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Can anyone give a road map for how Bhatt–Scholze's fancy recent p-adic work applies to questions in more general algebraic geometry and commutative algebra? I'm aware that it does, following Andre - La conjecture du facteur direct, but I still don't really get how the full picture works.

My question is simple, but broad:

How and when do broad problems in algebraic geometry reduce to p-adic problems which can be attacked using these methods?

Moreover: which problems can't be attacked this way? Why?

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    $\begingroup$ According to the abstract of that paper "M. Hochster conjectured that any finite extension of a regular commutative ring splits as a module. Building on his reduction to the case of an unramified complete regular local ring R of mixed characteristic" the conjecture was already reduced by the work of Hochster to the case of complete local mixed characteristic (i.e. p-adic) rings. So the fact that p-adic methods apply doesn't really have to do with the work of Bhatt and Scholze. Is the original reduction what you would like to understand? $\endgroup$
    – Will Sawin
    Commented Jun 25, 2023 at 14:08
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    $\begingroup$ Yes, I would like to understand how/when/why these reductions work. $\endgroup$
    – Patrick Elliott
    Commented Jun 25, 2023 at 14:09
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    $\begingroup$ I think at this point the reduction to the unramified case isn't needed (for instance, if you have big Cohen-Macaulay algebras). For direct summand, reducing to the complete case is easy, if $R \subseteq S$ is finite, it remains finite after completion and completion is faithfully flat. From a big picture perspective, showing the existence of big Cohen-Macaulay algebras is (if graded) dual to saying you can dominate your scheme where something like Kodaira vanishing holds. Thus for things where the missing piece in mixed characteristic is vanishing theorems, this is useful. $\endgroup$
    – Karl Schwede
    Commented Jun 25, 2023 at 16:18
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    $\begingroup$ Ok, but maybe you are asking where these Big Cohen-Macaulay algebras come from. How the ideas of Scholze, Bhatt-Scholze, etc are in the work of André, Gabbert, Bhatt-Scholze, etc on this stuff. The key thing is the perfectoid almost purity theorem and variants (also what Bhatt-Scholze call André’s flatness lemma is useful frequently). Now of course, Bhatt showed that the p-adic completion of $R^+$ is Cohen-Macaulay, via work of Bhatt-Lurie, via a different method. $\endgroup$
    – Karl Schwede
    Commented Jun 25, 2023 at 16:30

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