Is the push-forward of a quasi-coherent sheaf under open immersion still quasi-coherent?

Question

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ is noetherian or if $f: X\rightarrow Y$ is quasi-compact and separated then if $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $f_*\mathcal{F}$ is a quasi-coherent sheaf on $Y$.

Now let's consider an open immersion of schemes $i: U\rightarrow X$. In general $i$ is not quasi-compact. But is it still true that if $\mathcal{F}$ is a quasi-coherent sheaf on $U$, then $f_*\mathcal{F}$ is a quasi-coherent sheaf on $X$?

Answer 1

The answer to your question is no - i.e., if an open immersion $i \colon U \to X$ is not assumed to be quasi-compact, then it can happen that the direct image sheaf $i_* \mathcal{F}$ of a quasi-coherent $\mathcal{O}_U$-module $\mathcal{F}$ is not quasi-coherent. In fact, the answer by Georges Elencwajg to the very question you mentioned in your question contains a link to such a counterexample with $\mathcal{F} = \mathcal{O}_U$. As Georges wrote, it is described (even including a proof) on page 36 of the following paper:

A. Altman, R. Hoobler, S. Kleiman, A note on the base change map for cohomology, Compositio Math. 27 (1973), 25-38

On the other hand, the most general positive result I'm aware of is [Stacks Project, Lemma 25.24.1]:

For quasi-compact and quasi-separated $f$, the direct image sheaf functor $f_*$ maps quasi-coherent modules to quasi-coherent modules.

Note that this is a simulteaneous generalizations of both statements you quoted from Hartshorne.