and ${\sum }_i{e}_i=-n-1-d$.

The coefficient isomorphism. At total degree $-n-1$ the conditions $e_i<0$ force $e_i=-1$ for every $i$, so [F3] identifies $H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1))$ with $k$ by sending $x_0^{-1}\cdots x_n^{-1}$ to $1$.

The case $n=0$. By [F3], ${\mathbb{P}}_k^0=\mathrm{Spec}k$, $\mathcal{O}(d)\cong \mathcal{O}$ for every $d$, $H^0(\mathcal{O}(d))\cong k$, and all higher cohomology vanishes. The cup product in degree zero is the ordinary section product by [F1], so the pairing is multiplication $k\times k\to k$, perfect with dual basis $1$, and its compatibility identity is associativity of multiplication.

Cup product with a section. Let $s\in H^0(\mathcal{O}(d))$ and let $m\in \mathbb{Z}$. The section $s$ determines a sheaf morphism ${\sigma }_s:{\mathbb{Z}}_{{\mathbb{P}}^n}\to \mathcal{O}(d)$ and hence a multiplication morphism ${\mu }_s:\mathcal{O}(m)\to \mathcal{O}(m+d)$, obtained by composing ${\sigma }_s\otimes \mathrm{id}$ with the tensor multiplication pairing. Apply naturality in [F1] to this square of tensor pairings and the unit class $1\in H^0({\mathbb{Z}}_{{\mathbb{P}}^n})$: for every $\eta \in H^n(\mathcal{O}(m))$, the cup product $s\cup \eta$ equals $H^n({\mu }_s)(\eta )$. Under the natural Čech comparison [F4], the latter map is computed on the standard cover by multiplying each Laurent Čech representative by $s$ on its intersection. In particular, for $m=-n-1-d$ and monomials $s=x^{\underset{‾}{a}}$, $\eta =x^{\underset{‾}{e}}$, the value of the pairing is the coefficient of $(x_0\cdots x_n{)}^{-1}$ in the Laurent product $x^{\underset{‾}{a}}{x}^{\underset{‾}{e}}$.

Monomial duality. Mapping $\underset{‾}{a}=(a_0,\dots ,a_n)$ to $\underset{‾}{e}=(-1-a_0,\dots ,-1-a_n)$ is a bijection from the nonnegative exponent vectors of total degree $d$ to the all-negative exponent vectors of degree $-n-1-d$, with inverse $e_i\mapsto -1-e_i$. For a matched pair the Laurent product is $(x_0\cdots x_n{)}^{-1}$, so the pairing value is $1$ by step 2.1. For any other basis vector $x^{{\underset{‾}{e}}^{\mathrm{\prime }}}$, the product $x^{\underset{‾}{a}+{\underset{‾}{e}}^{\mathrm{\prime }}}$ has exponent vector different from $(-1,\dots ,-1)$, so its coefficient at that monomial is $0$. Thus the pairing matrix in the two finite monomial bases is the identity and the pairing is perfect.

Compatibility with multiplication. Let $g$ be homogeneous of degree $\delta \ge 0$, let $f=x^{\underset{‾}{a}}$, and let $\eta =x^{\underset{‾}{e}}$ be a basis element of $H^n(\mathcal{O}(-n-1-d-\delta ))$. By step 2.1 applied first to $gf$ and then to $g$ and $f$ successively, both $⟨g\cdot f,\eta ⟩$ and $⟨f,g\cdot \eta ⟩$ are the coefficient of $(x_0\cdots x_n{)}^{-1}$ in $g{x}^{\underset{‾}{a}}{x}^{\underset{‾}{e}}$; associativity and commutativity of polynomial multiplication identify the two products. Bilinearity [F1] extends the identity to arbitrary $f$, $g$ and $\eta$.

Conclusion. Steps 1.2–3.2 prove the perfect pairing and multiplication compatibility for $n\ge 1$, and step 1.4 covers $n=0$. The Axiom of Choice [F2] is inherited through the cup product [F1], the projective cohomology computation [F3], and the Čech comparison [F4]; no further choice is made.