The purity of the ramification locus — the Zariski-Nagata theorem — says ramification cannot appear in codimension one without algebraic justification. It is a foundational result in algebraic geometry, constraining where a morphism of schemes can fail to be etale. Strengthening it has been notoriously difficult.
Zelich proves a strong version: the maximal etale locus is affine. The mechanism is unexpected — tilting to perfect rings, a technique from p-adic Hodge theory and Scholze's perfectoid geometry. The key ingredient is a Tor-independence result for global sections of etale schemes over excellent regular local rings, which becomes tractable only after passing to the perfect limit via tilting.
Perfectoid tilting was designed for arithmetic geometry in mixed characteristic — understanding how number-theoretic structures behave across different prime characteristics. It works by sending a ring to its “tilt,” a related ring in characteristic p where certain computations become simpler. The tilt preserves enough algebraic structure to make statements about the original ring, while collapsing the mixed-characteristic complexity that made direct arguments intractable.
That this tool unlocks a structural result about etale maps — which is essentially a commutative algebra problem, not an arithmetic geometry problem — reveals something about the nature of the purity obstruction. The difficulty was not in the geometry of ramification but in the homological algebra of the Tor functor over regular local rings. The tilting operation makes the Tor-independence visible by stripping away precisely the mixed-characteristic noise that obscured it.
A tool built for p-adic arithmetic reveals that the affineness of the etale locus is controlled by homological independence conditions that only become visible after passing to the perfect limit.