Two-affine Mayer–Vietoris on the projective line

The two charts $U_0,U_{\mathrm{\infty }}$ are nonempty open subschemes of ${\mathbb{P}}_k^1$ covering it, their overlap $W=U_0\cap U_{\mathrm{\infty }}$ is nonempty and identified with $\mathrm{Spec}k[t,t^{-1}]$, and on $W$ the two coordinate functions are mutually inverse units $t,u$ with $tu=1$ [F2, F3]. The restriction $\mathcal{O}(n){\mathrm{\mid }}_{U_0}$ is the structure sheaf of $U_0$ with global frame $e_0$, and likewise $\mathcal{O}(n){\mathrm{\mid }}_{U_{\mathrm{\infty }}}$ is the structure sheaf of $U_{\mathrm{\infty }}$ with global frame $e_{\mathrm{\infty }}$ [F2].

By [F4] applied to the chart $U_0\cong \mathrm{Spec}k[t]$ and the element $f=1$, whose basic open is the whole chart, the global sections of the structure sheaf of $U_0$ are $k[t]$; the frame $e_0$ is a global generator, so $\mathrm{\Gamma }(U_0,\mathcal{O}(n))=k[t]e_0$. The same argument on $U_{\mathrm{\infty }}\cong \mathrm{Spec}k[u]$ gives $\mathrm{\Gamma }(U_{\mathrm{\infty }},\mathcal{O}(n))=k[u]e_{\mathrm{\infty }}$, and by [F3] the sections over the overlap are $\mathrm{\Gamma }(W,\mathcal{O}(n))=k[t,t^{-1}]e_0$ with $e_{\mathrm{\infty }}=t^n{e}_0$ there. Consequently every section of $\mathcal{O}(n)$ over the overlap has the form $c(t)e_0$ for a unique Laurent polynomial $c\in k[t,t^{-1}]$.

The underlying topological space of ${\mathbb{P}}_k^1$ is the union of the two open subsets $U_0$, $U_{\mathrm{\infty }}$, and $\mathcal{O}(n)$ is in particular a sheaf of abelian groups on it, its module structure being forgotten; the Axiom of Choice [F6] is available, as required by the Mayer–Vietoris theorem [F1]. Applying [F1] to this cover and substituting the three modules of [step 2.1] gives the long exact sequence of the statement, in which the second map is the difference of the two restrictions: for $a\in k[t]$ and $b\in k[u]$ the pair $(a{e}_0,b{e}_{\mathrm{\infty }})$ is sent to $b{e}_{\mathrm{\infty }}{\mathrm{\mid }}_W-a{e}_0{\mathrm{\mid }}_W=(t^{n}b(t^{-1})-a(t))e_0,$ using $e_{\mathrm{\infty }}=t^n{e}_0$ and $u=t^{-1}$ on $W$ [F2, F3]. The first term is $\mathrm{\Gamma }({\mathbb{P}}_k^1,\mathcal{O}(n))$ read as $H^0$ through the canonical identification of [F5], so the sequence begins $0\to H^0({\mathbb{P}}_k^1,\mathcal{O}(n))\to k[t]\oplus k[u]\stackrel{\beta }{\to }k[t,t^{-1}]$ with $\beta (a,b)=t^{n}b(t^{-1})-a(t)$.

An element $(a,b)$ lies in $\ker \beta$ exactly when $a(t)=t^{n}b(t^{-1})$. Writing $b={\sum }_{m\ge 0}{b}_m{u}^m$, the right-hand side is ${\sum }_{m\ge 0}{b}_m{t}^{n-m}$, which is a polynomial in $t$ exactly when $b_m=0$ for all $m>n$; this forces $b=0$ and $a=0$ when $n<0$, and for $n\ge 0$ leaves the $n+1$ free coefficients $b_0,\dots ,b_n$ with $a(t)={\sum }_{m=0}^n{b}_m{t}^{n-m}$ and $b(u)={\sum }_{m=0}^n{b}_m{u}^m$. Hence $\ker \beta \cong k^{n+1}$ for $n\ge 0$ and $\ker \beta =0$ for $n<0$.

Exactness of the sequence of [step 3.1] at the terms $k[t]\oplus k[u]$ and $k[t,t^{-1}]$ says $\ker \beta \cong H^0({\mathbb{P}}_k^1,\mathcal{O}(n))$ (the map out of $H^0$ being injective), and exactness at $H^1({\mathbb{P}}_k^1,\mathcal{O}(n))$ says that the image of $\mathrm{\partial }$ is the kernel of the restriction map to the two charts; since $\mathrm{\partial }$ induces an isomorphism from $k[t,t^{-1}]\mathrm{/}\mathrm{im}\beta$ onto that image, $\mathrm{coker}\beta \cong \ker (H^1({\mathbb{P}}_k^1,\mathcal{O}(n))\to H^1(U_0,\mathcal{O}(n){\mathrm{\mid }}_{U_0})\oplus H^1(U_{\mathrm{\infty }},\mathcal{O}(n){\mathrm{\mid }}_{U_{\mathrm{\infty }}}))\mathrm{.}$

Under the identification of $k[t,t^{-1}]$ with the Laurent polynomials, $\mathrm{im}\beta =k[t]+t^{n}k[t^{-1}]$ is the span of the monomials $t^j$ with $j\ge 0$ or $j\le n$, since $t^{n}k[t^{-1}]$ is spanned by $t^{n-m}$, $m\ge 0$ [step 2.1]. Hence $\beta$ is surjective exactly when every integer $j$ satisfies $j\ge 0$ or $j\le n$, that is exactly when $n\ge -1$; for $n\le -2$ the monomials $t^{n+1},\dots ,t^{-1}$ are not in the image and their classes form a basis of $\mathrm{coker}\beta$, so $\mathrm{coker}\beta \cong k^{-n-1}$.

Specialising [step 4.2] and [step 3.2]: for $n=0$ the differential is $\beta (a,b)=b(t^{-1})-a(t)$, it is surjective, and its kernel consists of the pairs with $a=b$ constant, so $H^0({\mathbb{P}}_k^1,{\mathcal{O}}_{{\mathbb{P}}_k^1})\cong k$ is the constants; for $n=1$ the differential is $\beta (a,b)=tb(t^{-1})-a(t)$, it is surjective, and its kernel is $\{(c_{0}t+c_1,c_0+c_{1}u):c_0,c_1\in k\}\cong k^2$ with the two generators $(t,1)$ and $(1,u)$; for $n=-1$ the differential is surjective with zero kernel; and for $n=-2$ the image misses exactly the multiples of $t^{-1}$, so $\mathrm{coker}\beta \cong k$ and $\beta$ is not surjective. In all cases the orientation of the transition enters through $e_{\mathrm{\infty }}=t^n{e}_0$, which converts a section $b(u)e_{\mathrm{\infty }}$ over $U_{\mathrm{\infty }}$ into $t^{n}b(t^{-1})e_0$ over the overlap.

The three chart modules of [step 2.1] and the sequence of [step 3.1], whose second map is $\beta$ of [step 4.2], prove assertions 1 and 2 of the statement; [step 4.1] gives assertion 3, and [step 3.2], [step 4.2] and [step 5.1] give assertion 4 with the two checks $n=0$ and $n=1$ of the transition orientation. The higher chart groups $H^i(U_0,\mathcal{O}(n){\mathrm{\mid }}_{U_0})$, $H^i(U_{\mathrm{\infty }},\mathcal{O}(n){\mathrm{\mid }}_{U_{\mathrm{\infty }}})$ and $H^i(W,\mathcal{O}(n){\mathrm{\mid }}_W)$ for $i\ge 1$ appear in the sequence as themselves and are not claimed to vanish; only the degree zero row $k[t]\oplus k[u]\to k[t,t^{-1}]$ and its immediate exactness consequences are computed. The Axiom of Choice of [F6] enters exactly through [F1], whose proof produces a functorial injective resolution, and through the cited construction of the structure sheaves of the affine charts in Two-affine projective line and its twists; no further choice is made, the modules and the map $\beta$ being given by explicit polynomials. ∎