Around Segal conjecture in p-adic geometry

This article records multiple results coming from interplay between de-completed topological periodic cyclic homology, Segal conjecture, and F-smoothness. We establish completeness of motivic filtration on de-completed topological periodic cyclic homology of commutative rings with weakly finitely generated absolute cotangent complex. When the ring in question is in addition F-smooth, we show that Segal conjecture holds for its topological Hochschild homology. We also identify our de-completed topological periodic cyclic homology with Manam's Frobenius untwisted topological periodic cyclic homology for quasiregular semiperfectoid rings. We find a crystalline degeneration of Segal conjecture which corresponds to such a statement for F-smoothness. On the other hand, inspired by constructions for topological Hochschild homology, the theory of cyclotomic synthetic spectra allows us to produce a relative conjugate filtration on Hodge--Tate cohomology and its variants, and in the same time, a relative conjugate filtration on topological Hochschild homology and its variants. As a consequence, we deduce transitivity of weak and strong F-smoothness.