Euler characteristic of line bundles on a projective line over a finite field extension

Fix a $\kappa$-isomorphism $\phi \text{\hspace{0.17em}⁣}:E\to {\mathbb{P}}_{\kappa }^1$ and set $N:={\phi }_{\ast }M$, so that $N$ is an invertible sheaf on ${\mathbb{P}}_{\kappa }^1$ and $\mathrm{\Gamma }(E,M)=\mathrm{\Gamma }({\mathbb{P}}_{\kappa }^1,N)$ by definition of the direct image. The direct image along an isomorphism is an exact equivalence of module categories with inverse ${\phi }^{\ast }$, so it preserves the global-section functors and their right derived functors; hence $\phi$ induces $\kappa$-linear isomorphisms $H^q(E,M)\cong H^q({\mathbb{P}}_{\kappa }^1,N)$ for all $q\ge 0$.

By [F2] applied to the field $\kappa$, the invertible sheaf $N$ on ${\mathbb{P}}_{\kappa }^1$ is isomorphic to ${\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(d)$ for a unique integer $d$, which we take as the definition of the degree $d$ of $M$ over $\kappa$.

By [F3] with $A=\kappa$ and $n=1$ the groups $H^q({\mathbb{P}}_{\kappa }^1,\mathcal{O}(d))$ vanish unless $q=0$ or $q=1$; the same description gives $H^0\cong \kappa [x,y{]}_d$, of dimension $h^0=d+1$ for $d\ge 0$ and $0$ for $d<0$, and $H^1$ free on the Laurent monomials $x^e{y}^f$ with $e,f<0$ and $e+f=d$, of dimension $h^1=-d-1$ for $d\le -2$ and $0$ otherwise, so $h^0-h^1=1+d$. Since cohomology depends only on the isomorphism class of the sheaf, [F1] gives ${\dim }_{\kappa }{H}^q({\mathbb{P}}_{\kappa }^1,N)=h^q$; combined with step 1.1 this yields $H^q(E,M)=0$ for $q\ge 2$ and ${\dim }_{\kappa }{H}^0(E,M)-{\dim }_{\kappa }{H}^1(E,M)=1+d$.

The structure morphism $E\to \mathrm{Spec}\kappa$ makes $\kappa \to \mathrm{\Gamma }(E,{\mathcal{O}}_E)$ a ring homomorphism, so each $H^q(E,M)$ is a $\kappa$-vector space; for a $\kappa$-vector space $V$ of dimension $h$ one has ${\dim }_{k}V=rh$, because if $v_1,\dots ,v_h$ is a $\kappa$-basis of $V$ and ${\mu }_1,\dots ,{\mu }_r$ is a $k$-basis of $\kappa$, then the products ${\mu }_j{v}_i$ span $V$ over $k$ and are $k$-independent. Applying this to the groups of step 3.1 gives ${\dim }_k{H}^0(E,M)-{\dim }_k{H}^1(E,M)=r(h^0-h^1)=r(1+d)$.

The scheme $E$ is proper over $\kappa$ because it is $\kappa$-isomorphic to ${\mathbb{P}}_{\kappa }^1$ and ${\mathbb{P}}_S^1\to S$ is proper for every $S$ [F5], and $M$ is coherent on the locally Noetherian scheme $E$ because it is invertible [F6]; thus the Euler characteristic of [F4], with base field $\kappa$, is the alternating sum over the finitely many nonzero cohomology groups. By step 3.1 only the terms $q=0,1$ occur, and passing to $k$-dimensions as in step 4.1 gives the $k$-Euler characteristic ${\chi }_k(E,M)={\dim }_k{H}^0(E,M)-{\dim }_k{H}^1(E,M)=r(1+d)$.

If ${\phi }^{\mathrm{\prime }}$ is another $\kappa$-isomorphism with associated integer $d^{\mathrm{\prime }}$, then step 5.1 applied to both gives $r(1+d)={\chi }_k(E,M)=r(1+d^{\mathrm{\prime }})$, and $r\ge 1$ because $\kappa \mathrm{/}k$ is a finite extension, so $d=d^{\mathrm{\prime }}$; thus the degree of $M$ over $\kappa$ is well defined. For $d=-j$ the formula reads ${\chi }_k(E,M)=r(1-j)$, which is the final assertion.