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I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of characteristic $p$ perfectoid spaces that can be written as a quotient of a perfectoid space by a proetale equivalence relation. What I don't understand is that why can't define diamonds over the whole category of perfectoid spaces (or at least perfectoid space over a base $S$).

Are there some important properties of diamonds that only work over characteristic $p$ or is this because we only need diamond in characteristic $p$ case?

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    $\begingroup$ Defining them in characteristic $p$ gives more flexibility (e.g. the existence of absolute products). All the other options you mentioned can be realized in this framework too. (Hint: The category of all perfectoid spaces is equivalent to the slice $\mathrm{Perf}_{ / \mathrm{Spd}\mathbf{Z}_p}$.) $\endgroup$ Sep 24 '20 at 15:26

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