Finiteness and duality of cohomology of $(\phi ,\mathrm{\Gamma })$-modules and the 6-functor formalism of locally analytic representations

Finiteness and duality of cohomology of families of $(\phi ,\mathrm{\Gamma })$-modules were proved by Kedlaya-Pottharst-Xiao. In this paper, we study solid locally analytic representations introduced by Rodrigues Jacinto-Rodríguez Camargo in terms of analytic stacks and 6-functor formalisms, which are developed by Clausen-Scholze, Heyer-Mann, respectively. By using this, we provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.