Euler characteristic is additive in short exact sequences

Setup and finiteness. By [F1] and [F2] applied to each of the coherent modules ${\mathcal{F}}^{\mathrm{\prime }}$, $\mathcal{F}$ and ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$, every group appearing below is a finite-dimensional $k$-vector space, the three Euler characteristics $\chi (X,{\mathcal{F}}^{\mathrm{\prime }})$, $\chi (X,\mathcal{F})$ and $\chi (X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ are well-defined integers, and there is an integer $M\ge 0$ with $H^q(X,\mathcal{G})=0$ for every $q>M$ and each $\mathcal{G}\in \{{\mathcal{F}}^{\mathrm{\prime }},\mathcal{F},{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\}$; moreover $H^q(X,\mathcal{G})=0$ for every $q<0$ by [F3]. Consequently each of the three sums defining $\chi$ is a finite sum over the finitely many degrees $q=0,1,\dots ,M$ in which a nonzero group can occur.

The long exact sequence. Applying [F3] to the short exact sequence of the statement gives the long exact sequence $\cdots \to H^{q-1}(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\stackrel{{\delta }_{q-1}}{\to }{H}^q(X,{\mathcal{F}}^{\mathrm{\prime }})\stackrel{{\alpha }_q}{\to }{H}^q(X,\mathcal{F})\stackrel{{\beta }_q}{\to }{H}^q(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\stackrel{{\delta }_q}{\to }{H}^{q+1}(X,{\mathcal{F}}^{\mathrm{\prime }})\to \cdots ,$ exact at every term, with all maps $k$-linear. Exactness gives $\mathrm{im}{\alpha }_q=\ker {\beta }_q,\mathrm{im}{\beta }_q=\ker {\delta }_q,\mathrm{im}{\delta }_{q-1}=\ker {\alpha }_q$ for every $q\in \mathbb{Z}$.

Dimension identities. Put $a_q={\dim }_k{H}^q(X,{\mathcal{F}}^{\mathrm{\prime }})$, $b_q={\dim }_k{H}^q(X,\mathcal{F})$ and $c_q={\dim }_k{H}^q(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$, finite numbers by 1.1. By [F4] applied to the $k$-linear maps ${\beta }_q$, ${\alpha }_q$ and ${\delta }_q$, using the exactness identifications of 1.2, $b_q={\dim }_k\mathrm{im}{\alpha }_q+{\dim }_k\mathrm{im}{\beta }_q,a_q={\dim }_k\mathrm{im}{\delta }_{q-1}+{\dim }_k\mathrm{im}{\alpha }_q,c_q={\dim }_k\mathrm{im}{\beta }_q+{\dim }_k\mathrm{im}{\delta }_q\mathrm{.}$ Indeed, for ${\beta }_q$ the kernel is $\mathrm{im}{\alpha }_q$; for ${\alpha }_q$ the kernel is $\mathrm{im}{\delta }_{q-1}$; for ${\delta }_q$ the kernel is $\mathrm{im}{\beta }_q$.

Alternating sum. Multiplying the three identities of 1.3 by $(-1{)}^q$ and summing over $q\in \mathbb{Z}$, a finite sum by 1.1, and using the linearity of the sum, ${\sum }_q(-1{)}^q{b}_q={\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\alpha }_q+{\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\beta }_q,$ while ${\sum }_q(-1{)}^q{a}_q+{\sum }_q(-1{)}^q{c}_q={\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\alpha }_q+{\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\beta }_q+{\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\delta }_{q-1}+{\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\delta }_q\mathrm{.}$ The two connecting-map sums cancel: the index shift $q\mapsto q-1$ gives ${\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\delta }_{q-1}=-{\sum }_q(-1{)}^q{\dim }_k\mathrm{im}{\delta }_q$, the $q=0$ term of the first sum being ${\dim }_k\mathrm{im}{\delta }_{-1}=0$ because $H^{-1}=0$ by 1.1 and the finite range $0,\dots ,M$ makes the shift legitimate. Hence ${\sum }_q(-1{)}^q{a}_q+{\sum }_q(-1{)}^q{c}_q={\sum }_q(-1{)}^q{b}_q$.

Conclusion. Restricting the sums of 1.4 to the degrees $q=0,\dots ,M$ on which the groups can be nonzero and replacing the dimension sums by the Euler characteristics through [F1] gives $\chi (X,\mathcal{F})={\sum }_{q\ge 0}(-1{)}^q{b}_q={\sum }_{q\ge 0}(-1{)}^q{a}_q+{\sum }_{q\ge 0}(-1{)}^q{c}_q=\chi (X,{\mathcal{F}}^{\mathrm{\prime }})+\chi (X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}),$ which is the asserted additivity.

Boundaries and choice. If $X=\mathrm{\varnothing }$ then every ${\mathcal{O}}_X$-module is zero, so ${\mathcal{F}}^{\mathrm{\prime }}=\mathcal{F}={\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}=0$, all cohomology vanishes and the identity reads $0=0+0$ by [F1]. If $\mathcal{F}=0$ then $\beta$ is the zero map and ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}=\mathrm{im}\beta =0$, while $\alpha$ is injective with image $\ker \beta =0$, so ${\mathcal{F}}^{\mathrm{\prime }}=0$, and again the identity reduces to $0=0+0$; the same argument applies when any one of the three terms is zero, the sequence then exhibiting an isomorphism between the other two and reducing the identity to the tautology $\chi (X,\mathcal{G})=\chi (X,\mathcal{G})+0$. Since $k$ is a field it is not the zero ring, and the trivial short exact sequence $0\to \mathcal{F}\to \mathcal{F}\to 0\to 0$ is included with $\chi (X,\mathcal{F})=\chi (X,\mathcal{F})+0$. The endpoint degrees are covered by 1.1 and 1.4: the sums are finite, the degree $q=0$ is included, and the terms with $q<0$ and $q>M$ vanish on both sides. The Axiom of Choice [F5] is inherited through the finiteness corollary of [F2], which uses it to produce the finite-dimensionality and the vanishing bound, and through the long exact sequence of [F3], which uses it to supply the injective resolutions defining sheaf cohomology and the connecting maps; no further selection is made, the data of the statement and the integer $M$ being fixed.