# Extension by zero and quasi-coherence

The following makes probably sense for any site, but I stick for concreteness to the etale one.

Let $f: U \to X$ be an etale morphism. As explained, for example, in Remark 8.16 of Milne's Lecture Notes on Etale Cohomology, there is a left adjoint $f_!$ to $f^*$ between the categories of (etale) module sheaves. For $\mathcal{F}$ a module sheaf on $U$, this can be defined as the sheafification of the presheaf $\mathcal{G}$, defined as follows: For $\phi: V \to X$ etale, define $\mathcal{G}(V) = \bigoplus_\alpha\mathcal{F}(V, \alpha)$, where $\alpha$ ranges over all (etale) maps $\alpha: V \to U$ such that $f\alpha = \phi$.

For $U\subset X$ an open subset, this coincides exactly with the extension by zero. Even in this case, $f_!$ does not need to preserve quasi-coherence: For example, let $X = Spec \mathbb{Z}_{(2)}$ and $f: Spec \mathbb{Q} \hookrightarrow X$ the inclusion of the open point. If we consider as $\mathcal{F}$ the module sheaf associated to $\mathbb{Q}$, then $\mathcal{G}(X)$ is zero, but $\mathcal{G}(Spec \mathbb{Q}) \cong \mathbb{Q}$. This cannot be quasi-coherent and no sheafification goes on.

On the other hand, for $f$ a Galois covering, $f_!$ agrees with $f_*$ and therefore preserves quasi-coherence.

My question is the following: What are more general conditions for $f_!$ preserving quasi-coherence?

• I have to teach in a few minutes, so I might be overlooking something, but $f_!$ should coincide with $f_*$ when $f$ is proper. So this case should be OK. – Donu Arapura Feb 7 '12 at 16:31
• If f^* has a left adjoint, then f^* is left exact. I.e., f is flat. – Moosbrugger Feb 7 '12 at 23:14
• @Donu: Do you have a reference for this? Ideally one that also works for algebraic stacks? – Lennart Meier Feb 8 '12 at 14:48