Degree is additive on invertible sheaves over a proper curve

Set-up and defect. For a coherent ${\mathcal{O}}_C$-module $\mathcal{F}$ the four tensor products appearing in $D(\mathcal{F})$ are coherent by [F2], so all four Euler characteristics are defined; by [F2] $D({\mathcal{O}}_C)=\chi (C,{\mathcal{O}}_C)-\chi (C,\mathcal{L})-\chi (C,\mathcal{M})+\chi (C,\mathcal{L}\otimes \mathcal{M})$, and it suffices for statements 1 and 3 to prove $D(\mathcal{F})=0$ for the single coherent module $\mathcal{F}={\mathcal{O}}_C$.

Additivity of the defect. Let $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of coherent ${\mathcal{O}}_C$-modules. Tensoring with $\mathcal{L}$, with $\mathcal{M}$ and with $\mathcal{L}\otimes \mathcal{M}$ preserves exactness by [F2], so all four sequences appearing in $D$ are short exact with coherent terms, and [F3] gives $\chi (C,\mathcal{G})=\chi (C,{\mathcal{G}}^{\mathrm{\prime }})+\chi (C,{\mathcal{G}}^{\mathrm{\prime }\mathrm{\prime }})$ for $\mathcal{G}\in \{\mathcal{F},\mathcal{L}\otimes \mathcal{F},\mathcal{M}\otimes \mathcal{F},\mathcal{L}\otimes \mathcal{M}\otimes \mathcal{F}\}$. Adding the untwisted and doubly twisted identities and subtracting the two singly twisted identities yields $D(\mathcal{F})=D({\mathcal{F}}^{\mathrm{\prime }})+D({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$; in particular $D$ has the two-of-three property, and $D(0)=0$.

Integral closed subschemes. By [F5] an integral closed subscheme $Z\subseteq C$ is either a closed point or of dimension one; in both cases the witness $\mathcal{G}:={\mathcal{O}}_Z$ pushed forward along $i$ is coherent on $C$, has support contained in $Z$, generic stalk $\kappa ({\xi }_Z)$ annihilated by the maximal ideal of ${\mathcal{O}}_{C,{\xi }_Z}$, of dimension one over $\kappa ({\xi }_Z)$.

The closed-point case. If $Z=\{p\}$ is a closed point, then $i_{\ast }{\mathcal{O}}_Z=i_{\ast }\kappa (p)$ and [F7] gives $D(i_{\ast }\kappa (p))=\chi (C,i_{\ast }\kappa (p))-\chi (C,i_{\ast }\kappa (p))-\chi (C,i_{\ast }\kappa (p))+\chi (C,i_{\ast }\kappa (p))=0$, because each of the twisted modules $\mathcal{L}\otimes i_{\ast }\kappa (p)$, $\mathcal{M}\otimes i_{\ast }\kappa (p)$ and $\mathcal{L}\otimes \mathcal{M}\otimes i_{\ast }\kappa (p)$ is isomorphic to $i_{\ast }\kappa (p)$.

The one-dimensional case. If $Z$ has dimension one, then $Z$ is an integral proper curve of dimension one over $k$, and by the projection formula of [F6] applied to the invertible modules ${\mathcal{O}}_C$, $\mathcal{L}$, $\mathcal{M}$, $\mathcal{L}\otimes \mathcal{M}$ and to $\mathcal{G}={\mathcal{O}}_Z$, $D(i_{\ast }{\mathcal{O}}_Z)=\chi (Z,{\mathcal{O}}_Z)-\chi (Z,i^{\ast }\mathcal{L})-\chi (Z,i^{\ast }\mathcal{M})+\chi (Z,i^{\ast }\mathcal{L}\otimes i^{\ast }\mathcal{M})\mathrm{.}$ The modules $i^{\ast }\mathcal{L}$ and $i^{\ast }\mathcal{M}$ are invertible on $Z$, and $i^{\ast }\mathcal{L}\otimes i^{\ast }\mathcal{M}\cong i^{\ast }(\mathcal{L}\otimes \mathcal{M})$ has generic rank one; the integral-curve twist identity of [F8] applied on $Z$ with invertible $\mathcal{F}=i^{\ast }\mathcal{M}$ gives $\chi (Z,i^{\ast }\mathcal{L}\otimes i^{\ast }\mathcal{M})={\mathrm{deg}}_Z(i^{\ast }\mathcal{L})+\chi (Z,i^{\ast }\mathcal{M})$, and ${\mathrm{deg}}_Z(i^{\ast }\mathcal{L})=\chi (Z,i^{\ast }\mathcal{L})-\chi (Z,{\mathcal{O}}_Z)$ by definition of the degree. Substituting, the right-hand side vanishes.

Devissage and the additive identity. By steps 1.2, 2.1 and 2.2 the defect $D$ holds for the zero module, satisfies the two-of-three property, and vanishes on the witness $i_{\ast }{\mathcal{O}}_Z$ of every integral closed subscheme $Z\subseteq C$; the devissage lemma [F4] therefore gives $D(\mathcal{F})=0$ for every coherent ${\mathcal{O}}_C$-module $\mathcal{F}$. Applying this to $\mathcal{F}={\mathcal{O}}_C$ and using $D({\mathcal{O}}_C)=\chi (C,{\mathcal{O}}_C)-\chi (C,\mathcal{L})-\chi (C,\mathcal{M})+\chi (C,\mathcal{L}\otimes \mathcal{M})$ yields statement 3, and rearranging gives $\chi (C,\mathcal{L}\otimes \mathcal{M})-\chi (C,{\mathcal{O}}_C)=(\chi (C,\mathcal{L})-\chi (C,{\mathcal{O}}_C))+(\chi (C,\mathcal{M})-\chi (C,{\mathcal{O}}_C))$, which is statement 1 by the definition of ${\mathrm{deg}}_C$.

Statement 2. Applying statement 1 of step 3.1 to the pair $(\mathcal{L},{\mathcal{L}}^{\vee })$ and using $\mathcal{L}\otimes {\mathcal{L}}^{\vee }\cong {\mathcal{O}}_C$ and ${\mathrm{deg}}_C({\mathcal{O}}_C)=0$ of [F9] gives $0={\mathrm{deg}}_C({\mathcal{O}}_C)={\mathrm{deg}}_C(\mathcal{L}\otimes {\mathcal{L}}^{\vee })={\mathrm{deg}}_C(\mathcal{L})+{\mathrm{deg}}_C({\mathcal{L}}^{\vee })$, hence ${\mathrm{deg}}_C({\mathcal{L}}^{\vee })=-{\mathrm{deg}}_C(\mathcal{L})$.

Conclusion and choice accounting. Step 3.1 gives statements 1 and 3 and step 4.1 gives statement 2. The Axiom of Choice enters only through the suppliers recorded in [F10]: the Euler-characteristic additivity [F3], the devissage lemma [F4], the projection formula and closed-immersion cohomology [F6], the closed-point twist invariance [F7] and the integral-curve twist identity [F8]; the curve, the sheaves and the point arguments are given, and no selection is made in steps 1.1–3.1.