Local normal form for a line bundle on a projective-line bundle

First check the fibre classification used below. Over any field $k$, an invertible sheaf on each affine chart of [F3] is free: a rank-one projective module over the Euclidean rings $k[t]$ and $k[t^{-1}]$ is free, since its corresponding invertible fractional ideal is generated by the greatest common divisor of finitely many generators. After choosing two frames, the overlap transition is a unit in $k[t,t^{-1}]$, necessarily $c{t}^m$ for $c\in k^{\times }$ and $m\in \mathbb{Z}$: comparison of the highest and lowest exponents in a Laurent polynomial and its inverse leaves one monomial. Rescaling a frame removes $c$, so [F3] identifies the line bundle with $\mathcal{O}(m)$. Its degree is $m$ by the transition convention. Consequently each geometric fibre of $F=L(-n)$ is $\mathcal{O}$, by the constant fibre degree hypothesis.

The projection ${\mathbb{P}}_A^1\to \mathrm{Spec}A$ is proper and of finite presentation. It is flat by [F3]. The invertible sheaf $F$ is locally free of rank one over ${\mathcal{O}}_{{\mathbb{P}}_A^1}$, so it is finitely presented and each stalk $F_x$ is flat over $A_{\pi (x)}$: locally $F_x\cong {\mathcal{O}}_{{\mathbb{P}}_A^1,x}$, which is flat over $A_{\pi (x)}$ by [F3]. These are the base-flatness hypotheses of [F1]. Apply [F1] to obtain a bounded finite projective complex $K$ with $H^q(K{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q({\mathbb{P}}_{A^{\mathrm{\prime }}}^1,F_{A^{\mathrm{\prime }}})$ for every $A$-algebra $A^{\mathrm{\prime }}$. For each prime $\mathfrak{p}$ choose an algebraic closure $k$ of $\kappa (\mathfrak{p})$. By step 1.1, $F_k\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}$, so [F2] gives $H^0(K{\otimes }_{A}k)=k$ and $H^q(K{\otimes }_{A}k)=0$ for $q\mathrm{\ne }0$. Field extension from $\kappa (\mathfrak{p})$ to $k$ is faithful and the terms of $K$ are finite projective; hence the same one-dimensional degree-zero pattern holds over $\kappa (\mathfrak{p})$.

Localize $A$ at $\mathfrak{p}$ and choose free bases for the finite projective terms of $K$. If a differential matrix has a nonzero entry modulo the maximal ideal, that entry is a unit in $A_{\mathfrak{p}}$; elementary row and column operations split off the two-term identity complex $A_{\mathfrak{p}}\stackrel{1}{\to }{A}_{\mathfrak{p}}$ without changing cohomology after any base change. Repeat from the highest degree down. The remaining differential matrices vanish modulo the maximal ideal. Their fibre cohomology is then the underlying graded vector space; by step 2.1 it has one basis vector in degree zero and none elsewhere. Thus the remaining complex is $A_{\mathfrak{p}}$ in degree zero. The finitely many inverted pivots remain units on a principal open $D(a)\ni \mathfrak{p}$, so the same splitting holds over $A_a$ and $H^0(K){\mathrm{\mid }}_{D(a)}$ is free of rank one, with all higher cohomology zero. The opens $D(a)$ cover $U$; hence $M:={\pi }_{\ast }F=H^0(K)$ is invertible, and the universal comparison of [F1] identifies $M{\otimes }_A{A}^{\mathrm{\prime }}$ with $H^0({\mathbb{P}}_{A^{\mathrm{\prime }}}^1,F_{A^{\mathrm{\prime }}})$ on each such open for every $A^{\mathrm{\prime }}$.

The adjunction evaluation ${\pi }^{\ast }M\to F$ is a morphism of invertible sheaves. On every geometric fibre its map on $H^0$ is the identity $k\to H^0({\mathbb{P}}_k^1,\mathcal{O})=k$ under step 3.1, hence it is the standard nonzero constant section of $\mathcal{O}$ and an isomorphism at every point of that fibre. A map of line bundles is locally multiplication by one function; if its residue in every geometric fibre is a unit, that function is outside every maximal ideal and is a unit. Thus evaluation is an isomorphism on $E_U$. Tensoring by $\mathcal{O}(n)$ yields the displayed normal form with $M={\pi }_{\ast }(L(-n))$. These constructions commute with restriction of $U$.

The universal finite-projective complex of [F1] supplies the arbitrary-base and arbitrary-ring-change comparison used in steps 2.1 and 3.1. The local degree-zero collapse of step 3.1 and evaluation argument of step 4.1 complete the normal form without a separate base-change theorem. AC is inherited through [F1]–[F3] and the choices of finite bases and field extensions.