Generator cocycle for H1 of O(-2)

Given: A field $k$; the projective line $X={\mathbb{P}}_k^1$ with the standard charts $U_0=D_{+}(x_0)$, $U_1=D_{+}(x_1)$ and the twisting sheaf ${\mathcal{O}}_X(-2)$; and the Axiom of Choice inherited from the cited suppliers.

Charts and affine cover (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, Standard opens are affine): $X={\mathbb{P}}_k^1\cong \mathrm{Proj}k[x_0,x_1]$; the standard charts $U_i=D_{+}(x_i)$ are affine open subschemes forming a cover of $X$, and the intersection $U_0\cap U_1=D_{+}(x_0{x}_1)$ is again affine; the index set $\{0,1\}$ is ordered by $0<1$.

Separatedness (The relative projective-space diagonal is closed): the diagonal ${\mathrm{\Delta }}_{X\mathrm{/}k}$ is a closed immersion, so the structure morphism of $X$ over $\mathrm{Spec}k$ is separated.

Twists and their sections (Twisting sheaf on Proj, Twists of a quasi-coherent sheaf, Sections of a graded-module sheaf on a standard open): ${\mathcal{O}}_X(-2)$ is the associated sheaf $\widetilde{S(-2)}$ of the graded module $S(-2)$, $S=k[x_0,x_1]$, it is quasi-coherent, and for a homogeneous $f$ of positive degree $\mathrm{\Gamma }(D_{+}(f),{\mathcal{O}}_X(-2))=S(-2{)}_{(f)}$ is the degree-zero part of the homogeneous localisation; for $f=x_0{x}_1$ this module has as $k$-basis the Laurent monomials $x_0^{e_0}{x}_1^{e_1}$ with $e_0+e_1=-2$, and the element $1\mathrm{/}(x_0{x}_1)$ is the member $e=(-1,-1)$.

Ordered Čech complex (Ordered Čech cochain complex of a cover, Fixed-cover Čech cohomology): for an open cover indexed by a linearly ordered set one has $C^p={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$ with the alternating Čech differential; for the two-member cover $U_0,U_1$ this gives $C^0=\mathcal{F}(U_0)\oplus \mathcal{F}(U_1)$, $C^1=\mathcal{F}(U_0\cap U_1)$ and $C^p=0$ for $p\ge 2$, with ${\delta }^0(s_0,s_1)=(s_1-s_0){\mathrm{\mid }}_{U_0\cap U_1}$.

Monomial decomposition of the Čech complex (Laurent-monomial decomposition of the projective Cech complex): for $X={\mathbb{P}}_A^n$ with the ordered standard cover and ${\mathcal{O}}_X(d)$ one has $C^{\bullet }(\mathcal{U},{\mathcal{O}}_X(d))={⨁}_e{K}^{\bullet }(e)$, the sum over $e\in {\mathbb{Z}}^{n+1}$ with ${\sum }_i{e}_i=d$, where $K^p(e)$ is free on basis elements $x_{\sigma }^e$ indexed by the $(p+1)$-element subsets $\sigma \supseteq N(e)$, $N(e)=\{i:e_i<0\}$; if $N(e)=\mathrm{\varnothing }$ then $H^0(K^{\bullet }(e))=A$ and all higher cohomology vanishes, if $N(e)=\{0,\dots ,n\}$ then $H^n(K^{\bullet }(e))=A$ and the other groups vanish, and if $N(e)$ is nonempty and proper then $K^{\bullet }(e)$ is contractible; cohomology of the total complex is the direct sum of the cohomologies of the summands.

Čech comparison (Cech cohomology computes quasi-coherent cohomology on a separated scheme): for a quasi-compact separated scheme $X$ with a finite affine open cover $U_0,\dots ,U_r$ and a quasi-coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, the canonical comparison ${\check{H}}^q(\mathcal{U},\mathcal{F})\to H^q(X,\mathcal{F})$ is an isomorphism for every $q\ge 0$.

Top cohomology of projective twists (Top cohomology of projective twists): for $n\ge 1$ the group $H^n({\mathbb{P}}_A^n,\mathcal{O}(d))$ is the free $A$-module on the Laurent monomials $x_0^{e_0}\cdots x_n^{e_n}$ with all $e_i<0$ and ${\sum }_i{e}_i=d$, and it is zero for $d>-n-1$; for $A=k$ a field, $n=1$ and $d=-2$ there is exactly one such monomial, $x_0^{-1}{x}_1^{-1}$, so $H^1({\mathbb{P}}_k^1,\mathcal{O}(-2))$ is free of rank one over $k$.

The Axiom of Choice (The Axiom of Choice): every family of nonempty sets has a choice function, inherited here from [F6] and [F7].