Why is Faltings' "almost purity theorem" a purity theorem?

Question

My understanding of purity theorems is that they come in several flavors:

1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose absolute values have size $p^{w/2}$". I don't think that this is the kind of purity I'm interested in.

2a) Purity in algebraic geometry 1: on a smooth algebraic variety the ramification locus of a morphism is a pure codimension 1 subvariety (Zariski, Nagata etc).

2b) Purity in algebraic geometry 2: absolute cohomological purity. Basically -- if $Y$ is a pure codimension $d$ subscheme of $X$ then the local cohomology groups $H^i_Y(X,\mathbf{Z}/n\mathbf{Z})$ should vanish away from $i=2d$ (under various hypotheses, e.g. $X$ locally Noetherian, $n$ invertible everywhere etc) and should be $\mathbf{Z}/n\mathbf{Z}$ etale locally if $i=2d$ (SGA5, Gabber etc).

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Faltings proved an "almost purity theorem" and I think that I'm supposed to be regarding it as some sort of analog of a purity theorem above. Faltings' work occurs in the context of "almost mathematics", where one is working over the integers $R$ in a certain type of (non-discrete) valuation ring $K$, so $R$ has a maximal ideal $m$ and the idea is that instead of working in the category of $R$-modules, one works in the category of $R$-modules up to $m$-torsion (some localised category); this is the category of "almost $R$-modules".

Faltings' almost purity theorem (or perhaps some beefed-up version due to Scholze) says something like this:

Theorem: If $K$ is a perfectoid field, $A$ is a perfectoid $K$-algebra, and $B/A$ is finite etale, then $B$ is also perfectoid and $B^o/A^o$ is almost finite etale.

Whatever does this have to do with the purity theorems mentioned at the beginning of this post?

Answer 1

This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions".

But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. In Faltings's original setup, it was formulated as follows. Consider the rings

$$ R_m = \mathbb{Z}_p[p^{1/p^m},T_1^{\pm 1/p^m},...,T_n^{\pm 1/p^m}] $$

Each of them is smooth over $\mathbb{Z}_p[p^{1/p^m}]$ (in particular regular), and the transition maps are finite and etale after inverting $p$. Let $S_0$ be a finite normal $R_0$-algebra, which is etale after inverting $p$, and let $S_m$ be the normalization of $S_0\otimes_{R_0} R_m$. The idea is that there is ramification of $S_0$ in the special fibre, and you want to get rid of it, by adjoining the chosen tower of highly ramified rings $R_m/R_0$. It is not hard to see that this actually works almost at the generic point of the special fibre: At the generic point, the local ring is a discrete valuation ring, and the statement is that the discriminant of the extension of discrete valuation rings becomes arbitrarily small as $m\to\infty$ (this boils down to some more or less classical ramification theory).

Now, assume that you were lucky, and for some $m$, the ramification at the generic point of the special fibre is not just very close to zero, but actually zero on the nose. Zariski-Nagata purity tells you that the ramification locus of $S_m$ over $R_m$ has to be pure of codimension $1$, but you know that there is no ramification at the codimension $1$ points (which are either in characteristic $0$, or equal to the generic point of the special fibre) -- thus, there is no ramification at all, and $S_m$ over $R_m$ is etale.

The almost purity theorem says that this result extends to the almost world: In the limit as $m\to \infty$, $S_\infty$ over $R_\infty$ is almost etale (in the technical sense defined in almost ring theory).