Topics on Geometric and Representation Theoretic Aspects of Period Rings I
aa r X i v : . [ m a t h . N T ] M a r TOPICS ON GEOMETRIC AND REPRESENTATION THEORETICASPECTS OF PERIOD RINGS I
XIN TONG
Abstract.
We consider more general framework than the corresponding one considered inour previous work on the Hodge-Iwasawa theory. In our current consideration we considerthe corresponding more general base spaces, namely the analytic adic spaces and analyticperfectoid spaces in Kedlaya’s AWS Lecture notes. We hope our discussion will also shedsome light on further generalization to even more general spaces such as those consideredby Gabber-Ramero namely one just considers certain topological rings which satisfy theFontaine-Wintenberger idempotent correspondence and calls them perfectoid generalizingthe notions from Scholze, Fontaine, Kedlaya-Liu and Kedlaya (AWS Lecture notes). Actu-ally some of the discussion we presented here is already in some more general form for thispurpose (although we have not made enough efforts to write all the things). Version: Feb 28 2021. Keywords and Phrases: Analytic Geometry, Period Rings, Relative p -adic Hodge Structures, Deformationof Representations of Fundamental Groups. ontents
1. Introduction 31.1. General Perfectoids and Preperfectoids 31.2. Some Consideration in the Future 42. Period Rings in General Setting with General Coefficients 53. The Frobenius Modules and Frobenius Sheaves over the Period Rings 84. Comparison in the Étale Setting 105. Comparison in the Non-Étale Setting 136. Discussion on the Generality of Gabber-Ramero 216.1. General Period Rings 216.2. Derived Algebraic Geometry of ∞ -Period Rings 27Acknowledgements 30References 31. Introduction
General Perfectoids and Preperfectoids.
Our previous work on Hodge-Iwasawatheory aimed at the corresponding deformation of the corresponding Hodge structure af-ter [KL1] and [KL2]. The corresponding application in our mind targets the correspondingnoncommutative Iwasawa theory and the corresponding p -adic local systems with generalBanach coefficients. The corresponding background of the foundations on the analytic ge-ometry comes from essentially [KL1] where Banach analog of Fontaine’s perfectoid spaces inthe Tate case was defined and studied extensively.The corresponding analytic Huber analog of the main results of [KL1] was initiated in[Ked1] where the analytic Huber analog of perfectoids are defined, and the analytic Huberanalog of the corresponding perfectoid correspondence is established and the correspondingdescent for vector bundles and stably-pseudocoherent sheaves is established in the analytictopology.In our current development of the corresponding Hodge-Iwasawa theory we consider thecorresponding generality at least parallel to [Ked1]. We now consider the correspondinganalytic analog of our previous consideration which generalized the corresponding [KL1] and[KL2] to the corresponding big coefficient situation. Without deformation in our sense [KL1]and [KL2] have already considered the corresponding free of trivial norm cases in certaincontext and considerations.Certainly inverting p takes back us to the Tate situation, but what we know is that thisis at least not included in our previous consideration, since the relative Robba rings are notthe same (even in our current situation many are still Tate as their owns). So we have tobasically establish the corresponding parallel theory, in order to treat the situation where thebase spaces or rings are not Tate. And more importantly the integral Robba rings are notTate, which needs to be discussed in order to apply to the situation where the base spacesor rings are not Tate.With the corresponding notations in Chapter 5, we have: Theorem 1.1.
Over A -relative preperfectoid Robba rings e Π ? ,R,A ( ? = , r, ∞ , [ r , r ] , { [ s, r ] } , < r ≤ r /p hk ) constructed over analytic base ( R, R + ) , we have the equivalence amonghe categories of the Frobenius étale-stably-pseudocoherent modules, which could be furthercompared in equivalence to the the pseudocoherent sheaves over adic Fargues-Fontaine curvesin both étale and pro-étale topology. This is basically the corresponding commutative version where we encode the correspond-ing ring A into the corresponding Huber spectrum in [KL1] and [KL2]. The correspondingnoncommutative version is also true but we have to do this slightly different by deformingthe structure sheaves. With the corresponding notations again in Chapter 5, we have: Theorem 1.2.
Over B -relative preperfectoid Robba rings e Π ? ,R,B ( ? = , r, ∞ , [ r , r ] , { [ s, r ] } , < r ≤ r /p hk ) constructed over analytic base ( R, R + ) , we have the equivalence amongthe categories of the Frobenius B -étale-stably-pseudocoherent modules, which could be furthercompared in equivalence to the the B -pseudocoherent sheaves over adic Fargues-Fontainecurves in étale topology. Our ultimate consideration will after [GR], where ( R, R + ) is not necessarily always as-sumed to be analytic, especially in the integral setting this is quite general. With thecorresponding notations again in Chapter 6, we have: Theorem 1.3.
Over A -relative preperfectoid Robba rings e Π ? ,R,A ( ? = [ r , r ] , { [ s, r ] } , Over B -relative preperfectoid Robba rings e Π ? ,R,B ( ? = [ r , r ] , { [ s, r ] } , More thorough applications of [Lu2] will beexpected in our study on the corresponding derived algebraic geometry of the correspondingperiod rings . We only touched this to some very transparent extent which will be somehowreflecting some further consideration along the corresponding applications in our mind. The original motivation comes from the study of the algebraic pseudocoherent sheaves over algebraicFargues-Fontaine curves originally in [KL2], and also in [XT1] and [XT2] where the corresponding functionalanalytic information around locally convex vector spaces (after Bourbaki) is not considered. ew robust developments from Clausen-Scholze [CS], Bambozzi-Ben-Bassat-Kremnizer[BBBK] and Bambozzi-Kremnizer [BK] may shed lights on the derived Galois (possibly inmore general setting for fundamental groups) deformation theory (and some unobstructedcrystalline substacks) such as in [GV] and possibly (we conjecture) the corresponding pos-sible theory of some derived eigenvarieties and derived Selmer varieties in Kim’s nonabelianChabauty. We hope to give more hard thorough study and understanding around this.Especially in our situation we will come across many technical issues where these new de-velopments could help.2. Period Rings in General Setting with General Coefficients In this section we start by defining Kedlaya-Liu’s style period rings with coefficients in Tateadic Banach rings which are of finite type. Setting 2.1. We consider the corresponding setting up which is as in the following. Firstwe consider a field E which is analytic nonarchimedean with normalized norm such thatthe corresponding uniformizer π E is of norm /p . And we assume that the correspondingresidue field of the field E takes the form of F p h . Then we fix a corresponding uniform adicBanach algebra ( R, R + ) over F p h which is not required to be contain topological nilpotentunit. However we need to assume that that this is analytic in the sense of [KL1] and[Ked1, Definition 1.1.2]. We recall that this means the set of the corresponding topologicalnilpotents generates the unit ideal. And let A be a general rigid analytic affinoid over Q p orover F p (( t )) .Then in this generality we define our large coefficient Robba rings following Kedlaya-Liu inthe Tate algebra situation namely we have A = Q p { X , ..., X d } or A = F p (( t )) { X , ..., X d } : Definition 2.2. (After Kedlaya-Liu [KL2, Definition 4.1.1] ) Now consider the followingconstructions. First we consider the corresponding Witt vectors coming from the corre-sponding adic ring ( R, R + ) . First we consider the corresponding generalized Witt vectorswith respect to ( R, R + ) with the corresponding coefficients in the Tate algebra with thegeneral notation W ( R + )[[ R ]] . The general form of any element in such deformed ring couldbe written as P i ≥ ,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d . Then we take the corresponding completionwith respect to the following norm for some radius t > : k . k t,A ( X i ≥ ,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d ) := max i ≥ ,i ≥ ,...,i d ≥ p − i k . k R ( y i ) (2.1)hich will give us the corresponding ring e Π int ,t,R,A such that we could put furthermore that: e Π int ,R,A := [ t> e Π int ,t,R,A . (2.2)Then as in [KL2, Definition 4.1.1], we now put the ring e Π bd ,t,R,A := e Π int ,t,R,A [1 /π ] and weset: e Π bd ,R,A := [ t> e Π bd ,t,R,A . (2.3)The corresponding Robba rings with respect to some intervals and some radius could bedefined in the same way as in [KL2, Definition 4.1.1]. To be more precise we consider thecompletion of the corresponding ring W ( R + )[[ R ]][1 /π ] with respect to the following normfor some t > where t lives in some prescribed interval I = [ s, r ] : k . k t,A ( X i,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d ) := max i ≥ ,i ≥ ,...,i d ≥ p − i k . k R ( y i ) . (2.4)This process will produce the corresponding Robba rings with respect to the given interval I = [ s, r ] . Now for particular sorts of intervals (0 , r ] we will have the corresponding Robbaring e Π r,R,A and we will have the corresponding Robba ring e Π ∞ ,R,A if the correspondinginterval is taken to be (0 , ∞ ) . Then in our situation we could just take the correspondingunion throughout all the radius r > to define the corresponding full Robba ring taking thenotation of e Π R,A . Remark 2.3. The corresponding Robba rings e Π bd ,R,A , e Π R,A , e Π I,R,A , e Π r,R,A , e Π ∞ ,R,A areactually themselves Tate adic Banach rings. However in many further application the non-Tateness of the ring R will cause some reason for us to do the corresponding modification,which is considered on this level in fact in [KL1] in the context therein. Definition 2.4. Then for any general affinoid algebra A over the corresponding base an-alytic field, we just take the corresponding quotients of the corresponding rings defined inthe previous definition over some Tate algebras in rigid analytic geometry, with the samenotations though A now is more general. Note that one can actually show that the definitiondoes not depend on the corresponding choice of the corresponding presentations over A . Remark 2.5. Again in this situation more generally, the corresponding Robba rings e Π bd ,R,A , e Π R,A , e Π I,R,A , e Π r,R,A , e Π ∞ ,R,A are actually themselves Tate adic Banach rings.ote that we can also as in the situation of [KL2] and [XT2] consider the correspondingthe corresponding property checking of the corresponding period rings defined aboave. Wecollect the corresponding statements here while the the proof could be found in [XT2]: Lemma 2.6. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For any two radii < r < r wehave the corresponding equality: e Π int ,r ,R, Q p { T ,...,T d } \ e Π [ r ,r ] ,R, Q p { T ,...,T d } = e Π int ,r ,R, Q p { T ,...,T d } . (2.5) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.13]. (cid:3) Lemma 2.7. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For any two radii < r < r wehave the corresponding equality: e Π int ,r ,R, F p (( t )) { T ,...,T d } \ e Π [ r ,r ] ,R, F p (( t )) { T ,...,T d } = e Π int ,r ,R, F p (( t )) { T ,...,T d } . (2.6) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.13]. (cid:3) Lemma 2.8. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For general affinoid A as above(over Q p or F p (( t )) ) and for any two radii < r < r we have the corresponding equality: e Π int ,r ,R,A \ e Π [ r ,r ] ,R,A = e Π int ,r ,R,A . (2.7) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.14]. (cid:3) Lemma 2.9. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.16]. (cid:3) Lemma 2.10. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.16]. (cid:3) Lemma 2.11. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.17]. (cid:3) . The Frobenius Modules and Frobenius Sheaves over the Period Rings We now consider the corresponding Frobenius actions and the corresponding action reflectedon the corresponding Hodge-Iwasawa modules over general analytic R . As in the Tatesituation we consider the corresponding ’arithmetic’ Frobenius as well. Setting 3.1. We are going to use the corresponding notation F to denote the correspondingrelative Frobenius induced from the corresponding p h -power absolute Frobenius from R . Thiswould mean that when we consider the corresponding Frobenius up to higher order.Furthermore in our situation we have the corresponding sheaves of period rings (carryingthe coefficient A ) over Spa( R, R + ) in the corresponding pro-étale site: e Π bd , Spa( R,R + ) proét ,A , e Π Spa( R,R + ) proét ,A , e Π I, Spa( R,R + ) proét ,A , e Π r, Spa( R,R + ) proét ,A , e Π ∞ , Spa( R,R + ) proét ,A (3.1)which we will also briefly write as: e Π bd , ∗ ,A , e Π ∗ ,A , e Π I, ∗ ,A , e Π r, ∗ ,A , e Π ∞ , ∗ ,A (3.2)when the corresponding topology is basically clear to the readers (one can also consider otherkinds of topology in the same fashion such as the corresponding étale situation in our currentcontext). Definition 3.2. (After Kedlaya-Liu [KL2, Definition 4.4.4] ) The corresponding Frobe-nius modules over the corresponding period sheaves will be defined to be finite projec-tive modules carrying the semilinear action from F k (where we allow some power k > )with some pullback isomorphic requirement which could be defined in the following way.Over the rings without intervals or radius this will mean that we have F k ∗ M ∼ → M .While over the corresponding Robba rings with radius r > this will mean that we have F k ∗ M ⊗ ∗ r/phk ∗ r/p hk ∼ → M ⊗ ∗ r ∗ r/p hk . While over the corresponding Robba rings with interval [ r , r ] this will mean that we have F k ∗ M ⊗ ∗ [ r /phk,r /phk ] ∗ [ r ,r /p hk ] ∼ → M ⊗ ∗ [ r ,r ∗ [ r ,r /p hk ] .Over the corresponding Robba rings without any intervals or radii we assume that they arebase change from some modules over the corresponding Robba rings with some radii. Definition 3.3. (After Kedlaya-Liu [KL2, Definition 4.4.4] ) The corresponding pseudo-coherent or fpd Frobenius modules over the corresponding period sheaves will be defined tobe pseudocoherent or fpd modules carrying the semilinear action from F k (where we allowsome power k > ) with some pullback isomorphic requirement which could be defined inthe following way. Over the rings without intervals or radius this will mean that we have k ∗ M ∼ → M . While over the corresponding Robba rings with radius r > this will meanthat we have F k ∗ M ⊗ ∗ r/phk ∗ r/p hk ∼ → M ⊗ ∗ r ∗ r/p hk . While over the corresponding Robbarings with interval [ r , r ] this will mean that we have F k ∗ M ⊗ ∗ [ r /phk,r /phk ] ∗ [ r ,r /p hk ] ∼ → M ⊗ ∗ [ r ,r ∗ [ r ,r /p hk ] . Over the corresponding Robba rings without any intervals or radii weassume that they are base changes from some modules over the corresponding Robba ringswith some radii. And then all the modules are required to be complete with respect to thenatural topology over any perfectoid subdomain within the corresponding pro-étale topology,and the corresponding modules over the corresponding Robba rings with respect to someradius r > will basically be required to be base change to any étale-stably pseudocoherent(over any specific chosen perfectoid subdomain) modules (note that this will include thecorresponding glueing along the space A with respect to the corresponding implicit étaletopology induced over A ).Then as in [KL2, Definition 4.4.4] we consider the corresponding Frobenius modules overthe period rings instead of period sheaves (and we do not either put topological conditions,which will be added later in specific consideration): Definition 3.4. (After Kedlaya-Liu [KL2, Definition 4.4.4] ) The corresponding Frobe-nius modules over the corresponding period rings will be defined to be finite projectivemodules carrying the semilinear action from F k (where we allow some power k > )with some pullback isomorphic requirement which could be defined in the following way.Over the rings without intervals or radius this will mean that we have F k ∗ M ∼ → M .While over the corresponding Robba rings with radius r > this will mean that we have F k ∗ M ⊗ ∗ r/phk ∗ r/p hk ∼ → M ⊗ ∗ r ∗ r/p hk . While over the corresponding Robba rings with interval [ r , r ] this will mean that we have F k ∗ M ⊗ ∗ [ r /phk,r /phk ] ∗ [ r ,r /p hk ] ∼ → M ⊗ ∗ [ r ,r ∗ [ r ,r /p hk ] .Over the corresponding Robba rings without any intervals or radii we assume that they arebase change from some modules over the corresponding Robba rings with some radii. Remark 3.5. As in [KL2, Definition 4.4.4] we did not out any topological conditions on thering theoretic and algebraic representation theoretic objects defined above, but this will bedefinitely more precise in later development. Certainly this is more relevant with respect tothe following definition. Definition 3.6. (After Kedlaya-Liu [KL2, Definition 4.4.4] ) The corresponding pseudo-coherent or fpd Frobenius modules over the corresponding period rings will be defined to bepseudocoherent or fpd modules carrying the semilinear action from F k (where we allow someower k > ) with some pullback isomorphic requirement which could be defined in the fol-lowing way. Over the rings without intervals or radius this will mean that we have F k ∗ M ∼ → M . While over the corresponding Robba rings with radius r > this will mean that we have F k ∗ M ⊗ ∗ r/phk ∗ r/p hk ∼ → M ⊗ ∗ r ∗ r/p hk . While over the corresponding Robba rings with interval [ r , r ] this will mean that we have F k ∗ M ⊗ ∗ [ r /phk,r /phk ] ∗ [ r ,r /p hk ] ∼ → M ⊗ ∗ [ r ,r ∗ [ r ,r /p hk ] .Over the corresponding Robba rings without any intervals or radii we assume that they arebase change from some modules over the corresponding Robba rings with some radii.Then we define over our current analytic base the corresponding pseudocoherent and fpdsheaves carrying the corresponding Frobenius modules. Definition 3.7. (After Kedlaya-Liu [KL2, Definition 4.4.6] ) Over the Robba rings withrespect to the corresponding all finite intervals contained in (0 , ∞ ) , we consider a family ( M [ s,r ] ) [ s,r ] of finite projective modules with respect to the corresponding intervals. Thenwe call this family a finite projective F k -bundle over e Π R,A if each member in the family isassumed to be a corresponding finite projective F k -modules. Definition 3.8. (After Kedlaya-Liu [KL2, Definition 4.4.6] ) Over the Robba rings withrespect to the corresponding all finite intervals contained in (0 , ∞ ) , we consider a family ( M [ s,r ] ) [ s,r ] of fpd or pseudocoherent modules with respect to the corresponding intervals.Then we call this family a fpd or pseudocoherent F k -sheaf over e Π R,A if each member inthe family is assumed to be a corresponding stably or étale-stably fpd or pseudocoherent F k -modules. Proposition 3.9. (After Kedlaya-Liu [KL2, Lemma 4.4.8] ) Any pseudocoherent F k -modulesover the corresponding period rings above admits a surjective morphism from some finite pro-jective F k -modules over the same types of period rings.Proof. This is the corresponding relative and analytic version of the corresponding [KL2,Lemma 4.4.8]. The proof is parallel, see [KL2, Lemma 4.4.8]. (cid:3) Comparison in the Étale Setting We now consider the corresponding parallel consideration to [KL2, Section 4.5]. Essentiallyin our current context, this is really the corresponding purely analytic setting especially whenwe do not have the chance to invert π . Certainly we will also consider the correspondingdeformed setting. The corresponding space we are going to work on is just the correspondingerfectoid spaces (in the corresponding analytic situation) taking the corresponding generalform of Spa( R, R + ) , and consider any topological ring Z and the corresponding sheaf Z . Wewill consider the corresponding finite projective, pseudocoherent and finite projective dimen-sion local systems of Z -modules. Note that again in this analytic setting the correspondingdefinitions of such modules are locally of the corresponding same types. This means in par-ticular in the situation of finite projective situation the corresponding definition is locallythe finite projective sheaves of modules over Z instead of the corresponding naive locallyfinite free ones especially in our large coefficient situation. Lemma 4.1. (After Kedlaya-Liu [KL2, Lemma 4.5.3] ) The corresponding taking elementto the corresponding element after the action of the corresponding operator F − will be torealize the following exact sequences as in [KL2, Lemma 4.5.3] : / / / / / / O E k / / / / / / e Ω int ,R / / / / / / e Ω int ,R / / / / / / , / / / / / / E k / / / / / / e Ω R / / / / / / e Ω R / / / / / / , / / / / / / O E k / / / / / / e Π int ,R / / / / / / e Π int ,R / / / / / / , / / / / / / E k / / / / / / e Π R / / / / / / e Π R / / / / / / . Proof. See [KL2, Lemma 4.5.3]. (cid:3) Proposition 4.2. (After Kedlaya-Liu [KL2, Lemma 4.5.4] ) We have the correspondingstrictly pseudocoherence of any corresponding finitely presented modules over e Ω int ,R or e Π int ,R .Therefore we do not have to consider the corresponding topological issue and the stability issuefurther in this certain special situation.Proof. See [KL2, Lemma 4.5.4]. (cid:3) The following result is then achievable: Proposition 4.3. (After Kedlaya-Liu [KL2, Theorem 4.5.7] ) Consider the following cat-egories. The first one is the corresponding category of all the finite projective or pseudocoher-ent O E k -local systems. The second one is the corresponding category of all the finite projectiveor stably-pseudocoherent F k - e Ω int ,R -modules. The third one is the corresponding category ofll the finite projective or stably-pseudocoherent F k - e Π int ,R -modules. The fourth one is thecorresponding category of all the finite projective or stably-pseudocoherent F k - e Ω int , ∗ -sheaves.The last one is the corresponding category of all the finite projective or stably-pseudocoherent F k - e Π int , ∗ -sheaves. Here we consider the corresponding analytic topology.Then we have that these categories are actually equivalent. Similar parallel statement couldthen be made to fpd objects. Remark 4.4. The corresponding stably-pseudocoherent indication in the previous propo-sition is actually not serious as in [KL2, Lemma 4.5.7], the reason is that by the previousproposition, everything is already complete with respect to the natural topology. Proof. This is just the corresponding analytic analog of the corresponding result in [KL2,Lemma 4.5.7]. We just briefly mention the corresponding functors which realize the corre-sponding equivalence. The corresponding one from the first category to the correspondingsheaves is naturally just the corresponding base change. The corresponding functor from thesheaves to the corresponding modules is naturally the corresponding global section functorafter the corresponding foundation in [Ked1, Theorem 1.4.2, Theorem 1.4.18]. (cid:3) Remark 4.5. Since we touched the corresponding foundations in [Ked1, Theorem 1.4.2,Theorem 1.4.18] on the corresponding glueing of the vector bundles and the correspondingstably-pseudocoherent modules in the situation of analytic topology. However the corre-sponding descent in the étale, pro-étale and v -topology situations are actually parallel tothe corresponding [KL2, Theorem 2.5.5, Theorem 2.5.14, Theorem 3.4.8, Theorem 3.5.8],which will basically upgrade naturally this proposition to the étale, pro-étale and v -topologysituations, although we have not established the chance to present.Now carrying the corresponding coefficient A we could have at least the correspondingfully faithfulness. Proposition 4.6. (After Kedlaya-Liu [KL2, Theorem 4.5.7] ) Consider the following cat-egories. The first one is the corresponding category of all the finite projective or pseudoco-herent O E k ,A -local systems. The second one is the corresponding category of all the finiteprojective or stably-pseudocoherent F k - e Ω int ,R,A -modules. The third one is the correspond-ing category of all the finite projective or stably-pseudocoherent F k - e Π int ,R,A -modules. Thefourth one is the corresponding category of all the finite projective or stably-pseudocoherent F k - e Ω int , ∗ ,A -sheaves. The last one is the corresponding category of all the finite projectiveor stably-pseudocoherent F k - e Π int , ∗ ,A -sheaves. Here we consider the corresponding analyticopology.Then we have that the first category could be embedded fully faithfully into the rest fourcategories and the corresponding modules could be compared to the sheaves under equivalencesof the corresponding categories as long as we assume the corresponding sousperfectoidness ofthe ring A which preserves the corresponding exactness of the exact sequences: / / / / / / O E k / / / / / / e Ω int ,R / / / / / / e Ω int ,R / / / / / / , / / / / / / E k / / / / / / e Ω R / / / / / / e Ω R / / / / / / , / / / / / / O E k / / / / / / e Π int ,R / / / / / / e Π int ,R / / / / / / , / / / / / / E k / / / / / / e Π R / / / / / / e Π R / / / / / / . Similar statement could be mede to the corresponding fpd objects.Proof. See [KL2, Theorem 4.5.7]. (cid:3) Comparison in the Non-Étale Setting We now consider the corresponding analog of the corresponding [KL2, Chapter 4.6] over Spa( R, R + ) as above which is just assumed to be analytic. Remark 5.1. As we mentioned before, the corresponding analyticity of the ring R will stillnot deduce from the analyticity of the corresponding rational Robba rings and sheaves. Theorem 5.2. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Assume the corresponding Robbarings with respect to closed intervals are sheafy. Consider the corresponding categories in thefollowing:A. The corresponding category of all the corresponding étale-stably-pseudocoherent sheavesover the adic Fargues-Fontaine curve (associated to { e Π I,R,A } { I ⊂ (0 , ∞ ) } ) in the correspondingétale topology;B. The corresponding category of all the corresponding étale-stably-pseudocoherent F k -sheavesover families of Robba rings { e Π I,R,A } { I ⊂ (0 , ∞ ) } ;C. The corresponding category of all the corresponding étale-stably-pseudocoherent modulesver any Robba rings e Π I,R,A such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk ;D. The corresponding category of all the corresponding pseudocoherent modules over anyRobba rings e Π R,A , but admitting models of strictly-pseudocoherent F k -modules over some e Π r,R,A for some radius r > whose base changes to some e Π [ r ,r ] ,R,A will produce correspond-ing F k -modules which are étale-stably-pseudocoherent;E. The corresponding category of all the corresponding strictly-pseudocoherent F k -modulesover some e Π ∞ ,R,A whose base changes to some e Π [ r ,r ] ,R,A will provide corresponding F k -modules which are étale-stably-pseudocoherent;F. The corresponding category of all the corresponding étale-stably-pseudocoherent sheavesover the adic Fargues-Fontaine curve (associated to { e Π I,R,A } { I ⊂ (0 , ∞ ) } ) in the correspondingpro-étale topology.Then we have that they are actually equivalent.Proof. This is just parallel [KL2, Theorem 4.6.1], see [XT2, Theorem 4.11]. (cid:3) Theorem 5.3. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Assume A is sousperfectoid.Consider the corresponding categories of sheaves over Spa( R, R + ) pét in the following:A. The corresponding category of all the corresponding étale-stably-pseudocoherent modulesover any Robba rings e Π I, ∗ ,A such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk ;B. The corresponding category of all the corresponding pseudocoherent modules over anyRobba rings e Π ∗ ,A , but admit models of strictly-pseudocoherent F k -modules over some e Π r, ∗ ,A for some radius r > whose base changes to some e Π [ r ,r ] ,R,A will provide corresponding F k -modules which are étale-stably-pseudocoherent;C. The corresponding category of all the corresponding étale-strictly-pseudocoherent F k -modules over some e Π ∞ , ∗ ,A whose base changes to some e Π [ r ,r ] , ∗ ,A will provide corresponding F k -modules which are étale-stably-pseudocoherent.Then we have that they are actually equivalent.Proof. This is parallel to [KL2, Theorem 4.6.1]The proof is the same as the one where ∗ isjust a ring R . See [XT2, Theorem 4.11]. (cid:3) Theorem 5.4. (After Kedlaya-Liu [KL2, Corollary 4.6.2] ) The corresponding analogs ofthe previous two theorems hold in the corresponding finite projective setting. Here we assumethat A is in the same hypotheses as above.Proof. See [KL2, Corollary 4.6.2]. (cid:3) ow we drop the corresponding condition on the sheafiness on A by using the correspond-ing derived spectrum Spa h ( A ) from [BK] where we have the corresponding ∞ -sheaf O Spa h ( A ) ,we now apply the corresponding construction to the Robba rings (in the rational setting over Q p ) ∗ R,A and denote the corresponding derived version by ∗ hR,A . Here A is assumed to bejust Banach over Q p or F p (( t )) , which is certainly in a very general situation. Remark 5.5. One can apply the corresponding results of [CS] to achieve so as well, whichwe believe will be robust as well. Theorem 5.6. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Take a derived rational local-ization of e Π I S I ,R,A , which we denote it by e Π h, ∗ I S I ,R,A , where we choose two overlapped closedintervals I and I . Now take the corresponding base changes of this localization along: e Π I S I ,R,A → e Π I ,R,A , (5.1) e Π I S I ,R,A → e Π I ,R,A , (5.2) which we will denote by e Π h, ∗ I ,R,A and e Π h, ∗ I ,R,A . Consider the corresponding categories in thefollowing:A. The corresponding product of the category of all the corresponding finite projective F k -modules over the Robba ring e Π h, ∗ I ,R,A and the the category of all the corresponding finiteprojective F k -modules over the Robba ring e Π h, ∗ I ,R,A , over the category of all the correspondingfinite projective F k -modules over e Π h, ∗ I ,R,A b ⊗ L e Π h, ∗ I ,R,A ;B. The corresponding category of all the corresponding finite projective modules over theRobba ring e Π h, ∗ I S I ,R,A .Then we have that they are actually equivalent.Proof. We consider a pair of overlapped intervals I = [ r , r ] , I = [ s , s ] , and we have thecorresponding ∞ -Robba rings which form the desired situation, where we have the sequencewhich is basically exact up to higher homotopy: e Π h, ∗ I S I ,R,A / / / / / / / / e Π h, ∗ I ,R,A L e Π h, ∗ I ,R,A / / / / / / / / e Π h, ∗ I ,R,A b ⊗ L e Π h, ∗ I ,R,A . Now we consider the corresponding two finite projective module spectra (which admit re-tracts from finite free module spectra as in [Lu1, Proposition 7.2.2.7]) M , M , M over therings in the middle and the rightmost positions, and assume that they form a correspondingglueing datum. Then we consider the corresponding presentation diagram for the modulepectra: F / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) F L F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G L G r ⊕ r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / G r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) M / / / / / / M L M / / / / / / M by just taking the corresponding fibers, here G, G , G , G are the finite free modules andwe have the corresponding retracts r , r , r , but note that the corresponding map r ⊕ r might not be a priori a retract. However consider the following commutative diagram: (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F L π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π G / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π G L π G π ( r ) ⊕ π ( r ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π G r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π M / / / / / / π M L π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / 00 0 . e then consider the corresponding the argument of Kedlaya as in [XT3, Proposition 5.11]which gives us modification on sections π s and π s of π r and π r respectively such thatwe have (with the same notations) the new π s and π s give the corresponding new lifts s and s whose coproduct actually restricts to M which produce a section for M for the map G → M , which proves the desired finite projectivity on M . From A to B we just take thecorresponding projection while going back we consider the corresponding binary glueing totackle the corresponding interval which cannot be reached by taking Frobenius translationsbut could be covered by two ones reachable by taking Frobenius translations. The lifts couldbe also modified by directly on the derived level. (cid:3) In [GV], Galatius and Venkatesh considered some derived Galois deformation theory. Nowwe make some discussion around this point by using the corresponding simplicial Banachrings (over Q p or over F p (( t )) ) in [BK], namely we use the notation A h to denote any localcharts coming from the corresponding Bambozzi-Kremnizer spectrum in [BK] attached tosome Banach algebra over (over Q p or over F p (( t )) ). Over the analytic field specified we takethe corresponding completed tensor products the period rings involved with A h , which wewill denote by ∗ R,A h . Then we have the following similar results as above: Conjecture 5.7. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the correspondingcategories in the following:A. The corresponding category of all the corresponding locally finite free sheaves over theadic Fargues-Fontaine curve (associated to { e Π I,R,A h } { I ⊂ (0 , ∞ ) } ) in the corresponding homo-topy Zariski topology from [BK] and [BBBK] induced by Koszul derived rational localization;B. The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I,R,A h } { I ⊂ (0 , ∞ ) } ;C. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I,R,A h such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk .Then we have that they are actually equivalent. Theorem 5.8. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the corresponding cat-egories in the following: Even one can take the more general simplicial Banach rings as in [BBBK], especially one would like tofocus on the corresponding analytification of derived Galois deformation rings as in [GV], at least our feelingis that [BBBK] will allow one to take the corresponding derived adic generic fiber to produce some desiredsimplicial Banach rings to tackle derived Galois deformation problems in [GV]. . The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I,R,A h } { I ⊂ (0 , ∞ ) } ;B. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I,R,A h such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk .Then we have that they are actually equivalent.Proof. Note that in our situation we do have the following nice short exact sequences: / / / / / / π k e Π I S I ,R,A h / / / / / / π k e Π I ,R,A h L π k e Π I ,R,A h / / / / / / π k e Π I T I ,R,A h / / / / / / , k = 0 , , .... We consider a pair of overlapped intervals I = [ r , r ] , I = [ s , s ] , and we have the corre-sponding ∞ -Robba rings which form a corresponding glueing sequence in the sense [KL1,Definition 2.7.3]: / / / / / / π e Π I S I ,R,A h / / / / / / π e Π I ,R,A h L π e Π I ,R,A h / / / / / / π e Π I T I ,R,A h / / / / / / . Now we consider the corresponding two finite projective module spectra (which admitretracts from finite free module spectra as in [Lu1, Proposition 7.2.2.7]) M , M , M over therings in the middle and the rightmost position, and assume that they form a correspondingglueing datum. Then we consider the corresponding presentation diagram for the modulespectra: F / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) F L F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G L G r ⊕ r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / G r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) M / / / / / / M L M / / / / / / M by just taking the corresponding fibers, here G, G , G , G are the finite free modules andwe have the corresponding retracts r , r , r , but note that the corresponding map r ⊕ r might not be a priori a retract. However consider the following commutative diagram (whichdmits more exactness than in the previous theorem): (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F L π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π G / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π G L π G π ( r ) ⊕ π ( r ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π G r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / π M / / / / / / π M L π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / 00 0 . We then consider the corresponding the argument of Kedlaya as in [XT3, Proposition 5.11]which gives us modification on sections π s and π s of π r and π r respectively suchthat we have (with the same notations) the new π s and π s give the corresponding newlifts s and s whose coproduct actually restricts to M which produce a section for M forthe map G → M , which proves the desired finite projectivity on M . From A to B we justtake the corresponding projection while going back we consider the corresponding binaryglueing to tackle the corresponding interval which cannot be reached by taking Frobeniustranslations but could be covered by two ones reachable by taking Frobenius translations.The lifts could be also modified by directly on the derived level. Or one can just show theflatness on the derived level directly by using derived Tor, which will directly proves that M is finite projective as a module spectrum. (cid:3) Now we even drop the hypothesis (by using the corresponding techniques in [XT3] and[XT4]) on the commutativity on A , namely now A could be allowed to be strictly quotientcoming from the free Tate algebras Q p h Z , ..., Z d i and F p (( t )) h Z , ..., Z d i , which we will useome different notation B to denote this. Note certainly that A is some special case of such B ,therefore now we are going to indeed generalize the discussion above to the noncommutativesetting. Remark 5.9. One can definitely consider more general coefficients in the Banach setting,but for Hodge-Iwasawa theory, we prefer to locate our discussion in the area where we couldget the interesting rings by taking admissible and strictly quotient from free Tate algebrasover analytic field, which certainly includes the corresponding situation of rigid analyticaffinoids. Setting 5.10. By the work [XT3] and [XT4] we can freely translate between the languageof B -stably-pseudocoherent sheaves over analytic or étale topology and étale setting and thecorresponding B -stably pseudocoherent modules or B -étale stably pseudocoherent modules,which makes the discussion in our context possible. What was happening indeed when weare in the corresponding sheafy tensor product situation is that one can direct read off suchresult in the next theorem by using the descent for the pseudocoherent modules with desiredstability in [KL2] and [Ked1]. Theorem 5.11. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Assume the correspondingRobba rings with respect to closed intervals are sheafy. Consider the corresponding cate-gories in the following (all modules are left):A. The corresponding category of all the corresponding B -étale-stably-pseudocoherent sheavesover the adic Fargues-Fontaine curve (associated to { e Π I,R,B } { I ⊂ (0 , ∞ ) } ) in the correspondingétale topology;B. The corresponding category of all the corresponding B -étale-stably-pseudocoherent F k -sheaves over families of Robba rings { e Π I,R,B } { I ⊂ (0 , ∞ ) } ;C. The corresponding category of all the corresponding B -étale-stably-pseudocoherent modulesover any Robba rings e Π I,R,B such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk ;D. The corresponding category of all the corresponding B -pseudocoherent modules over anyRobba rings e Π R,B , but admit models of B -strictly-pseudocoherent F k -modules over some e Π r,R,B for some radius r > whose base changes to some e Π [ r ,r ] ,R,B will provide corre-sponding F k -modules which are B -étale-stably-pseudocoherent;E. The corresponding category of all the corresponding B -strictly-pseudocoherent F k -modulesover some e Π ∞ ,R,B whose base changes to some e Π [ r ,r ] ,R,B will provide corresponding F k -modules which are B -étale-stably-pseudocoherent.Then we have that they are actually equivalent.roof. This is just parallel [KL2, Theorem 4.6.1], see [XT2, Theorem 4.11]. The correspond-ing modules could be regarded as sheaves over the deformed sites as in our previous work[XT3] and [XT4]. (cid:3) Theorem 5.12. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the correspondingcategories of sheaves over Spa( R, R + ) ét in the following (all the modules are left over thecorresponding rings):A. The corresponding category of all the corresponding B -étale-stably-pseudocoherent modulesover any Robba rings e Π I, ∗ ,B such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk ;B. The corresponding category of all the corresponding B -pseudocoherent modules over anyRobba rings e Π ∗ ,B , but admit models of B -strictly-pseudocoherent F k -modules over some e Π r, ∗ ,B for some radius r > whose base changes to some e Π [ r ,r ] ,R,B will provide corresponding F k -modules which are B -étale-stably-pseudocoherent;C. The corresponding category of all the corresponding B -strictly-pseudocoherent F k -modulesover some e Π ∞ , ∗ ,B whose base changes to some e Π [ r ,r ] , ∗ ,B will provide corresponding F k -modules which are B -étale-stably-pseudocoherent.Then we have that they are actually equivalent.Proof. This is parallel to [KL2, Theorem 4.6.1]The proof is the same as the one where ∗ isjust a ring R . See [XT2, Theorem 4.11]. (cid:3) Theorem 5.13. (After Kedlaya-Liu [KL2, Corollary 4.6.2] ) The corresponding analogsof the previous two theorems hold in the corresponding finite projective setting (one has toconsider the bimodules). Here we assume that B is in the same hypotheses as above.Proof. See [KL2, Corollary 4.6.2] and [XT3, Proposition 5.12]. (cid:3) Discussion on the Generality of Gabber-Ramero General Period Rings. We can encode now the corresponding discussion in the pre-vious section actually, but we choose to separately discuss the corresponding results in detailhere. Certainly many results in [KL1] and [KL2], and [XT1], [XT2], [XT3] and [XT4] relyon 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]. etting 6.1. Drop the condition on the analyticity on ( R, R + ) and we switch to the corre-sponding notation ( S, S + ) .After Kedlaya-Liu [KL2, Definition 4.1.1], we consider the following constructions. Firstwe consider the corresponding Witt vectors coming from the corresponding adic ring ( S, S + ) .First we consider the corresponding generalized Witt vectors with respect to ( S, S + ) with thecorresponding coefficients in the Tate algebra with the general notation W ( S + )[[ S ]] . The gen-eral form of any element in such deformed ring could be written as P i ≥ ,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d .Then we take the corresponding completion with respect to the following norm for some ra-dius t > : k . k t,A ( X i ≥ ,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d ) := max i ≥ ,i ≥ ,...,i d ≥ p − i k . k S ( y i ) (6.1)which will give us the corresponding ring e Π int ,t,S,A such that we could put furthermore that: e Π int ,S,A := [ t> e Π int ,t,S,A . (6.2)Then as in [KL2, Definition 4.1.1], we now put the ring e Π bd ,t,S,A := e Π int ,t,S,A [1 /π ] and we set: e Π bd ,S,A := [ t> e Π bd ,t,S,A . (6.3)The corresponding Robba rings with respect to some intervals and some radius could bedefined in the same way as in [KL2, Definition 4.1.1]. To be more precise we consider thecompletion of the corresponding ring W ( S + )[[ S ]][1 /π ] with respect to the following norm forsome t > where t lives in some prescribed interval I = [ s, r ] : k . k t,A ( X i,i ≥ ,...,i d ≥ π i [ y i ] X i ...X i d d ) := max i ≥ ,i ≥ ,...,i d ≥ p − i k . k S ( y i ) . (6.4)This process will produce the corresponding Robba rings with respect to the given interval I = [ s, r ] . Now for particular sorts of intervals (0 , r ] we will have the corresponding Robbaring e Π r,S,A and we will have the corresponding Robba ring e Π ∞ ,S,A if the corresponding intervalis taken to be (0 , ∞ ) . Then in our situation we could just take the corresponding unionthroughout all the radius r > to define the corresponding full Robba ring taking thenotation of e Π S,A .The corresponding Robba rings e Π bd ,S,A , e Π S,A , e Π I,S,A , e Π r,S,A , e Π ∞ ,S,A are actually themselvesTate adic Banach rings. However in many further application the non-Tateness of the ring S will cause some reason for us to do the corresponding modification.hen for any general affinoid algebra A over the corresponding base analytic field, we justtake the corresponding quotients of the corresponding rings defined in the previous definitionover some Tate algebras in rigid analytic geometry, with the same notations though A nowis more general. Note that one can actually show that the definition does not depend on thecorresponding choice of the corresponding presentations over A .Again in this situation more generally, the corresponding Robba rings e Π bd ,S,A , e Π S,A , e Π I,S,A , e Π r,S,A , e Π ∞ ,S,A are actually themselves Tate adic Banach rings. Lemma 6.2. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For any two radii < r < r wehave the corresponding equality: e Π int ,r ,S, Q p { T ,...,T d } \ e Π [ r ,r ] ,S, Q p { T ,...,T d } = e Π int ,r ,S, Q p { T ,...,T d } . (6.5) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.13]. (cid:3) Lemma 6.3. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For any two radii < r < r wehave the corresponding equality: e Π int ,r ,S, F p (( t )) { T ,...,T d } \ e Π [ r ,r ] ,S, F p (( t )) { T ,...,T d } = e Π int ,r ,S, F p (( t )) { T ,...,T d } . (6.6) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.13]. (cid:3) Lemma 6.4. (After Kedlaya-Liu [KL2, Lemma 5.2.6] ) For general affinoid A as above(over Q p or F p (( t )) ) and for any two radii < r < r we have the corresponding equality: e Π int ,r ,S,A \ e Π [ r ,r ] ,S,A = e Π int ,r ,S,A . (6.7) Proof. See [KL2, Lemma 5.2.6] and [XT2, Proposition 2.14]. (cid:3) Lemma 6.5. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.16]. (cid:3) Lemma 6.6. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.16]. (cid:3) emma 6.7. (After Kedlaya-Liu [KL2, Lemma 5.2.10] ) For any four radii < r < r See [KL2, Lemma 5.2.10] and [XT2, Proposition 2.17]. (cid:3) Definition 6.8. All the Frobenius finite projective, pseudocoherent and fpd modules un-der F k could be defined in the exact same way as in section 3. We will not repeat thecorresponding definition, but note that we change the notation for R to be just S . Conjecture 6.9. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the correspondingcategories in the following:A. The corresponding category of all the corresponding locally finite free sheaves over theadic Fargues-Fontaine curve (associated to { e Π I,S,A h } { I ⊂ (0 , ∞ ) } ) in the corresponding homo-topy Zariski topology from [BK] and [BBBK] induced by Koszul derived rational localization;B. The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I,S,A h } { I ⊂ (0 , ∞ ) } ;C. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I,S,A h such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk .Then we have that they are actually equivalent. Theorem 6.10. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the correspondingcategories in the following:A. The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I,S,A h } { I ⊂ (0 , ∞ ) } ;B. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I,S,A h such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk .Then we have that they are actually equivalent.Proof. As in the situation in the analytic setting we consider the following argument whichwill show the desired result. In our situation we do have the following nice short exactsequences: / / / / / / π k e Π I S I ,S,A h / / / / / / π k e Π I ,S,A h L π k e Π I ,S,A h / / / / / / π k e Π I T I ,S,A h / / / / / / , k = 0 , , .... We consider a pair of overlapped intervals I = [ r , r ] , I = [ s , s ] , and we have the corre-sponding ∞ -Robba rings which form a corresponding glueing sequence in the sense [KL1,efinition 2.7.3]: / / / / / / π e Π I S I ,S,A h / / / / / / π e Π I ,S,A h L π e Π I ,S,A h / / / / / / π e Π I T I ,S,A h / / / / / / . Now we consider the corresponding two finite projective module spectra (which admitsretracts from finite free module spectra as in [Lu1, Proposition 7.2.2.7]) M , M , M over therings in the middle and the rightmost position, and assume that they form a correspondingglueing datum. Then we consider the corresponding presentation diagram for the modulespectra: L / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) L L L (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / L (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) N / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) N L N r ⊕ r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / N r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) M / / / / / / M L M / / / / / / M by just taking the corresponding fibers, here N, N , N , N are the finite free modules andwe have the corresponding retracts r , r , r , but note that the corresponding map r ⊕ r ight not be a priori a retract. However consider the following commutative diagram: (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π F L π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π F (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π G / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) π G L π G π ( r ) ⊕ π ( r ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π G r (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / π M / / / / / / π M L π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / π M (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / 00 0 . We then consider the corresponding the argument of Kedlaya as in [XT3, Proposition 5.11]which gives us modification on sections π s and π s of π r and π r respectively such thatwe have (with the same notations) the new π s and π s give the coresponding new lifts s and s whose coproduct actually restricts to M which produce a section for M for the map N → M , which proves the desired finite projectivity on M . From A to B we just take thecorresponding projection while going back we consider the corresponding binary glueing totackle the corresponding interval which cannot be reached by taking Frobenius translationsbut could be covered by two ones reachable by taking Frobenius translations. One can actu-ally derive this on the derived level directly by taking the corresponding lift of the differenceof the base changes of s and s to the right in Ext ( M , L ) to Ext ( M L M , L L L ) to modify the corresponding sections s and s in order to restrict to M , which proves theresult by the argument of Kedlaya as in [XT3, Proposition 5.11]. (cid:3) Now again we even drop the hypothesis (by using the corresponding techniques in [XT3]and [XT4]) on the commutativity on A , namely now A could be allowed to be strictlyuotient coming from the free Tate algebras Q p h Z , ..., Z d i and F p (( t )) h Z , ..., Z d i , whichwe will use some different notation B to denote this. Note certainly that A is some specialcase of such B , therefore now we are going to indeed generalize the discussion above to thenoncommutative setting. Theorem 6.11. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Assume the correspondingRobba rings with respect to closed intervals are sheafy. Consider the corresponding cate-gories in the following (all modules are left):A. The corresponding category of all the corresponding finite locally projective sheaves overthe adic Fargues-Fontaine curve (associated to { e Π I,S,B } { I ⊂ (0 , ∞ ) } ) in the corresponding étaletopology;B. The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I,S,B } { I ⊂ (0 , ∞ ) } ;C. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I,S,B such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk .Then we have that they are actually equivalent.Proof. This is just parallel [KL2, Theorem 4.6.1], see [XT2, Theorem 4.11]. The correspond-ing modules could be regarded as sheaves over the deformed sites as in our previous work[XT3] and [XT4]. (cid:3) Theorem 6.12. (After Kedlaya-Liu [KL2, Theorem 4.6.1] ) Consider the correspondingcategories of sheaves over Spa( S, S + ) ét in the following (all the modules are bimodules overthe corresponding rings):A. The corresponding category of all the corresponding finite projective F k -sheaves over fam-ilies of Robba rings { e Π I, ∗ ,B } { I ⊂ (0 , ∞ ) } ;B. The corresponding category of all the corresponding finite projective modules over anyRobba rings e Π I, ∗ ,B such that the interval I = [ r , r ] satisfies that ≤ r ≤ r /p hk ;Then we have that they are actually equivalent.Proof. Note that in this situation the corresponding space is not even analytic adic space,but the Robba rings over some certain perfectoid domains are analytic. This is parallel to[KL2, Theorem 4.6.1]The proof is the same as the one where ∗ is just a ring R . See [XT2,Theorem 4.11]. (cid:3) Derived Algebraic Geometry of ∞ -Period Rings. We now consider the contactwith the corresponding algebraic sheaves over the corresponding schemes associated to thecorresponding period rings in [KL1], [KL2] as in [XT1], [XT2], [XT3], [XT4] along [KL1]. emark 6.13. Certainly we are now dropping the corresponding topology and functionalanalyticity. This is motivated by the corresponding discussion on the algebraic quasi-coherentsheaves over schematic Fargues-Fontaine curves in [KL2]. Setting 6.14. A is assumed to be now general Banach (commutative at this moment) overthe base analytic field, and keep R as in the situation before in this section (in the mostgeneral setting we considered so far). Now we apply the whole machinery in [Lu1] to thecorresponding ∞ -Bambozzi-Kremnizer rings ∗ hR,A attached to the period rings ∗ R,A carrying A . Then we regard ∗ hR,A as a corresponding E -ring in [Lu1] . Proposition 6.15. The corresponding category of all the perfect, almost perfect (after [Lu1,Section 7.2.4] , namely pseudocoherent) F k -equivariant sheaves over the families of ∞ -Robbaétale ∞ -Deligne-Mumford toposes (Spec e Π hI,R,A , O ) { I ⊂ (0 , ∞ ) } (as in [Lu2, Chapter 1.4] ) isequivalent to the category of all the perfect, almost perfect (after [Lu1, Section 7.2.4] , namelypseudocoherent) F k -equivariant sheaves over some ∞ -Robba étale ∞ -Deligne-Mumford topos (Spec e Π hI,R,A , O ) (as in [Lu2, Chapter 1.4] ) where I = [ r , r ] ( < r ≤ r /p hk ). Here weneed to assume the corresponding conjecture 6.20 holds.Proof. This is actually quite transparent that we just take the corresponding Frobeniustranslation to compare the corresponding sheaves. (cid:3) Proposition 6.16. The corresponding category of all the locally finite free (after [Lu1, Sec-tion 7.2.2, 7.2.4] , namely pseudocoherent and flat) F k -equivariant sheaves over the familiesof ∞ -Robba étale ∞ -Deligne-Mumford toposes (Spec e Π hI,R,A , O ) { I ⊂ (0 , ∞ ) } (as in [Lu2, Chap-ter 1.4] ) is equivalent to the category of all the locally finite free (after [Lu1, Section 7.2.2,7.2.4] , namely pseudocoherent and flat) F k -equivariant sheaves over some ∞ -Robba étale ∞ -Deligne-Mumford topos (Spec e Π hI,R,A , O ) (as in [Lu2, Chapter 1.4] ) where I = [ r , r ] ( < r ≤ r /p hk ).Proof. This is again actually quite transparent that we just take the corresponding Frobeniustranslation to compare the corresponding sheaves. (cid:3) Example 6.17. Everything will be certainly more interesting when we maintain in thecorresponding noetherian situation, where we do have the corresponding derived analyticconsideration. But since our current goal is to study the corresponding derived algebraicgeometry carrying relative p -adic Hodge structure we will not explicitly mention the corre-sponding analytic context. Note that the original convention of [BK] is cohomological and vanishing in positive degree. imilarly as what we did in the corresponding derived analytic geometry above, we considerthe corresponding contact with [GV] in the derived algebraic geometric setting. We keepthe notation as above. Then we have the following results with the same arguments as inthe above on derived deformation of Hodge structures (note that this is more related to thenon-derived situation): Proposition 6.18. The corresponding category of all the perfect, almost perfect (after [Lu1,Section 7.2.4] , namely pseudocoherent) F k -equivariant sheaves over the families of ∞ -Robbaétale ∞ -Deligne-Mumford toposes (Spec e Π I,R,A h , O ) { I ⊂ (0 , ∞ ) } (as in [Lu2, Chapter 1.4] ) isequivalent to the category of all the perfect, almost perfect (after [Lu1, Section 7.2.4] , namelypseudocoherent) F k -equivariant sheaves over some ∞ -Robba étale ∞ -Deligne-Mumford topos (Spec e Π I,R,A h , O ) (as in [Lu2, Chapter 1.4] ) where I = [ r , r ] ( < r ≤ r /p hk ). Here weneed to assume the corresponding conjecture 6.20 holds. Proposition 6.19. The corresponding category of all the locally finite free (after [Lu1, Sec-tion 7.2.2, 7.2.4] , namely pseudocoherent and flat) F k -equivariant sheaves over the familiesof ∞ -Robba étale ∞ -Deligne-Mumford toposes (Spec e Π I,R,A h , O ) { I ⊂ (0 , ∞ ) } (as in [Lu2, Chap-ter 1.4] ) is equivalent to the category of all the locally finite free (after [Lu1, Section 7.2.2,7.2.4] , namely pseudocoherent and flat) F k -equivariant sheaves over some ∞ -Robba étale ∞ -Deligne-Mumford topos (Spec e Π I,R,A h , O ) (as in [Lu2, Chapter 1.4] ) where I = [ r , r ] ( < r ≤ r /p hk ). The following is expect to hold in full generality. Conjecture 6.20. In the two settings in this subsection above, we conjecture that the corre-sponding descent holds (along binary rational coverings) for perfect, almost perfect and finiteprojective modules (which will allow us to compare sheaves and the global sections) in thenoetherian situation. Example 6.21. The corresponding descent could happen when we have the noetherianness,where the corresponding rational localization is actually automatically flat, which will implythat one can certainly glue coherent sheaves in our situation, see [KL1, Theorem 1.3.9].Beyond the corresponding noetherianness it might be not safe to conjecture so. Any verified ∞ -descent results in derived algebraic geometry certainly apply. cknowledgements. This is really rooted in our previous works and our previous consid-eration around the corresponding analytic geometry over relative period rings from [KL1]and [KL2]. We would like to thank Professor Kedlaya for the key discussion around somevery subtle points herein. eferences [Ked1] Kedlaya, K. "Sheaves, stacks, and shtukas, lecture notes from the 2017 Arizona Winter School:Perfectoid Spaces." Math. Surveys and Monographs 242.[KL1] Kedlaya, Kiran Sridhara, and Ruochuan Liu. Relative p -adic Hodge theory: foundations. Sociétémathématique de France, 2015.[KL2] Kedlaya, Kiran S., and Ruochuan Liu. "Relative p F1