Analytic Geometry and Hodge-Frobenius Structure Continued

Abstract

This is our sequel to our previous work on the corresponding generalized Frobenius modules over some big multivariate Robba rings. We will go beyond our previous discussion where we focused on the corresponding analytic functions on polydiscs and polyannuli in the strictly affinoid situation, and general Hodge-Frobenius structures which are admissible in the corresponding context in our previous work.However we will in this paper to consider some more complicated version of the rings. We will use some partial Frobenius to perfectize partially the rings defined above. Therefore we will in some more uniform way to denote the rings in the following different way:Π an,r I ,I,I,∅,A := Π an,r I ,I,A (2.2) Π an,con,I,I,∅,A := Π an,con,I,∅,A .(2.3) Now we follow the idea in [Ked2, Definition 5.2.1] to define some extended version of the rings. We will have the following rings to be: Π [s I ,r I ],I, J,I\J,A , (2.4) Π [s I ,r I ],I, J,I\J,A , (2.5) Π [s I ,r I ],I,J, Ȋ \J,A , (2.6) Π [s I ,r I ],I, J, Ȋ \J,A , (2.7) Π [s I ,r I ],I, J, Ȋ \J,A , (2.8) Π [s I ,r I ],I,J, I\J,A , (2.9) Π [s I ,r I ],I, J, I\J,A , (2.10) Π [s I ,r I ],I, J, I\J,A .