Proper cohomology need not be finite for noncoherent sheaves

For an open $W\subseteq X$ let $\mathcal{F}(W)$ be the set of locally finite families $(s_r{)}_{r\ge 1}$ with $s_r\in {\mathcal{O}}_X(W)$, meaning that every $x\in W$ has an open neighbourhood $V\subseteq W$ with $s_r{\mathrm{\mid }}_V=0$ for all but finitely many $r$, equipped with componentwise restrictions and componentwise ${\mathcal{O}}_X(W)$-module operations: restrictions of locally finite families are locally finite, compatible families glue componentwise because the components glue in the sheaf ${\mathcal{O}}_X$ and the glued family is locally finite on each member of the cover, and the module axioms are inherited componentwise, so $\mathcal{F}$ is a sheaf of ${\mathcal{O}}_X$-modules as in the statement.

For every prime $\mathfrak{p}\subseteq B$ the summand $B_{\mathfrak{p}}$ is nonzero: $B=k[x_1^{(0)}]$ is a domain, so its localisation $B_{\mathfrak{p}}$ at a prime is a domain with $1\mathrm{\ne }0$; consequently ${⨁}_{r\ge 1}{B}_{\mathfrak{p}}$ is an infinite direct sum of nonzero modules.

The coprojections ${\iota }_r:{\mathcal{O}}_X\to \mathcal{F}$, whose sections are concentrated in the single slot $r$, make this sheaf the direct sum ${⨁}_{r\ge 1}{\mathcal{O}}_X$: for an ${\mathcal{O}}_X$-module $\mathcal{G}$ and morphisms ${\varphi }_r:{\mathcal{O}}_X\to \mathcal{G}$ the prescription ${\mathrm{\Phi }}_W((s_r))={\sum }_r{\varphi }_r{}_{,W}(s_r)$ glues the finite sums ${\sum }_{r\in S}{\varphi }_r{}_{,W^{\mathrm{\prime }}}(s_r{\mathrm{\mid }}_{W^{\mathrm{\prime }}})$ over a cover of $W$ by opens $W^{\mathrm{\prime }}$ on which the family is finitely supported, giving a well-defined ${\mathcal{O}}_X$-linear morphism with $\mathrm{\Phi }\circ {\iota }_r={\varphi }_r$, and it is unique because every section of $\mathcal{F}$ is locally a finite sum of its summands, so a morphism agreeing with $\mathrm{\Phi }$ on all ${\iota }_r$ agrees with it everywhere.

On a quasi-compact open $W\subseteq X$ every locally finite family is finitely supported, since finitely many of the neighbourhoods witnessing local finiteness cover $W$, and a component vanishing on each of them vanishes on $W$; hence for such $W$ the identity is an isomorphism $\mathcal{F}(W)\cong {⨁}_{r\ge 1}{\mathcal{O}}_X(W)$, and for a distinguished open $D(f)\subseteq U_0$ this reads $\mathcal{F}(D(f))={⨁}_{r\ge 1}{B}_f$ with the componentwise localisation maps as restrictions.

Put $N={⨁}_{r\ge 1}B$, so that $N_f={⨁}_{r\ge 1}{B}_f$ canonically for every $f\in B$ by [F8]; by step 2.2 the distinguished-open data and restrictions of $\mathcal{F}{\mathrm{\mid }}_{U_0}$ and of $\tilde{N}$ agree, so the uniqueness of the extension of distinguished-open data [F7] gives an isomorphism $\mathcal{F}{\mathrm{\mid }}_{U_0}\cong \tilde{N}$, and symmetrically $\mathcal{F}{\mathrm{\mid }}_{U_1}$ is an associated sheaf on the affine chart $U_1$; since the affine opens $U_0$ and $U_1$ cover $X$, quasi-coherence of $\mathcal{F}$ follows by its definition [F9].

Fix $x\in U_0$ with corresponding prime $\mathfrak{p}\subseteq B$, so that ${\mathcal{O}}_{X,x}\cong B_{\mathfrak{p}}$ by [F6]; every germ of $\mathcal{F}$ at $x$ is represented on some distinguished open $D(f)$ with $f\mathrm{\notin }\mathfrak{p}$, where the family is finitely supported by step 2.2, and the map $\theta :{\mathcal{F}}_x\to {⨁}_{r\ge 1}{B}_{\mathfrak{p}}$ sending such a germ to the tuple $(s_{r,x}{)}_r$ of germs of its components is well defined, because two representatives agree on a smaller distinguished open and hence componentwise, injective, because a tuple of vanishing germs is annihilated on a common smaller distinguished open, and surjective, because finitely many denominators $f_r\mathrm{\notin }\mathfrak{p}$ can be cleared on the single distinguished open $D(f)$, $f={\prod }_r{f}_r\mathrm{\notin }\mathfrak{p}$, giving a finitely supported family with the prescribed germs; hence ${\mathcal{F}}_x\cong {⨁}_{r\ge 1}{B}_{\mathfrak{p}}$.

By [F10] one has $H^0(X,\mathcal{F})\cong \mathrm{\Gamma }(X,\mathcal{F})=\mathcal{F}(X)$, and $\mathcal{F}(X)={⨁}_{r\ge 1}{\mathcal{O}}_X(X)$: a locally finite family over $X$ restricts to locally finite families over the quasi-compact opens $U_0$ and $U_1$ by [F3], and by step 2.2 only finitely many components are nonzero over $U_0$ and only finitely many over $U_1$, so only finitely many components are nonzero on all of $X$.

An infinite direct sum $M={⨁}_{i\in I}{M}_i$ of nonzero modules is not finitely generated: if $m_1,\dots ,m_n$ generate $M$, each $m_j$ is supported in a finite set $S_j$ by [F4], and for any $i\mathrm{\notin }{S}_1\cup \cdots \cup S_n$, a nonempty complement when $I$ is infinite, the $i$-th component of a combination ${\sum }_j{a}_j{m}_j$ is ${\sum }_j{a}_j(m_j{)}_i=0$, so a nonzero element of $M_i$ does not lie in the generated submodule; with $I=\{r\ge 1\}$ and $M_r=B_{\mathfrak{p}}\mathrm{\ne }0$ this shows that ${\mathcal{F}}_x\cong {⨁}_{r\ge 1}{B}_{\mathfrak{p}}$ is not finitely generated over ${\mathcal{O}}_{X,x}=B_{\mathfrak{p}}$.

The module $\mathcal{F}$ is not of finite type: if it were, [F9] would provide an affine open $V=\mathrm{Spec}A$ containing $x$ and a finitely generated $A$-module $M$ with $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$, and passing to the stalk at the prime of $A$ corresponding to $x$ would give ${\mathcal{F}}_x\cong (\tilde{M}{)}_{{\mathfrak{p}}^{\mathrm{\prime }}}\cong M_{{\mathfrak{p}}^{\mathrm{\prime }}}$ by [F6], a module finitely generated over the local ring $A_{{\mathfrak{p}}^{\mathrm{\prime }}}={\mathcal{O}}_{X,x}$ because the localisations of a finite generating set of $M$ generate $M_{{\mathfrak{p}}^{\mathrm{\prime }}}$, contradicting step 4.1; hence $\mathcal{F}$ is not coherent either, since a coherent module is of finite type by definition [F9].

By [F10] one has ${\mathcal{O}}_X(X)=H^0(X,{\mathcal{O}}_X)\cong k[x_0,x_1{]}_0=k$, so $H^0(X,\mathcal{F})\cong {⨁}_{r\ge 1}k$; since $k\mathrm{\ne }0$ and the index set is infinite, step 4.1 shows that this module is not finitely generated over $k$, that is, $H^0(X,\mathcal{F})$ is infinite-dimensional over $k$, while $\pi :X\to \mathrm{Spec}k$ is proper by [F11]; this refutes the finiteness statement and exhibits the quasi-coherent noncoherent witness.

Boundary and degenerate cases: $k$ is a field, so $k\mathrm{\ne }0$, the scheme $X$ and both charts $U_0,U_1$ are nonempty and the empty and zero-ring bases are excluded; $\mathcal{F}\mathrm{\ne }0$ because each summand has nonzero sections over $U_0$; the cover has the two charts as members and the summand index set $\{r\ge 1\}$ is infinite, which is what step 4.1 uses; the field $k={\mathbb{F}}_2$ and fields of every characteristic are allowed, no Noetherian, separatedness or finiteness hypothesis being used; only the degree $q=0$ of cohomology is computed, the higher cohomology of $\mathcal{F}$ being left unasserted; and the Axiom of Choice is inherited from the cited associated-sheaf, stalk and cohomology suppliers [A1], only finitely many selections (of the elements $f_r$ and of denominators) occurring in step 3.2.