Depending on a geometer's needs, they may use the Zariski/etale/syntomic/etc. topology on the spaces they consider. I know some settings where etale topology is better suited for the task than the Zariski topology and where fppf or fpqc topology is better than the etale topology. However, I do not know a situation where fppf topology is better than fpqc or vice versa. Do such situations happen in algebraic geometry?
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9$\begingroup$ Well, fppf is nice if you need to sheafify or stackify anything, since you know this always exists (not so for fpqc). But I guess there are objects that really do satisfy fpqc descent 'natively', and this may be useful; I'll let an algebraic geometer answer that one. $\endgroup$– David Roberts ♦Commented Nov 10, 2019 at 23:42
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2$\begingroup$ Usually fppf suffices, and it's easier to work with - sheafifications exist, fppf maps are open, etc. However, schemes and algebraic spaces do satisfy fpqc descent (the latter is a non-trivial theorem of Gabber). One place I've seen the fpqc topology appear is when working in the category of perfect schemes, where maps are rarely finitely presented. Then again, it seems that in this setting, more exotic topologies like the v-topology or arc-topology might be preferable. (See arxiv.org/abs/1407.8519, arxiv.org/abs/1507.06490, arxiv.org/pdf/1807.04725.pdf) $\endgroup$– dorebellCommented Nov 12, 2019 at 2:15
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