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Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful resolution in char $0$, or offer an alternative view of alterations. Has this been attempted, and if not, why is it not a plausible line of approach?

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    $\begingroup$ While we wait for Peter to answer, my impression is that the perfectoid approach allows us to mimic techniques in char $p>0$ that were used to make up for the lack of resolution of singularity such as Frobenius and existence of big Cohen-Macaulay algebras (which allow us to prove vanishing results up to finite covers). $\endgroup$ Mar 26, 2021 at 19:43
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    $\begingroup$ @HailongDao, I too am waiting for him, like Gogo for Godot. $\endgroup$
    – Arna
    Mar 26, 2021 at 20:02
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    $\begingroup$ I think a lot of interest is in resolutions over fields algebraic over their prime subfields and I doubt perfectoids would help in that setting. $\endgroup$
    – Faris
    Mar 27, 2021 at 6:19

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Somehow that question slipped my radar, sorry!

The truth is that shamefully I'm not able to say much, as I don't have a strong knowledge of resolution of singularities. But at least so far, the flow of information has been from characteristic $p$ to characteristic $0$, or more specifically to mixed characteristic. So maybe I wouldn't be surprised if once resolution of singularities has been proved in characteristic $p$, one can use some perfectoid(-inspired) things in order to generalize these results to mixed characteristic. But I would be very surprised if the known results in characteristic $0$ can be used to make progress in characteristic $p$.

As Hailong Dao hints in the comments, this scenario has recently played out in the somewhat related field of the homological conjectures, and related questions in birational geometry. There, "everything" was known in characteristic $p$ building on the work of Hochster, Huneke, etc. An interesting feature here is that one measures singularities in characteristic $p$ not by comparison to a resolution of singularities, but by using Frobenius (in particular, by the relation to their perfection -- for example, by a theorem of Kunz a noetherian $\mathbb F_p$-algebra is regular if and only if its Frobenius morphism is flat, if and only if the map to its perfection is flat). Only very recently all of this has been generalized to mixed characteristic, by Andre, Bhatt, etc., using critically perfectoid methods. One can then measure singularities by comparing to perfectoid algebras: for example, by a theorem of Bhatt-Iyengar-Ma, a noetherian $p$-complete $\mathbb Z_p$-algebra is regular if and only if it admits a faithfully flat map to a perfectoid ring. The most recent results of Bhatt, on Cohen-Macaulayness of absolute integral closures and Kodaira vanishing up to finite covers, have in turn been used to establish the minimal model program for arithmetic threefolds, adapting previous work of Hacon--Xu in characteristic $p$.

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