Coherence is essential for proper finiteness

Assume the Axiom of Choice (The Axiom of Choice), used here through the universal-closedness of projective space and the associated-sheaf construction.

The finiteness theorem Coherent higher direct images under proper morphisms is a coherence assertion, and coherence cannot be weakened to quasi-coherence: the conclusion fails already for the projective line over a field. Let $k$ be a field and let $X={\mathbb{P}}_k^1$ with structure morphism $\pi :X\to \mathrm{Spec}k$. The morphism $\pi$ is proper: it is separated (The relative projective-space diagonal is closed), of finite type (Projective space is of finite type over its base) and universally closed (Projective-space projection is universally closed by finite graded pieces), and these three properties are what properness means (Proper morphisms). Let $\mathcal{F}={⨁}_{m\ge 1}{\mathcal{O}}_X$ be the direct sum of countably many copies of the structure sheaf, formed in the category of ${\mathcal{O}}_X$-modules (Modules on a ringed space): over a quasi-compact open these are the finite-support families of sections of ${\mathcal{O}}_X$, over a general open the locally finite such families, and the coprojections into the slots exhibit the sheaf as the direct sum (The direct sum of an indexed family of modules).

Then $\mathcal{F}$ is quasi-coherent (Quasi-coherent module on a scheme). Indeed, on the standard chart $U_0=\mathrm{Spec}k[x_1]$ the restriction $\mathcal{F}{\mathrm{\mid }}_{U_0}$ is the direct sum of countably many copies of the structure sheaf of $U_0$, and direct sums of modules commute with localisation by Localisation commutes with quotient modules and arbitrary direct sums, so on the distinguished-open basis of $U_0$ this restriction is described by $({⨁}_{m\ge 1}k[x_1]{)}_f\cong {⨁}_{m\ge 1}k[x_1{]}_f$, which is the associated sheaf of ${⨁}_{m\ge 1}k[x_1]$ (The associated module sheaf exists); the same holds on the second chart $U_1$ with $k[x_1^{-1}]$ in place of $k[x_1]$, and the two charts cover $X$.

The sheaf $\mathcal{F}$ is not coherent (Coherent module sheaves), indeed not even of finite type (Finite type and finitely presented module sheaves): its sections over the affine chart $U_0$ are $\mathcal{F}(U_0)\cong {⨁}_{m\ge 1}k[x_1]$, and the unit elements of the successive summands form an infinite family that is linearly independent over $k[x_1]$, so no finite set of sections generates.

Its degree-zero cohomology is $H^0(X,\mathcal{F})\cong \mathrm{\Gamma }(X,\mathcal{F})\cong {⨁}_{m\ge 1}k,$ computed on the two-chart cover (Degree-zero sheaf cohomology is global sections): the sheaf axiom identifies global sections with the pairs of finite-support families $(s_0,s_1)$, with $s_0$ over $U_0$, $s_1$ over $U_1$ and equal images in ${⨁}_{m\ge 1}k[x_1,x_1^{-1}]$ on the overlap; agreement in a direct sum is componentwise, and a polynomial in $x_1$ that equals a polynomial in $x_1^{-1}$ is constant, so each summand contributes one copy of $k$, identified with the constants $\mathrm{\Gamma }({\mathbb{P}}_k^1,{\mathcal{O}}_X)=H^0({\mathbb{P}}_k^1,{\mathcal{O}}_X)\cong k$ (Global sections of projective twists). This module is not finitely generated over $k$, so $H^0(X,\mathcal{F})$ is infinite-dimensional even though $\pi$ is proper and $\mathcal{F}$ is quasi-coherent: the coherence hypothesis on the coefficient sheaf in Coherent higher direct images under proper morphisms is essential and not a technical convenience, and the failure is caused by the coefficient sheaf alone, the morphism being as good as a proper morphism over a field can be.

The companion examples page develops this witness in full detail, including the locally finite family model of the direct sum, the computation of its stalks and the verification that it is not of finite type.