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math.AG2025
The -adic Galois Cohomology of Valuation Fields
Tongmu He
We compute the Galois cohomology of any -adic valuation field extension of a pre-perfectoid field. Moreover, we obtain a generalization and also a new proof of the classical res…
math.AG2024
Perfectoidness via Sen Theory and Applications to Shimura Varieties
Tongmu He
Sen's theorem on the ramification of a -adic analytic Galois extension of -adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmeti…
math.AG2024
Purity for Perfectoidness
Tongmu He
There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over…