Smooth-Projective Serre Duality and Flag-Variety Line Bundles

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.

The pairing is well defined. Sheaf multiplication $\mathcal{O}(d){\otimes }_{\mathbb{Z}}\mathcal{O}(-d-n-1)\to \mathcal{O}(-n-1)$ gives by [F2] a bilinear cup product into $H^n(\mathcal{O}(-n-1))$; composing with the $k$-linear residue trace [F1] gives the displayed pairing. It is $k$-bilinear: multiplication by $\lambda \in k$ on either twist sheaf commutes with the tensor pairing, so naturality of the cup product [F2] carries the scalar action on either argument to multiplication by $\lambda$ on the target.

Vanishing in the middle degrees. By [F5], for $n\ge 1$ both $H^q(\mathcal{O}(d))$ and $H^{n-q}(\mathcal{O}(-d-n-1))$ vanish whenever $0<q<n$, because both cohomological degrees lie strictly between $0$ and $n$. Their zero-space pairing is perfect. The remaining degrees are $q=0$ and $q=n$.

The case $q=0$. If $d\ge 0$, this is precisely the perfect residue pairing of [F3]. If $d<0$, then $H^0(\mathcal{O}(d))=0$ by [F5], while $-d-n-1>-n-1$, so $H^n(\mathcal{O}(-d-n-1))=0$ by [F5]; the pairing of two zero spaces is perfect.

The case $q=n$. If $d\le -n-1$, put $d^{\mathrm{\prime }}=-d-n-1\ge 0$. For $s\in H^0(\mathcal{O}(d^{\mathrm{\prime }}))$ and $\eta \in H^n(\mathcal{O}(-n-1-d^{\mathrm{\prime }}))$, [F3] identifies $t_{{\mathbb{P}}^n}(s\cup \eta )$ with the perfect residue pairing. The exchanged cup product $\eta \cup s$ has the same image: the section $s$ defines a sheaf map ${\sigma }_s:{\mathbb{Z}}_{{\mathbb{P}}^n}\to \mathcal{O}(d^{\mathrm{\prime }})$ and multiplication ${\mu }_s:\mathcal{O}(-n-1-d^{\mathrm{\prime }})\to \mathcal{O}(-n-1)$; naturality in [F2] applied to $\mathrm{id}\otimes {\sigma }_s$ and the right-unit class gives $\eta \cup s=H^n({\mu }_s)(\eta )$, while the left-unit argument of [F3] gives $s\cup \eta =H^n({\mu }_s)(\eta )$ because sheaf multiplication is commutative. Thus the exchanged pairing is perfect. If $d>-n-1$, then $H^n(\mathcal{O}(d))=0$ and $-d-n-1<0$ gives $H^0(\mathcal{O}(-d-n-1))=0$ by [F5], so the pairing is perfect vacuously.

The case $n=0$. By [F5], ${\mathbb{P}}_k^0=\mathrm{Spec}k$ and every twist has $H^0\cong k$ with no higher cohomology. The trace [F1] and degree-zero cup product [F2] make the pairing ordinary multiplication $k\times k\to k$, perfect with dual basis $1$, as also recorded in [F3].

Conclusion. Steps 1.1–1.5 cover bilinearity, the middle degrees, both extremes for $n\ge 1$, and $n=0$. AC [F4] is inherited through the cup product [F2] and the projective-space cohomology and residue suppliers [F3, F5]; no additional selection is made.

The section module. For $i=0,\dots ,n$ the section $x_i\in \mathrm{\Gamma }({\mathbb{P}}_k^n,\mathcal{O}(1))=S_1$ is homogeneous of degree one and its nonvanishing locus is the standard open $D_{+}(x_i)$; these finitely many affine charts cover ${\mathbb{P}}_k^n$. Put $M_F={⨁}_{d\ge 0}\mathrm{\Gamma }({\mathbb{P}}_k^n,F(d)),F(d)=F{\otimes }_{{\mathcal{O}}_{{\mathbb{P}}^n}}\mathcal{O}(d),$ a graded $S$-module whose degree-$d$ part is $\mathrm{\Gamma }({\mathbb{P}}_k^n,F(d))$, with $S$-action induced by the multiplication maps $\mathcal{O}(e)\otimes \mathcal{O}(d)\to \mathcal{O}(e+d)$ of [F1]. The sheaf $F$ is quasi-coherent by [F2] and ${\mathbb{P}}_k^n$ is quasi-compact and quasi-separated by [F3], as is each affine chart.

Chartwise comparison. Apply the section-extension lemma [F4] on the quasi-compact quasi-separated scheme ${\mathbb{P}}_k^n$ to the quasi-coherent sheaf $F$, the invertible sheaf $L=\mathcal{O}(1)$ and the section $s=x_i\in \mathrm{\Gamma }({\mathbb{P}}_k^n,\mathcal{O}(1))$, so that $d=1$ and $X_s=D_{+}(x_i)$. Since ${\mathrm{\Gamma }}_{\ast }({\mathbb{P}}_k^n,F,\mathcal{O}(1))={⨁}_{r\ge 0}\mathrm{\Gamma }({\mathbb{P}}_k^n,F\otimes \mathcal{O}(1{)}^r)=M_F$ by [F1], the lemma gives a canonical isomorphism of abelian groups ${\theta }_i:M_F[x_i^{-1}{]}_0=(M_F{)}_{(x_i)}⟶\mathrm{\Gamma }(D_{+}(x_i),F),a\mathrm{/}{x}_i^r⟼a\otimes x_i^{-r}{\mathrm{\mid }}_{D_{+}(x_i)}\mathrm{.}$

The same group on the sheaf side. By [F5] the global sections of the associated sheaf on the standard open are $\mathrm{\Gamma }(D_{+}(x_i),\widetilde{M_F})=(M_F{)}_{(x_i)}$, and restriction from $D_{+}(x_i)$ to $D_{+}(x_i{x}_j)$ is the localisation $(M_F{)}_{(x_i)}\to (M_F{)}_{(x_i{x}_j)}$.

Finite generation after truncation. By [F6] there is $m_0\ge 0$ such that $M:={⨁}_{d\ge m_0}\mathrm{\Gamma }({\mathbb{P}}_k^n,F(d))$ is a finitely generated graded $S$-module; it is the degree-$\ge m_0$ truncation of $M_F$.

Graded free covers and finite generation of syzygies. Since $S$ is Noetherian and $M$ is finitely generated and graded, $M$ has a finite homogeneous generating set; the degree-preserving surjection $F_0={⨁}_{j}S(-d_j)\to M$ from the finite graded free module on those generators has a kernel $K_1=\ker (F_0\to M)$ which is a graded submodule of the finitely generated module $F_0$, hence is again finitely generated. Repeating this construction with $K_0:=M$ and $K_{i+1}:=\ker (F_i\to K_i)$, each $F_i$ a finite graded free module and each map degree-preserving, produces a graded free resolution $\cdots ⟶F_1⟶F_0⟶M⟶0.$ Every $F_i$ is a finite direct sum of shifted free modules $S(d)$, every kernel is graded and finitely generated, and all maps have degree zero.

Gluing the chart isomorphisms. For every $i$ the map ${\theta }_i$ of step 1.2 is the canonical identification of $\mathrm{\Gamma }(D_{+}(x_i),F)$ with the degree-zero localisation of $M_F$, and by step 1.3 the same description holds for $\widetilde{M_F}$. On an overlap $D_{+}(x_i{x}_j)$ both identifications restrict to the common localisation $(M_F{)}_{(x_i{x}_j)}$: the restriction of ${\theta }_i$ is induced by inverting $x_j$, and this is exactly the restriction of $\widetilde{M_F}$ by [F5]. The charts $D_{+}(x_i)$ cover ${\mathbb{P}}_k^n$ by [F3] and [F5], so the chart isomorphisms agree on overlaps and glue to an isomorphism of ${\mathcal{O}}_{{\mathbb{P}}^n}$-modules $\theta :\widetilde{M_F}⟶F,$ quasi-coherent on both sides. The overlap condition holds by the displayed identification of both restrictions with the same localisation map, whose two composites to $(M_F{)}_{(x_i{x}_j{x}_l)}$ agree on triple overlaps by the universal property of localisation.

Termination at $n+1$ by graded Nakayama. By [F7] the regular ring $S$ has global dimension $n+1$, so the finitely generated module $M$ has projective dimension at most $n+1$. The syzygy criterion [F8] applied to the resolution of step 1.5 makes $K_{n+1}$ projective; it is finitely generated and graded. We prove that any finitely generated graded projective $S$-module $K$ is graded free. Put $\mathfrak{m}=S_{+}=(x_0,\dots ,x_n)$. First, if a finitely generated graded $S$-module $Q$ satisfies $Q=\mathfrak{m}Q$, then $Q=0$: by [F9] its nonzero homogeneous degrees are bounded below, and a nonzero homogeneous element of least degree cannot be a sum ${\sum }_j{x}_j{q}_j$ with each nonzero $q_j$ of one smaller degree. This is graded Nakayama and does not require $\mathfrak{m}\subseteq J(S)$. Choose homogeneous lifts of a homogeneous $k$-basis of the finite-dimensional graded vector space $K\mathrm{/}\mathfrak{m}K$, and let $u:G\to K$ be the resulting degree-preserving map from a finite sum of shifted copies of $S$. Its cokernel $Q$ is finite graded and $Q\mathrm{/}\mathfrak{m}Q=0$, so graded Nakayama makes $u$ surjective. Projectivity of $K$ gives an ungraded section $s$ of $u$; taking, for each homogeneous $v\in K$, the component of $s(v)$ in degree $\mathrm{deg}v$ gives a degree-zero section, because $u$ preserves degrees and $u(s(v))=v$. Thus $G\cong K\oplus L$ as graded modules, where $L=\ker u$ is finite graded by [F9]. Since $u$ is an isomorphism modulo $\mathfrak{m}$ and the splitting is graded, $L\mathrm{/}\mathfrak{m}L=0$; graded Nakayama gives $L=0$. Hence $u$ is a graded isomorphism and $K$ is a finite direct sum of shifted free modules. Apply this to $K_{n+1}$, including $K_{n+1}=0$, to obtain the exact finite graded free sequence $0⟶F_{n+1}⟶F_n⟶\cdots ⟶F_0⟶M⟶0,$ with $F_{n+1}=K_{n+1}$. If an earlier syzygy is free, the sequence may terminate earlier; its length is at most $n+1$.

The sheaf $\widetilde{M_F}$ is quasi-coherent by [F5], so $\theta$ is a morphism of quasi-coherent modules and its construction used only the canonical localisation maps, not a choice of trivialisations: the identifications ${\theta }_i$ are the maps supplied by [F4].

Truncation does not change the associated sheaf. The inclusion $M\hookrightarrow M_F$ of graded $S$-modules induces for every $i$ a map of localisations $(M{)}_{(x_i)}\to (M_F{)}_{(x_i)}$, which is injective because localisation is exact [F10]. It is surjective: a fraction $a\mathrm{/}{x}_i^r$ with $a\in \mathrm{\Gamma }({\mathbb{P}}_k^n,F(r))$ homogeneous of degree $r$ satisfies $x_i^{N}a\in \mathrm{\Gamma }({\mathbb{P}}_k^n,F(r+N))\subseteq M$ for every $N\ge m_0$, and $a\mathrm{/}{x}_i^r=(x_i^{N}a)\mathrm{/}{x}_i^{r+N}$ in the degree-zero localisation. Hence $(M{)}_{(x_i)}\cong (M_F{)}_{(x_i)}$ for every $i$, the two associated sheaves agree on each standard chart, and since the charts cover ${\mathbb{P}}_k^n$ the canonical morphism $\tilde{M}\to \widetilde{M_F}$ is an isomorphism. Composing with $\theta$ of step 2.1 gives $\tilde{M}\cong F\mathrm{.}$

Sheafification is exact. The sequence of step 2.2 is an exact sequence of graded $S$-modules. Localising at $\{x_i^r\}$ is exact [F10], so for each $i$ the sequence $0⟶(F_{n+1}{)}_{(x_i)}⟶(F_n{)}_{(x_i)}⟶\cdots ⟶(F_0{)}_{(x_i)}⟶(M{)}_{(x_i)}⟶0$ is exact; on the affine chart $D_{+}(x_i)=\mathrm{Spec}{S}_{(x_i)}$ the associated-sheaf functor is exact, being a quasi-inverse equivalence [F10], and it intertwines restriction to the chart with the identifications of [F5]. Hence the sheafified complex restricted to each chart is exact, and its stalks are the localisations of the exact localised sequences. Since the standard charts cover ${\mathbb{P}}_k^n$, exactness of the sheafified complex is checked stalkwise [F11], so $0⟶{\tilde{F}}_{n+1}⟶{\tilde{F}}_n⟶\cdots ⟶{\tilde{F}}_0⟶\tilde{M}⟶0$ is an exact sequence of ${\mathcal{O}}_{{\mathbb{P}}_k^n}$-modules. By [F1] each ${\tilde{F}}_i$ is a finite direct sum ${⨁}_j\mathcal{O}(d_{ij})$, and $\tilde{M}\cong F$ by step 3.2.

Conclusion. Step 3.2 produces a finitely generated graded $S$-module $M$ with $\tilde{M}\cong F$, step 2.2 a finite exact graded free resolution of $M$ of length at most $n+1$, and step 4.1 its exact sheafification by finite direct sums of twisting sheaves. For the zero sheaf take $M=0$ and the zero resolution; for $n=0$ the regular ring $S=k[x_0]$ has global dimension one, and the same graded argument applies. The Axiom of Choice [A1] is inherited through the section-extension lemma [F4], finite generation [F6], the regular-ring global-dimension theorem [F7], and the syzygy criterion [F8]. The graded Nakayama argument in step 2.2 uses a least homogeneous degree and does not invoke ordinary Nakayama at the irrelevant ideal.