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In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5):

-- any ind-etale map is weakly etale,

-- weakly etale maps satisfy fpqc-descent.

Moreover, it is stated (right after the proof of Prop.2.3.3) that the second claim fails for ind-etale maps. Do someone know

  1. An example of a weakly etale map (of affine schemes), which is not ind-etale, resp.

  2. An example of an ind-etale map (of affine schemes), which violates fpqc-descent?

Note that such examples must be of non-finite presentation, as weakly etale + of finite presentation is equivalent to etale, and etale is fpqc-local on the target.

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    $\begingroup$ (1) is answered in Example 4.1.12 (due to de Jong) in the linked paper. This is an example of a weakly etale map that is Zariski locally (on the base) ind-etale but not ind-etale itself. I didn't understand (2) but perhaps the same example helps there? $\endgroup$
    – Anonymous
    Commented Oct 21, 2021 at 13:23
  • $\begingroup$ Thank you! This answers (1). But, as far as I can see, not (2). By (2), I meant an example of an fpqc-cover $A \rightarrow A'$ and a non-(ind-etale) --but necessarily weakly etale-- map $A \rightarrow B$, such that $A' \rightarrow B\otimes_A A'$ is ind-etale. $\endgroup$
    – AlexIvanov
    Commented Oct 21, 2021 at 13:34
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    $\begingroup$ Unless I'm mistaken, the example actually gives such an example with A ---> A' being a Zariski cover. $\endgroup$
    – Anonymous
    Commented Oct 21, 2021 at 13:54

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