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).

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?