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In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.

The most virtuosic pages in Scholze's papers generally involve finding ways to reduce constructions that appear to be hopelessly infinite to comprehensible (finite type) ring theory.

I'm inquiring as to whether or not someone can elaborate on this sentence and give some more details. Thanks.

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  • $\begingroup$ I know very little but Scholze answers the question What are perfectoid spaces? $\endgroup$ Aug 13, 2017 at 1:28
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    $\begingroup$ The sentiment of that sentence seems to be backwards since really one is doing the exact opposite: reducing problems hopeless to attack at "finite level" to tasks that can (with hard work and fearlessness) be solved in a suitable way at "infinite level". It is the whole point of perfectoid spaces, after all: one can do analytic geometry up there! Conclusions can then often be passed back down to finite level by approximation procedures that require effort but are often not the "most virtuosic" part (IMHO; study survey papers which discuss some actual proofs to decide for yourself). $\endgroup$
    – nfdc23
    Aug 13, 2017 at 3:35
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    $\begingroup$ Here is an analogy to convey why reducing tasks with "huge" structures to those of a "finitistic" nature can be very useful but not where the main difficulties arise (once one knows the approximation methods!). EGA IV$_3$ exhaustively develops methods to reduce "non-noetherian" tasks to the noetherian case. Anyone who knows that material appreciates its non-triviality and utility, even for proofs about noetherian schemes (!). But in the end, it is just a tool; the most virtuosic pages in EGA are elsewhere (coherence of higher direct images, formal GAGA, excellence, Zariski's Main Theorem,...). $\endgroup$
    – nfdc23
    Aug 13, 2017 at 4:29

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