Euler characteristic of a coherent sheaf

Assume the Axiom of Choice, inherited from the finiteness and vanishing corollary cited below (The Axiom of Choice). Let $k$ be a field (Field), let $X$ be a scheme proper over $k$ (Proper morphisms), and let $\mathcal{F}$ be a coherent ${\mathcal{O}}_X$-module (Coherent module sheaves) with sheaf cohomology groups $H^q(X,\mathcal{F})$ (Sheaf cohomology as right derived global sections).

By Finite-dimensional coherent cohomology over a field each group $H^q(X,\mathcal{F})$ is a finite-dimensional $k$-vector space (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) and only finitely many of the groups are nonzero: if $X$ admits a finite affine open cover with $n\ge 0$ members, then $H^q(X,\mathcal{F})=0$ for every $q\ge n$. Consequently the alternating sum $\chi (X,\mathcal{F}):={\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F})$ has only finitely many nonzero terms and defines an integer, the Euler characteristic of $\mathcal{F}$ on $X$. Equivalently, for any finite affine open cover of $X$ with $n$ members one has $\chi (X,\mathcal{F})={\sum }_{q=0}^{n-1}(-1{)}^q{\dim }_k{H}^q(X,\mathcal{F}),$ the value being independent of the cover because the definition uses only the cohomology groups.

If $X=\mathrm{\varnothing }$ then all groups $H^q(X,\mathcal{F})$ vanish, the cover has $n=0$ members, and we set $\chi (X,\mathcal{F})=0$; this is the empty sum in the displayed formula. The zero sheaf $\mathcal{F}=0$ likewise has $\chi (X,0)=0$.