# Quasicoherent analogue of a theorem on fiberwise acyclicity for etale cohomology

I am interested in knowing what (if any) is the quasicoherent analogue of the following result that I have paraphrased from SGA 4, exposé xv, Théorème 1.15:

Let $g \colon X \to Y$ be quasicompact, quasiseparated, and locally acyclic for $\ell$-torsion sheaves on $Y$. Let $\mathcal{F}$ be an $\ell$-torsion sheaf on $Y$, and consider the "unit" map for $g$, $H^i_\text{et}(Y, \mathcal{F}) \to H^i_\text{et}(X, g^* \mathcal{F})$. If, for every algebraic geometric point $y$ of $Y$ with $g$-fiber $g_y \colon X_y \to y$, the unit map for $g_y$ (applied to $\mathcal{F}|_y$) is an isomorphism, then so is that for $g$, and conversely.

I believe it has the following more abstract reformulation:

With $g$ as before, let $\mathcal{F}$ denote an object of the derived category of $\ell$-torsion sheaves on $Y$, and consider the unit map $\mathcal{F} \to Rg_* g^* \mathcal{F}$. If for every algebraic geometric point $y$ of $Y$, the corresponding unit map is an isomorphism, $y^* \mathcal{F} \xrightarrow{\sim} Rg_{y{*}} g_y^* y^* \mathcal{F}$, then the global unit map is also an isomorphism, and conversely.

(Someone tell me if I'm wrong about that.) I have used plain $g_*$ and $g^*$ in place of $R g_*$ and $L g^*$.

I would like to know if the same statement is true when $g$ is replaced by a flat map of locally noetherian schemes and $\mathcal{F}$ denotes a complex of quasicoherent sheaves. I have looked all over the standard references but not found one that proves specifically this theorem, nor even really discusses it. Is it obviously false, or perhaps obviously true? For some reason, the subject of acyclicity and local acyclicity is never mentioned in the coherent setting.

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