  We establish the flat cohomology version of the Gabber-Thomason purity for
\'{e}tale cohomology: for a complete intersection Noetherian local ring $(R,
\mathfrak{m})$ and a commutative, finite, flat $R$-group $G$, the flat
cohomology $H^i_{\mathfrak{m}}(R, G)$ vanishes for $i < \mathrm{dim}(R)$. For
small $i$, this settles conjectures of Gabber that extend the
Grothendieck-Lefschetz theorem and give purity for the Brauer group for schemes
with complete intersection singularities. For the proof, we reduce to a flat
purity statement for perfectoid rings, establish $p$-complete arc descent for
flat cohomology of perfectoids, and then relate to coherent cohomology of
$\mathbb{A}_{\mathrm{inf}}$ via prismatic Dieudonn\'{e} theory. We also present
an algebraic version of tilting for \'{e}tale cohomology, use it to reprove the
Gabber-Thomason purity, and exhibit general properties of fppf cohomology of
(animated) rings with finite, locally free group scheme coefficients, such as
excision, agreement with fpqc cohomology, and continuity.
