Extensions of line bundles on the projective line split after ordering

The induction statement. We prove, for every $r\ge 0$, the assertion $P(r)$: for every $c\in \mathbb{Z}$ and every short exact sequence $0\to {\mathcal{O}}_X(c)\stackrel{i}{\to }E\stackrel{p}{\to }F\to 0$ of finite locally free ${\mathcal{O}}_X$-modules with $F\cong {⨁}_{i=1}^r{\mathcal{O}}_X(c_i)$ and $c_i\le c$ for all $i$, one has $E\cong {\mathcal{O}}_X(c)\oplus F$. Since $X\cong \mathrm{Proj}k[x_0,x_1]$ and ${\mathcal{O}}_X(1)$ generates the twisting sheaves with ${\mathcal{O}}_X(m)\otimes {\mathcal{O}}_X(n)\cong {\mathcal{O}}_X(m+n)$ by [F1], the reindexing of the $c_i$ and the canonical identifications of direct sums do not change the conclusion, and $P(r)$ for all $r$ and all $c$ gives both assertions of the Statement: the first with $c=0$.

Base case. For $r=0$ one has $F=0$, so $p=0$ and $i$ is an isomorphism; hence $E\cong {\mathcal{O}}_X(c)\cong {\mathcal{O}}_X(c)\oplus F$.

Vanishing of the relevant $H^1$. Assume $r\ge 1$ and that $P(r-1)$ holds. Choose an enumeration with $c_r={\min }_i{c}_i$, put $F^{\mathrm{\prime }}={⨁}_{i<r}{\mathcal{O}}_X(c_i)$, so that $F\cong F^{\mathrm{\prime }}\oplus {\mathcal{O}}_X(c_r)$, and let $q:F\to {\mathcal{O}}_X(c_r)$ be the projection. Twist the given sequence by ${\mathcal{O}}_X(-c_r)$: by [F1] and [F5] the result is the short exact sequence $0\to {\mathcal{O}}_X(c-c_r)\to E(-c_r)\to F(-c_r)\to 0$, with $F(-c_r)\cong F^{\mathrm{\prime }}(-c_r)\oplus {\mathcal{O}}_X$. Since $c_i-c_r\ge 0$ for all $i<r$, each summand ${\mathcal{O}}_X(c_i-c_r)$ has vanishing $H^1$ by [F2], and $H^1$ vanishes on the finite direct sum $F^{\mathrm{\prime }}(-c_r)$ by induction on the number of summands: for a summand split injection $0\to G\to G\oplus {\mathcal{O}}_X(c_j-c_r)\to {\mathcal{O}}_X(c_j-c_r)\to 0$ of [F5] the long exact sequence [F3] yields the exact portion $H^1(X,G)\to H^1(X,G\oplus {\mathcal{O}}_X(c_j-c_r))\to H^1(X,{\mathcal{O}}_X(c_j-c_r))$ with both outer groups zero. Also $c-c_r\ge 0\ge -1$, so $H^1(X,{\mathcal{O}}_X(c-c_r))=0$ by [F2].

A lift of the unit section. The long exact sequence [F3] of the twisted sequence begins $H^0(X,E(-c_r))\to H^0(X,F(-c_r))\to H^1(X,{\mathcal{O}}_X(c-c_r))=0,$ so the first map is surjective. The projection $q$ twisted by ${\mathcal{O}}_X(-c_r)$ is a surjection $F(-c_r)\to {\mathcal{O}}_X$ whose kernel is $F^{\mathrm{\prime }}(-c_r)$; by step 1.3 and the long exact sequence of $0\to F^{\mathrm{\prime }}(-c_r)\to F(-c_r)\to {\mathcal{O}}_X\to 0$, the induced map on $H^0$ is surjective. Composing the two surjections there is $s\in H^0(X,E(-c_r))$ with image the unit section $1\in H^0(X,{\mathcal{O}}_X)\cong k$. By [F4] the section $s$ corresponds to a morphism $\sigma =s^{\mathrm{♯}}:{\mathcal{O}}_X\to E(-c_r)$ with $p^{\mathrm{\prime }}\circ \sigma =\mathrm{i}\mathrm{d}$, where $p^{\mathrm{\prime }}:E(-c_r)\to {\mathcal{O}}_X$ is the composite $q\circ p$ twisted.

Splitting off the minimal summand. Let $K=\ker p^{\mathrm{\prime }}\subseteq E(-c_r)$. The morphism $\mathrm{\Psi }:K\oplus {\mathcal{O}}_X\to E(-c_r)$ defined on the summands by the inclusion of $K$ and by $\sigma$ is an isomorphism. Indeed, on stalks at a point $x$ the argument is the elementary module argument: if $k+\sigma (l)=0$ with $k\in K_x$, $l\in {\mathcal{O}}_{X,x}$, then applying $p_x^{\mathrm{\prime }}$ gives $l=0$ and then $k=0$, so ${\mathrm{\Psi }}_x$ is injective; and for $m\in E(-c_r{)}_x$ one has $m-\sigma (p_x^{\mathrm{\prime }}(m))\in K_x$, so ${\mathrm{\Psi }}_x$ is surjective. By [F5] a morphism of sheaves that is stalkwise bijective is an isomorphism. Hence $E(-c_r)\cong K\oplus {\mathcal{O}}_X$.

The complement is again an extension of the same shape. Let $r^{\mathrm{\prime }}:F(-c_r)\to F^{\mathrm{\prime }}(-c_r)$ be the projection of step 1.3. The sequence $0\to {\mathcal{O}}_X(c-c_r)\to K\to F^{\mathrm{\prime }}(-c_r)\to 0$ is exact, where the first map is the restriction of the inclusion of ${\mathcal{O}}_X(c-c_r)$ and the second is the restriction of $r^{\mathrm{\prime }}\circ p$ to $K$. Stalks at $x$: the sequence $0\to {\mathcal{O}}_X(c-c_r{)}_x\to E(-c_r{)}_x\to F(-c_r{)}_x\to 0$ is exact and $F(-c_r{)}_x=F^{\mathrm{\prime }}(-c_r{)}_x\oplus {\mathcal{O}}_{X,x}$; an element of $F^{\mathrm{\prime }}(-c_r{)}_x$ lifted to $E(-c_r{)}_x$ can be corrected by an element with the same ${\mathcal{O}}_{X,x}$-component to lie in $K_x$, because $E(-c_r{)}_x\to {\mathcal{O}}_{X,x}$ is surjective, so $K_x\to F^{\mathrm{\prime }}(-c_r{)}_x$ is surjective; and its kernel is the kernel of $E(-c_r{)}_x\to F(-c_r{)}_x$, namely ${\mathcal{O}}_X(c-c_r{)}_x$, the inclusion being injective. Exactness of the displayed sequence follows from [F5]. Also $K$ is finite locally free: near any point, trivialize the kernel line bundle and the finite locally free quotient $F^{\mathrm{\prime }}(-c_r)$, lift the finitely many quotient basis germs to sections of $K$, and shrink so that their images equal the basis sections. These lifts define a local splitting. Together with a frame of the kernel they identify $K$ locally with a finite free sheaf, as required for $P(r-1)$.

Induction step. In the exact sequence of step 4.1 the quotient $F^{\mathrm{\prime }}(-c_r)\cong {⨁}_{i<r}{\mathcal{O}}_X(c_i-c_r)$ has $r-1$ summands. Since $c_r={\min }_i{c}_i$ and $c_i\le c$, their exponents satisfy $0\le c_i-c_r\le c-c_r$. Thus $P(r-1)$, applied with kernel exponent $c-c_r$, gives $K\cong {\mathcal{O}}_X(c-c_r)\oplus F^{\mathrm{\prime }}(-c_r)$. Combining with step 3.1 and twisting back by ${\mathcal{O}}_X(c_r)$, which preserves direct sums and isomorphisms by [F1] and inverts $(-c_r)$, $E\cong E(-c_r)(c_r)\cong K(c_r)\oplus {\mathcal{O}}_X(c_r)\cong {\mathcal{O}}_X(c)\oplus F^{\mathrm{\prime }}\oplus {\mathcal{O}}_X(c_r)\cong {\mathcal{O}}_X(c)\oplus F\mathrm{.}$ This proves $P(r)$.

Conclusion. By steps 1.2 and 5.1 the assertion $P(r)$ holds for every $r\ge 0$ and every $c\in \mathbb{Z}$; taking $c=0$ gives $M\cong {\mathcal{O}}_X\oplus W$ for the first sequence of the Statement, and the general assignment $c=b$ gives $E\cong {\mathcal{O}}_X(b)\oplus F$ whenever all $b_i\le b$. Every selection made was the choice of one lift of a specified element in step 2.1 and finitely many such selections occur, so the Axiom of Choice enters through the Proj, twisting-sheaf and cohomology suppliers recorded in [F6].