I have a simple question about the conservativity of stalks of ind-constructible sheaves. Let $X$ be a topologically noetherian scheme, $S$ a set of geometric points of $X$ corresponding bijectively to the scheme-theoretic points of $X$, $\Lambda$ an algebraic extension of $\mathbf{Q}_\ell$ or the ring of integers of such, $I$ a filtered category (which I will assume is a directed poset), $\operatorname{Cons}_X(\Lambda)$ the abelian category of constructible $\Lambda$-sheaves on $X_{\text{pro-étale}}$ in the sense of Bhatt-Scholze §6.8, and let $(\mathcal F_i)_{i\in I}$ correspond to a functor $I\to \operatorname{Cons}_X(\Lambda)$ with colimit $\mathcal F$ computed in $\operatorname{Mod}(X_{\text{pro-étale}},\Lambda)$, the category of sheaves of $\Lambda$-modules, where $\Lambda$ here and often below also denotes the sheaf $\Lambda_X$ associated to the topological ring $\Lambda$.
My question is, if $\mathcal F_x=0$ for every $x\in S$, is $\mathcal F=0$? (Here $\mathcal F_x:=\Gamma(x_{\text{pro-étale}},x^*\mathcal F$.)
I believe I can prove this is true (argument below), but what is giving me some doubts is Warning 2.3.4.10 of the paper Weil’s conjecture for function fields [PDF], which says that there is a non-zero ind-constructible sheaf which vanishes at all closed points. This is relevant since Corollary 3.51 of the recent paper Constructible sheaves on schemes and a categorical Künneth formula connects $\operatorname{Ind}(D_{\mathrm{cons}}(X,\Lambda))$ to filtered colimits of pro-étale sheaves. Admittedly, this warning only deals with closed points, but so far I haven’t been able to construct a non-zero ind-constructible sheaf that vanishes at all closed points. Here is my purported (simple) ‘proof’ that the answer to my question is ‘yes’:
Proof It would suffice to show that for every $i\in I$, there is a $j\geq i$ so that $\varphi_{ij}:\mathcal F_i\to\mathcal F_j$ is zero. Suppose we could find a nonempty open $U\subset X$ over which this is true. Then we could replace $i$ by $j$ and $X$ by $\operatorname{supp}\operatorname{coker}(\ker\varphi_{ij}\to\mathcal F_i)$, and we'd be done by noetherian induction.
To find $U$: let $\eta$ be a geometric generic point of $X$. We can find a $j\geq i$ so that $\varphi_{ij}:\mathcal F_i\to\mathcal F_j$ induces zero on stalks at $\eta.^\dagger$ There is some connected neighborhood $U$ of $\eta$ so that both $\mathcal F_i$ and $\mathcal F_j$ are locally constant of finite presentation over $U$. Then the same is true of $\ker\varphi_{ij}$, and as $(\ker\varphi_{ij})_\eta=(\mathcal F_i)_\eta$, $\ker\varphi_{ij}|_U=\mathcal F_i|_U.^\ddagger$ $\square$
$\dagger$: Both $\eta^*$ and $\Gamma(\eta_\text{pro-étale},-)$ commute with colimits, the latter since $\eta$ is connected and w-contractible. For each $i$, there is some neighborhood $U_i$ of $\eta$ restricted to which $\mathcal F_i$ is (pro-étale) locally of the form $M_i\otimes_\Lambda\Lambda_X$, where $M$ is a $\Lambda$-module of finite presentation and I've dropped the underlines for constant sheaves ($\Lambda_X$ is not literally a constant sheaf). (This is what I mean by ‘locally constant of finite presentation.’) Note $(M_i\otimes_\Lambda\Lambda_X)_\eta=M_i$ as $\Lambda_X(\eta)=\Lambda$ and $\eta$ is w-contractible.
$\ddagger$: When $\Lambda$ is a field, this follows from Corollary 6.8.5 of [BS] and that $(-)_x$ is an exact functor (of abelian categories). Otherwise, modulo possibly shrinking $U$, replace Corollary 6.8.5 with Proposition 6.8.11.
It remains unclear to me how this doesn’t contradict Warning 2.3.4.10. Thanks for any clarification of the matter.
