The $p$-adic monodromy theorem over algebraic-affinoid algebras
In the previous paper of the author, motivated by the categorical $p$-adic local Langlands correspondence, the author studied families of $G_K$-equivariant vector bundles over the Fargues-Fontaine curve parametrized by algebraic-affinoid $\mathbb{Q}_{p,\square}$-algebras (e.g., $\mathbb{Q}_p[T]$). In this paper, we study the $p$-adic Hodge theoretic properties of such families. More precisely, we define the notions of Hodge-Tate, de Rham, and semistable representations for such families, and then prove the $p$-adic monodromy theorem ("de Rham" implies "potentially semistable") in this setting. This is a generalization of the work of Berger-Colmez. As an application, we prove the classification of families of $G_K$-equivariant line bundles. While a similar classification was previously obtained in the previous paper under a certain freeness condition by relying on the results of Kedlaya--Pottharst--Xiao, the approach in the present paper removes this freeness condition and entirely bypasses their results.