Hodge-Iwasawa Theory II

Abstract

We continue our study on the Hodge-Iwasawa theory which is a continuation of our previous work on Hodge-Iwasawa theory, which is aimed at higher dimensional deformation of higher dimensional Hodge structures over general analytic spaces or adic spaces.We still follow closely the approaches of Kedlaya-Liu to study our Frobenius modules over the different kinds of period rings including more generalized perfect period rings and the corresponding imperfect period rings. It is desirable that one can globalize the deformed information in order to organize the deformed period rings into the corresponding sheaves with respect to the coefficient sheaves, which will build up some foundations on further application to moduli of Frobenius sheaves.

Period Rings with Big Coefficients and Applications III

“…The situation where we deform from the corresponding E ∞ by some noncommutative deformation could be implied by [TX2,Lemma 2.14]. While in general this follows from [TX4,Lemma 6.83]. Note that we achieve the finiteness for each π n (M) just by application of the results we know in the classical deformed situation, which is because the connecting homomorphism vanishes on each level.…”

“…We studied the corresponding glueing in the fashion of [KL1] and [KL2] carrying sufficiently large coefficients. Although we have some conditions on both the corresponding adic spaces we are considering and the corresponding coefficients, but the descent results on their own are already general enough to tackle some specific situations in the Hodge-Iwasawa theoretic consideration and noncommutative analytic geometry such as in [TX3] and [TX4].…”

Topics on Geometric and Representation Theoretic Aspects of Period Rings I

“…Note that we can also as in the situation of [KL2] and [XT2] consider the corresponding the corresponding property checking of the corresponding period rings defined aboave. We collect the corresponding statements here while the the proof could be found in [XT2]: Lemma 2.6.…”

“…We can encode now the corresponding discussion in the previous section actually, but we choose to separately discuss the corresponding results in detail here. Certainly many results in [KL1] and [KL2], and [XT1], [XT2], [XT3] and [XT4] rely on the corresponding topologically nilpotent units and systems of topologically nilpotents. Therefore we will discuss the corresponding admissible and reasonable generalization after [KL1], [KL2], [XT1], [XT2], [XT3], [XT4] and [GR].…”

“…We only touched this to some very transparent extent which will be somehow reflecting some further consideration along the corresponding applications in our mind. 1 The original motivation comes from the study of the algebraic pseudocoherent sheaves over algebraic Fargues-Fontaine curves originally in [KL2], and also in [XT1] and [XT2] where the corresponding functional analytic information around locally convex vector spaces (after Bourbaki) is not considered.…”