A non-quasi-coherent module with H1 on an affine scheme

The sheaf $\mathcal{F}=j_{!}{\mathcal{O}}_U$ carries the structure of an ${\mathcal{O}}_X$-module. Indeed ${\mathcal{O}}_U$ is the restriction ${\mathcal{O}}_X{\mathrm{\mid }}_U$ (Extension by zero for abelian sheaves on an open subspace, Modules on a ringed space), and on an open $V\subseteq X$ the group $\mathcal{F}(V)\subseteq {\mathcal{O}}_X(V\cap U)$ consists of the sections whose support is closed in $V$; multiplying such a section by the restriction of a section $a\in {\mathcal{O}}_X(V)$ preserves the support condition, and the restriction maps of $\mathcal{F}$ are those of ${\mathcal{O}}_X$, so the presheaf-level multiplication makes $\mathcal{F}$ a sheaf of ${\mathcal{O}}_X$-modules by [F6]. The inclusion of $\mathcal{F}$ into ${\mathcal{O}}_X$ and the quotient map to $i_{\ast }({\mathcal{O}}_X{\mathrm{\mid }}_Z)$ are ${\mathcal{O}}_X$-linear.

The closed subset $Z=\{(t),(t-1)\}$ is discrete: the two points are the maximal ideals $(t)$ and $(t-1)$, their defining closed sets $V(t)$ and $V(t-1)$ are disjoint because $t$ and $t-1$ generate the unit ideal of $k[t]$, and $Z=V(t(t-1))$ is closed. Hence $\mathrm{\Gamma }(X,i_{\ast }({\mathcal{O}}_X{\mathrm{\mid }}_Z))={\mathcal{O}}_{X,(t)}\times {\mathcal{O}}_{X,(t-1)}$, the product of the two stalks at the points of $Z$, and by [F5] this is $A_{(t)}\times A_{(t-1)}$. The structure sheaf is quasi-coherent with $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=A$, so $H^1(X,{\mathcal{O}}_X)=0$ and $H^0(X,{\mathcal{O}}_X)=A$ by [F4].

Applying [F2] to the abelian sheaf $\mathcal{G}={\mathcal{O}}_X$ and the open inclusion $j:U\hookrightarrow X$ with closed complement $Z$ gives the short exact sequence of abelian sheaves $0\to \mathcal{F}\to {\mathcal{O}}_X\to i_{\ast }({\mathcal{O}}_X{\mathrm{\mid }}_Z)\to 0$.

The long exact cohomology sequence of step 1.3 begins $0\to H^0(X,\mathcal{F})\to H^0(X,{\mathcal{O}}_X)\to H^0(X,i_{\ast }({\mathcal{O}}_X{\mathrm{\mid }}_Z))\to H^1(X,\mathcal{F})\to H^1(X,{\mathcal{O}}_X)$. Substituting the identifications of step 1.2 and $H^1(X,{\mathcal{O}}_X)=0$, this reads $0\to H^0(X,\mathcal{F})\to A\stackrel{\mathrm{\Delta }}{\to }{A}_{(t)}\times A_{(t-1)}\to H^1(X,\mathcal{F})\to 0$, where $\mathrm{\Delta }(f)=(f\mathrm{/}1,f\mathrm{/}1)$ is the diagonal embedding; exactness at the last two terms gives $H^1(X,\mathcal{F})\cong (A_{(t)}\times A_{(t-1)})\mathrm{/}\mathrm{\Delta }(A)$.

The quotient of step 2.1 is nonzero: the class of the element $(0,1)$ is not in the image of $\mathrm{\Delta }$, since $\mathrm{\Delta }(f)=(0,1)$ would force $f=0$ in $A_{(t)}$ (as $A\to A_{(t)}$ is injective, $A$ being a domain) and simultaneously $f\mathrm{/}1=1$ in $A_{(t-1)}$, a contradiction. Hence $H^1(X,\mathcal{F})\mathrm{\ne }0$.

Finally, $\mathcal{F}$ is not quasi-coherent. If it were, then since $X=\mathrm{Spec}A$ is affine the AC-qualified affine vanishing theorem [F4], licensed by [F7], would give $H^1(X,\mathcal{F})=0$, contradicting step 3.1. Thus the displayed ${\mathcal{O}}_X$-module on the affine scheme $X$ has nonvanishing $H^1$, so the quasi-coherence hypothesis of affine vanishing cannot be dropped; the module is nonzero because $U\mathrm{\ne }\mathrm{\varnothing }$ and $(j_{!}{\mathcal{O}}_U)(U)={\mathcal{O}}_X(U)\mathrm{\ne }0$ by [F1]. The Axiom of Choice is inherited from [F3] and [F4], and the only selections made are the two closed points already named.