Topologization and Functional Analytification I: Intrinsic Morphisms of Commutative Algebras
aa r X i v : . [ m a t h . N T ] F e b TOPOLOGIZATION AND FUNCTIONAL ANALYTIFICATION I:INTRINSIC MORPHISMS OF COMMUTATIVE ALGEBRAS
XIN TONG
Abstract.
Eventually after Dieudonné-Grothendieck, we give intrinsic definitions of étale,lisse and non-ramifié morphisms for general adic rings and general locally convex rings. Andwe investigate the corresponding étale-like, lisse-like and non-ramifié-like morphisms for gen-eral ∞ -Banach, ∞ -Borné and ∞ -ind-Fréchet ∞ -rings and ∞ -functors into ∞ -groupoid (asin the work of Bambozzi-Ben-Bassat-Kremnizer) in some intrinsic way by using the corre-sponding infinitesimal stacks and crystalline stacks. The two directions of generalizationwill intersect at Huber’s book in the strongly noetherian situation. Contents
1. Introduction 21.1. Main Consideration 21.2. Further Consideration 32. Affinoid Morphisms of Huber Rings 43. Affinoid Morphisms of Banach Rings 104. Naive Étale Morphisms and Intrinsic Étale Morphisms 185. Properties 226. Étale-Like Morphisms of ∞ -Banach Rings and the ∞ -Analytic Stacks 246.1. Approach through De Rham Stacks 246.2. Approach through Crystalline Stack and PD-morphisms 277. Lisse-Like and Non-Ramifié-Like Morphisms of ∞ -Banach Rings and the ∞ -Analytic Stacks 287.1. Approach through De Rham Stacks 287.2. Approach through Crystalline Stack and PD-morphisms 298. Perfectization and Fontainisation of ∞ -Analytic Stacks Situation 29 Version: Feb 21 2021. Keywords and Phrases: Intrinsic Morphisms, Topologization, Functional Analysis of Locally Convexspaces, Geometric Grothendieck Sites, Stackization. .1. Perfectization, Fontainisation and Crystalline Stacks 298.2. Perfectization, Fontainisation and Robba Stacks 30Acknowledgements 32References 331. Introduction
Main Consideration.
Scholze’s diamond is actually very general notion beyond thecorresponding perfectoid spaces, partially because it contains the corresponding diamantinespaces after Hansen-Kedlaya. This point of view certainly gives the motivation for this no-tion from Hansen-Kedlaya. Therefore one could regard to some extent diamantine spaces asgiving some (maybe better to say more ring theoretic) analogs of the corresponding Scholze’sdiamonds, moreover they behave as if they are perfectoid spaces. Similar discussion couldbe made to suosperfectoid space, which should be more ’perfectoid’ generalization.Hansen-Kedlaya [HK] have given the definition of naive étale morphisms among any TateHuber pairs namely these are locally composites of the rational localizations and finite étalemorphisms (very importantly with strongly sheafy domains and targets). This is becauseone definitely believes that correct notion of étale morphism should admit such admissibledecomposition and factorization. However what should be the correct intrinsic one has notbeen given in full detail yet. In the significant strongly sheafy situation, we are going to tryto answer this question as proposed in [Ked1, Appendix 5]. In this paper, we try to studythe corresponding properties of the corresponding naive étale morphisms along the ideas of[Ked1, Appendix 5] and [HK]. The goal is to accurately characterize the corresponding naiveétale morphisms in some intrinsic way.Sheafiness plays a very crucial role in the discussion above. However suppose we do nothave to worry about the sheafiness at all (in fact in some sense we really do not have toworry about this at all by the work of Clausen-Scholze [CS] and Bambozzi-Kremnizer [BK]),then one might want to believe that the robust definitions could be made even more robust.Therefore we investigate the corresponding morphisms of the corresponding ring objectswhere sheafiness could be replaced by ∞ -sheafiness (namely sheafiness up to higher homo-topy) after [BBBK] and [BK]. One should be able to consider Clausen-Scholze’s foundation[CS] as well, however we will mainly focus on the ∞ -locally convex objects in [BBBK] andBK], as in the corresponding schematic situation in [Lu1], [Lu2], [TV1] and [TV2]. Weconsider the corresponding interesting approaches through the corresponding formal andPD completions just as in the ∞ -schematic situation in [R] which is very related to thecorresponding Drinfeld’s stacky construction [Dr1] and [Dr2] by using the Čech-Alexandercomplex on the corresponding crystalline cohomology and prismatic cohomology.The current list of definitions will be established for discrete E ∞ -ring objects and E ∞ -ringobjects in suitable locally convex ∞ -categories after after Bambozzi-Ben-Bassat-Kremnizer[BBBK]:D1. Localized intrinsic étale morphisms of open mapping Huber rings;D2. Localized intrinsic étale morphisms of open mapping adic Banach rings;D3. Localized intrinsic lisse morphisms of open mapping Huber rings;D4. Localized intrinsic lisse morphisms of open mapping adic Banach rings;D5. Localized intrinsic non-ramifié morphisms of open mapping Huber rings;D6. Localized intrinsic non-ramifié morphisms of open mapping adic Banach rings; ∞
1. De Rham intrinsic étale-like morphisms of ∞ -analytic functors; ∞
2. De Rham intrinsic lisse-like morphisms of ∞ -analytic functors; ∞
3. De Rham intrinsic non-ramifié-like morphisms of ∞ -analytic functors; ∞
4. PD (crystalline) intrinsic étale-like morphisms of ∞ -analytic functors; ∞
5. PD (crystalline) intrinsic lisse-like morphisms of ∞ -analytic functors; ∞
6. PD (crystalline) intrinsic non-ramifié-like morphisms of ∞ -analytic functors.Certainly for general locally convex spaces producing nice ring structures we really haveto be very precise and accurate in any sorts of characterization. However we have not unfor-tunately achieve this due to some very subtle issues, mainly coming from the correspondingissues in very general functional analytification. That being otherwise all said, we still actu-ally could literally talk about the desired definitions for simplicial noetherian Banach ringsin certain situations.1.2. Further Consideration.
Our ultimate goal is certainly to study the correspondinggeometric sites (étale, pro-étale, crystalline and prismatic [SGAIV],[Gro1],[Sch1],[KL1],[KL2],[BS],[Dr1],[Dr2]) and the corresponding cohomologies (étale, pro-étale, crystalline and prismatic)for really general ∞ -analytic spaces (possibly also noncommutative analogs of those in [KR1])ver F and try to apply to the locally noetherian situations, the strongly noetherian situa-tions (such as in [G1], [GL]), the strongly sheafy situations under the foundation of ∞ -locallyconvex spaces (as in [HK], [KL1] and more general situations), although our very beginningcorresponding motivation for this article is an attempt to answer some questions in [Ked1,Appendix A5]. 2. Affinoid Morphisms of Huber Rings
We start with the discussion on the corresponding intrinsic definition of étale morphisms.
Setting 2.1.
We start with an analytic uniform Huber pair ( A, A + ) . And we will considerthe category of all such rings. We assume the corresponding completeness for the Huberpairs. Definition 2.2. (Hansen-Kedlaya [HK, Definition 5.1] ) We call a map of Huber rings ( A, A + ) → ( B, B + ) naive étale after [HK, Definition 5.1] if it admit a factorization intorational localizations and finite étale morphisms. Here we assume ( A, A + ) is strong sheafyand we assume that ( B, B + ) is strongly sheafy. Definition 2.3. (Kedlaya [Ked1, Definition A5.2] ) Recall from [Ked1, Definition A5.2], wehave the corresponding affinoid morphism from any strongly sheafy Huber ring A , namelya morphism A → B , such that B admits some surjective covering from A h T , ..., T d i andthrough this map we have that B is a stably-pseudocoherent sheaf over A h T , ..., T d i and thecorresponding ring B is assumed to be sheafy .The belief (as proposed in [Ked1, Problem A5.3, Problem A5.4]) is that somehow the cor-responding affinoid morphisms in the definition should be directly used in the correspondingdefinitions of lisse morphisms and unramified morphisms, as well as certainly the étale mor-phisms. To investigate this kind of idea, we are going to first investigate the corresponding Certainly one needs to be careful since we are now considering more general context than [HK] withoutassuming the corresponding Tateness. Cetainly one needs to be more careful since this is also slightly different from the original definition[Ked1, Definition A5.2], thanks Professor Kedlaya for telling me this should be better. We want to mentionthat this is a quite subtle point around the sheafiness (see [Ked1, Theorem 1.4.20]), the point here is thatwe do not know the kernel of an affinoid morphism is closed or not, if it is closed then we could keep theknowledge that B being stably-pseudocoherent is equivalent to B being sheafy. However if this is not closed,then we do not have this sort of equivalence to our knowledge. aive étale morphisms along this idea. Lemma 2.4.
Let f : Γ → Γ and f : Γ → Γ be two affinoid morphisms, then thecomposition f ◦ f is also affinoid.Proof. Straightforward. (cid:3)
Lemma 2.5. (Kedlaya)
For any standard binary rational localization of A with respect to f, g ∈ A , suppose we know that there are two surjective morphisms: s : A (cid:28) fg (cid:29) h T , ..., T n i → B (cid:28) fg (cid:29) , (2.1) s : A (cid:28) gf (cid:29) h T , ..., T n i → B (cid:28) gf (cid:29) . (2.2) Then we have that there is a surjective morphism: s : A h T , ..., T n ′ i → B. (2.3) Proof.
The following argument is due to Kedlaya , we work out it for the convenience ofthe readers. First, we have the following short exact sequence: / / / / / / B / / / / / / B D fg E L B D gf E / / / / / / B D fg , gf E / / / / / / . Take any b ∈ B , and use the notation ( b , b ) for the image in the middle. By the surjectivityof the maps s , s we have that there exist some element a ∈ A D fg E h T , ..., T n i and someelement a ∈ A D fg E h T , ..., T n i such that we have: s ( a ) = b , (2.4) s ( a ) = b . (2.5)With more explicit expression we have the following: s ( X i ,...,i n X i a i,i ,...,i n u i T i ...T i n n ) = X i b i u i , (2.6) s ( X i ,...,i n X i a i,i ,...,i n v i T i ...T i n n ) = X i b i v i , (2.7) Thanks Professor Kedlaya for mentioning the similarity of this to the corresponding locality of morphismsof finite type as in Grothendieck’s EGA I and II. nder the corresponding presentations up to liftings: B (cid:28) fg (cid:29) = B h u i / ( gu − f ) , (2.8) B (cid:28) gf (cid:29) = B h v i / ( f v − g ) . (2.9)(2.10)Then to finish we only have to take some finite sum in the summation to make approxima-tion. We first claim that such finite sum approximation and modification will not changethe corresponding surjectivity of the map s and s . Namely for each k = 1 , the map s k will maintain surjective once we modify the image of T , ..., T n infinitesimally aroundsome neighbourhood U of , in other words it will maintain to be surjective even if we set s k ( T ) , ..., s k ( T n ) to be x , ..., x n whenever x − s k ( T ) , ..., x n − s k ( T n ) lives in the neighbour-hood U , and moreover we have that the corresponding modification could be assumed totake T i to x i with i = 1 , ..., n . By open mapping, we have that the corresponding lifts ofthe corresponding differences x − s k ( T ) , ..., x n − s k ( T n ) could be made to be living in somearbitrarily chosen neighbourhood V of . Then we only have to consider the following mapfactoring through the corresponding map s k : h : A k h T , ..., T n i → A k h T , ..., T n i (2.11) T i T i + lifts of x i − s k ( T k ) (2.12)where A is the ring A D fg E while we have A is the ring A D gf E , which basically proves theclaim. Then this will indicate that one can find some joint finite subset T := { T , ..., T n ′ } for B D fg E and B D fg E such that the modified s : A (cid:28) fg (cid:29) h T , ..., T n ′ i → B (cid:28) fg (cid:29) , (2.13) s : A (cid:28) gf (cid:29) h T , ..., T n ′ i → B (cid:28) gf (cid:29) , (2.14)re basically surjective and they fit into the following commutative diagram: / / / / / / A h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg E h T , ..., T n ′ i L A D gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg , gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / B / / / / / / B D fg E L B D gf E / / / / / / B D fg , gf E / / / / / / , where the middle and the rightmost vertical arrows are surjective. Then claim is thenthat the left vertical one is also surjective. The kernels K ⊕ K in the middle is mappedsurjectively to the kernel K of the rightmost vertical map. So the snake lemma will forcethe cokernel of the left vertical arrow to be zero which shows the corresponding exactness atthe corresponding location ? in 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) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K L K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / A h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg E h T , ..., T n ′ i L A D gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg , gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / B ? (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / B D fg E L B D gf E (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / B D fg , gf E (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / /
00 0 0 where K , K , K are pseudocoherent, which implies that the corresponding module K isalso pseudocoherent. (cid:3) roposition 2.6. Any naive étale morphism is affinoid.Proof.
First let f : ( A, A + ) → ( B, B + ) be any naive étale morphism. Then locally thisis basically composition of the corresponding rational localizations and finite étale maps.Locally rational localizations involved are actually affinoid, and locally the correspondingfinite étale maps from strongly sheafy rings will have strongly sheafy target, which willimply that locally finite étale maps are affinoid. Then this could be globalized to force theglobal map f to be affinoid. The properties of factoring through a surjection globally couldbe proved by glueing local ones through lemma 3.6, by considering [KL1, Proposition 2.4.20].And globally the corresponding ring B is stably-pseudocoherent over A h T , ..., T n i for some n since this is a local property. (cid:3) Therefore we have proved that the corresponding étale maps in the corresponding naivesense is actually affinoid in the above sense. Therefore it is now natural to try to find thecorresponding properties which may completely characterize the corresponding naive étalemorphisms which are affinoid.Certainly we may have the corresponding conjectures that all the naive étale morphismswill satisfy the corresponding properties of algebraically étale ones (such as in [EGAIV4,Chapitre 17], [SP, Tag 00U1]). We now discuss the corresponding completed cotangentcomplex after Huber [Hu1, 1.6.2]. Recall for our current B the corresponding completed dif-ferential Ω B/A, topo (see [Hu1, 1.6.2] for the construction for any f -adic rings in the noetheriansetting). Therefore we consider the corresponding topological naive cotangent complex: τ ≤ L B/A, topo (2.15)for any naive étale map f : ( A, A + ) → ( B, B + ) . We now discuss the construction withoutthe corresponding strongly noetherian requirement in our current situation. First we knowthat B is of topologically finite type over A : B = A h X , ..., X n i T ,...,T n /I. (2.16)Then we could first define the topological free differentials: Ω := A h X , ..., X n i T ,...,T n dX + ... + A h X , ..., X n i T ,...,T n dX n . (2.17)Then we have: Ω B/A, topo := Ω / ( I [ d ( I ))Ω . (2.18)Here everything is assumed to be basically complete with respect to the correspondingnatural topology. Namely we need to take the corresponding completion always with respecto the corresponding induced topology. Certainly here Ω is already complete due to thefact that it is finitely projective. Recall that a map f : Γ → Γ is called étale in the schemetheory if the naive cotangent complex (truncated and could be regarded as an ∞ -modulespectrum) is quasi-isomorphic to zero. The corresponding underlying complex reads: [ I/I / / / / / / Ω / Γ , topo ] . In the situation where we consider A → B is affinoid, the corresponding ideal I is actuallystably-pseudocoherent over A h X , ..., X n i . It is peudocoherent by the corresponding two outof three property. The stability holds locally, so we have the case. And if the morphism iffurthermore naive étale then we have I/I is also stably-pseudocoherent, see lemma 4.8. Remark 2.7.
Note that we are considering the very general and complicated non-noetheriansituation, modules will need to be endowed with the natural topology and complete, althoughfinite projective modules are complete automatically. This will have nontrivial things to dowith the corresponding definition of Ω B/A, topo .One can actually generalize the corresponding full cotangent complexes and derived deRham complexes to this topological context following [III1], [III2] and [B1]. First for thecorresponding topological cotangent complex we consider the following definition (note thatwe have to assume the corresponding topologically finite type condition). We start with thecorresponding algebraic ones for B h = A [ X , ..., X n ] T ,...,T n /I , under the topologization wehave the corresponding derived cotangent complex: L B h /A, alg , (2.19)by taking the usual algebraic one. Then we take the corresponding completion with respectto the corresponding topologization which gives rise to the following topological one: L B/A, topo . (2.20)We define the corresponding de Rham complex in the following parallel way. What ishappen is that consider the presentation B h = A [ X , ..., X n ] T ,...,T n /I which gives rise to thecorresponding algebraic de Rham complex: / / / / / / B h / / / / / / Ω B h /A, alg / / / / / / Ω B h /A, alg / / / / / / ... / / / / / / Ω • B h /A, alg / / / / / / ..., which will give rise to the corresponding topological one if we take the corresponding com-pletion induced from the subset T , ..., T n : / / / / / / B / / / / / / Ω B h /A, topo / / / / / / Ω B h /A, topo / / / / / / ... / / / / / / Ω • B h /A, topo / / / / / / .... rom our construction for Ω B h /A, topo , one can actually define: Ω • , f B h /A, topo := M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • (2.21)and then define: Ω • B/A, topo := M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • / (2.22) ( I [ dI [ d • I ) M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • , (2.23)after taking suitable completion when needed .3. Affinoid Morphisms of Banach Rings
We now consider the parallel situation of Banach rings.
Setting 3.1.
We start with a uniform adic Banach ring ( A, A + ) in the general sense of [KL1]and [KL2] (without assumption on the topologically nilpotent units being existing, but weassume this is open mapping). And we will consider the category of all such rings. Weassume the corresponding completeness as well. Definition 3.2. (Hansen-Kedlaya [HK, Definition 5.1] ) We call a map of adic Banachrings ( A, A + ) → ( B, B + ) naive étale after [HK, Definition 5.1] if it admit a factorization intorational localizations and finite étale morphisms. Here we assume ( A, A + ) is strong sheafyand we assume that ( B, B + ) is sheafy. Remark 3.3.
Certainly this is in more general situation than the corresponding context of[HK].
Definition 3.4. (Kedlaya [Ked1, Definition A5.2] ) Recall from [Ked1, Definition A5.2],we have the corresponding affinoid morphism from any strongly sheafy adic Banach ring A ,namely a morphism A → B , such that B admits some surjective covering from A h T , ..., T d i and through this map we have that B is a stably-pseudocoherent sheaf over A h T , ..., T d i and we assume that ( B, B + ) is strongly sheafy. As in [B1] and [GL] where one takes the corresponding derived p -completion out from the algebraiccotangent complex and the corresponding derived algebraic de Rham complex. emark 3.5. Of course the corresponding foundation is not the same but parallel to suchsituation we are considering now, however it is definitely reasonable and parallel to follow[Ked1, Appendix A5] to give the definition here.The belief (as proposed in [Ked1, Problem A5.3, Problem A5.4]) is that somehow the cor-responding affinoid morphisms in the definition should be directly used in the correspondingdefinitions of lisse morphisms and unramified morphisms, as well as certainly the étale mor-phisms. To investigate this kind of idea, we are going to first investigate the correspondingnaive étale morphisms along this idea.
Lemma 3.6. (Kedlaya)
For any standard binary rational localization of A with respect to f, g ∈ A , suppose we know that there are two surjective morphisms: s : A (cid:28) fg (cid:29) h T , ..., T n i → B (cid:28) fg (cid:29) , (3.1) s : A (cid:28) gf (cid:29) h T , ..., T n i → B (cid:28) gf (cid:29) . (3.2) Then we have that there is a surjective morphism: s : A h T , ..., T n ′ i → B. (3.3) Proof.
The following argument is due to Kedlaya, we work out it for the convenience of thereaders. And this is the corresponding Banach analog of corresponding parallel result in theHuber ring situation. First, we have the following short exact sequence: / / / / / / B / / / / / / B D fg E L B D gf E / / / / / / B D fg , gf E / / / / / / . Take any b ∈ B , and use the notation ( b , b ) for the image in the middle. By the surjectivityof the maps s , s we have that there exist some element a ∈ A D fg E h T , ..., T n i and someelement a ∈ A D fg E h T , ..., T n i such that we have: s ( a ) = b , (3.4) s ( a ) = b . (3.5)ith more explicit expression we have the following: s ( X i ,...,i n X i a i,i ,...,i n u i T i ...T i n n ) = X i b i u i , (3.6) s ( X i ,...,i n X i a i,i ,...,i n v i T i ...T i n n ) = X i b i v i , (3.7)under the corresponding presentations up to liftings: B (cid:28) fg (cid:29) = B h u i / ( gu − f ) , (3.8) B (cid:28) gf (cid:29) = B h v i / ( f v − g ) . (3.9)(3.10)Then to finish we only have to take some finite sum in the summation to make approxima-tion. We first claim that such finite sum approximation and modification will not changethe corresponding surjectivity of the map s and s . Namely for each k = 1 , the map s k will maintain surjective once we modify the image of T , ..., T n infinitesimally aroundsome neighbourhood U of , in other words it will maintain to be surjective even if we set s k ( T ) , ..., s k ( T n ) to be x , ..., x n whenever k x − s k ( T ) k ≤ δ, ..., k x n − s k ( T n ) k ≤ δ for someprescribed constant δ < and moreover we have that the corresponding modification couldbe assumed to take T i to x i with i = 1 , ..., n . By open mapping in this current context, wehave that the corresponding lifts of the corresponding differences x − s k ( T ) , ..., x n − s k ( T n ) could be made to be living in some arbitrarily chosen neighbourhood V of namely, we canfind lifts y , ..., y n of these differences such that k y k < , ..., k y n k < . (3.11)Then we only have to consider the following map factors through the corresponding map s k : h : A k h T , ..., T n i → A k h T , ..., T n i (3.12) T i T i + lifts of x i − s k ( T k ) (3.13)where A is the ring A D fg E while we have A is the ring A D gf E , which basically proves theclaim. Then this will indicate that one can find some joint finite subset T := { T , ..., T n ′ } for D fg E and B D fg E such that the modified s : A (cid:28) fg (cid:29) h T , ..., T n ′ i → B (cid:28) fg (cid:29) , (3.14) s : A (cid:28) gf (cid:29) h T , ..., T n ′ i → B (cid:28) gf (cid:29) , (3.15)are basically surjective and they fit into the following commutative diagram: / / / / / / A h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg E h T , ..., T n ′ i L A D gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg , gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / B / / / / / / B D fg E L B D gf E / / / / / / B D fg , gf E / / / / / / , where the middle and the rightmost vertical arrows are surjective. Then claim is thenthat the left vertical one is also surjective. The kernels K ⊕ K in the middle is mappedsurjectively to the kernel K of the rightmost vertical map. So the snake lemma will forcethe cokernel of the left vertical arrow to be zero which shows the corresponding exactness athe corresponding location ? in 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) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K L K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / K (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / A h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg E h T , ..., T n ′ i L A D gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / A D fg , gf E h T , ..., T n ′ i (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / / / / / / / B ? (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / B D fg E L B D gf E (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / / B D fg , gf E (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / / / / /
00 0 0 where K , K , K are pseudocoherent, which implies that the corresponding module K isalso pseudocoherent. (cid:3) Lemma 3.7.
Let f : Γ → Γ and f : Γ → Γ be two affinoid morphisms, then thecomposition f ◦ f is also affinoid.Proof. Straightforward. (cid:3)
Proposition 3.8.
Any naive étale morphism is affinoid.Proof.
See proposition 3.8. (cid:3)
Therefore as in the corresponding Huber pair situation, we have proved that the corre-sponding étale maps in the corresponding naive sense is actually affinoid in the above sense.Therefore it is now natural to try to find the corresponding properties which may completelycharacterize the corresponding naive étale morphisms which are affinoid.ertainly we may have the corresponding conjectures that all the naive étale morphismswill satisfy the corresponding properties of algebraically étale ones (such as in [EGAIV4,Chapitre 17], [SP, Tag 00U1]). We now discuss the corresponding Banach completed cotan-gent complex after Huber [Hu1, 1.6.2]. Recall for our current B the corresponding completeddifferential Ω B/A, topo (see [Hu1, 1.6.2] for the construction for any f -adic rings). Certainly inthe context of adic Banach rings we have the parallel completed version of the differentialsby taking Banach completion, which is in some trival way the current situation. Thereforewe consider the corresponding topological naive cotangent complex: τ ≤ L B/A, topo (3.16)for any naive étale map f : ( A, A + ) → ( B, B + ) . We consider the construction without thecorresponding strongly noetherian requirement in our current situation. First we know that B is of topologically finite type over A : B = A h X , ..., X n i T ,...,T n /I. (3.17)Then we could first define the Banach free differentials: Ω := A h X , ..., X n i T ,...,T n dX + ... + A h X , ..., X n i T ,...,T n dX n . (3.18)Then we have: Ω B/A, topo := Ω / ( I [ d ( I ))Ω . (3.19)Here everything is assumed to be basically complete with respect to the correspondingnatural topology. Namely we need to take the corresponding completion always with respectto the corresponding induced norms. Certainly here Ω is already complete due to the factthat it is finitely projective. Recall that a map f : Γ → Γ is called étale in the schemetheory if the naive cotangent complex (truncated and could be regarded as an ∞ -modulespectrum) is quasi-isomorphic to zero. The corresponding underlying complex reads: [ I/I / / / / / / Ω / Γ , topo ] . In the situation where we consider A → B is affinoid, the corresponding ideal I is actuallystably-pseudocoherent over A h T , ..., T n i . It is peudocoherent by the corresponding two outof three property. The stability holds locally, so we have the case. And if the morphism iffurthermore naive étale then we have I/I is also stably-pseudocoherent, see lemma 4.8. Remark 3.9.
Note that we are considering the very general and complicated non-noetheriansituation, modules will need to be endowed with the natural topology coming from theanach structures on the Banach rings and complete, although in this situation as well finiteprojective modules are complete automatically. This will have nontrivial things to do withthe corresponding definition of Ω B/A, topo .In the Banach world, one can actually generalize the corresponding full cotangent com-plexes and de Rham complex to this context. First for the corresponding topological cotan-gent complex we consider the following definition (note that we have to assume the corre-sponding topologically finite type condition). We start with the corresponding algebraic onesfor B h = A [ X , ..., X n ] T ,...,T n /I , under the topologization we have the corresponding derivedcotangent complex: L B h /A, alg , (3.20)by taking the usual algebraic one. Then we take the corresponding completion with respectto the corresponding topologization which gives rise to the following topological one: L B/A, topo . (3.21)We define the corresponding de Rham complex in the following way parallely. What ishappen is that consider the presentation B h = A [ X , ..., X n ] T ,...,T n /I which gives rise to thecorresponding algebraic de Rham complex: / / / / / / B h / / / / / / Ω B h /A, alg / / / / / / Ω B h /A, alg / / / / / / ... / / / / / / Ω • B h /A, alg / / / / / / ..., which will give rise to the corresponding topological one if we take the corresponding com-pletion under ( k . k , Ban) induced from the subset T , ..., T n : / / / / / / B h k . k , Ban / / / / / / Ω B h /A, alg , k . k , Ban / / / / / / Ω B h /A, alg , k . k , Ban / / / / / / ... / / / / / / Ω • B h /A, alg , k . k , Ban / / / / / / ..., / / / / / / B / / / / / / Ω B h /A, topo / / / / / / Ω B h /A, topo / / / / / / ... / / / / / / Ω • B h /A, topo / / / / / / .... From our construction for Ω B h /A, topo , one can actually define: Ω • , f B h /A, topo := M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • (3.22)nd then define: Ω • B/A, topo := M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • / (3.23) ( I [ dI [ d • I ) M i ,...,i • ∈{ ,...,n } A h X , ..., X n i T ,...,T n dX i ∧ dX i ∧ ... ∧ dX i • , (3.24)after taking suitable completion under Banach norms when needed. One can also followthe construction in [III1], [III2], [B1] and [GL] to first consider the corresponding polyno-mial resolution P • for B , then consider the corresponding algebraic derived de Rham complex Ω ∗ P • /A, alg , then take the corresponding Banach completion to produce the corresponding topo-logical one Ω ∗ P • /A, topo . Then as in [III1], [III2], [GL] and [B1] around analytic derived p -adicde Rham complex we can take the corresponding suitable derived filtered completion to getthe complex b Ω • B/A, topo (certainly we need to consider Banach version of some filtered derivedcategory of simplicial Banach rings). In the pro-étale site theoretic setting for rigid spaces(namely the corresponding p-complete context) this recovers the corresponding constructionin [GL]. Recall in [GL] the corresponding analytic derived p -adic de Rham complex is con-structed by first define the integral version Ω • B /A , topo , and then take the colimit throughoutall such rings of definition, and invert p , and then take the filtered completion. Remark 3.10.
Certainly after this previous discussion we can construct the correspondingBanach derived de Rham complex, Banach cotangent complex and Banach André-Quillenhomology for any morphism A → B of Banach rings admissible in our situation. Recall from[III1], [III2], [B1] we have the corresponding algebraic p -adic derived cotangent complex: Ω A [ B ] • /A ⊗ A [ B ] • B (3.25)where A [ B ] • is just the corresponding standard cofibrant replacement (in the topology the-oretic language) of B/A . Then we take the corresponding derived completion under theinduced norm to achieve the corresponding topological one: L B/A, topo := (Ω A [ B ] • /A ⊗ A [ B ] • B ) ∧k . k . (3.26)We have the corresponding algebraic p -adic derived de Rham complex: Ω • A [ B ] • /A (3.27) The derived completion in our Banach situation is the derived Banach completion which for instance couldhappen by using the ’completion’ functor from
Simp(Ind(NormSets)) to Simp(Ind(BanachSets)) literally in[BBBK]. here A [ B ] • is just the corresponding standard cofibrant replacement (in the topology the-oretic language) of B/A . Then we take the corresponding completion under the inducednorm to achieve the corresponding topological one: dR B/A, topo := (Ω • A [ B ] • /A ) ∧k . k , (3.28)carrying certain filtration Fil ∗ dR B/A, topo , which allows one take the corresponding filtered com-pletion to achieve the corresponding final object: dR ∧ B/A, topo := (Ω • A [ B ] • /A ) ∧k . k , Fil ∗ dR B/A, topo . (3.29) 4. Naive Étale Morphisms and Intrinsic Étale Morphisms
As discussed above we now study the corresponding properties of naive étale morphismsaiming at the corresponding characterization of the corresponding correct definitions of in-trinsic étale morphisms. We now assume the corresponding analyticity of the adic rings.
Lemma 4.1.
Let f : ( A, A + ) → ( B, B + ) be any rational localization map. Then we havethat B is of topologically finite type.Proof. Straightforward. (cid:3)
Lemma 4.2.
Let f : ( A, A + ) → ( B, B + ) be any rational localization map. Then we havethat B is affinoid over A in the sense definition 3.4.Proof. See the proof of proposition 3.8. (cid:3)
Lemma 4.3.
Let f : ( A, A + ) → ( B, B + ) be any rational localization map. Then we have τ ≤ L B/A, topo (4.1) is quasi-isomorphic to zero.Proof.
It suffices to reduce to standard binary localization such as simple Laurent or balancedlocalization, where we give a proof in the case of simple Laurent one: A → A { T } / ( T − f ) (4.2)for some f ∈ A . Then we have that actually the corresponding topological cotangent com-plex will be the corresponding completion of the corresponding algebraic ones (see [Hu1,roposition 1.6.3]). The corresponding quasi-isomorphism could be defined directly. Onejust considers the following algebraic differential map: I = ( T − f ) → A [ T ] / ( T − f ) dT (4.3)(4.4)which is actually surjective since for any: X i ≥ a i T i + g ( T )( T − f ) dT ∈ A [ T ] / ( T − f ) dT (4.5)one takes the corresponding integration of: Z Tf X i ≥ a i T i + g ( T )( T − f ) dT = Z Tf X i ≥ a i T i + ( X i ≥ g i T i )( T − f ) dT (4.6) = Z Tf X i ≥ a i T i + Z Tf ( X i ≥ g i T i )( T − f ) dT (4.7) = Z Tf X i ≥ a i T i + Z Tf X i ≥ g i T i +1 − f Z Tf X i ≥ g i T i (4.8) = X i ≥ a i i + 1 T i +1 | Tf + X i ≥ g i i + 2 T i +2 | Tf − f X i ≥ g i i + 1 T i +1 | Tf (4.9) = ( ∗ )( T − f ) , (4.10)where we only have finite sums here since we are considering the corresponding topologizedpolynomial. For this algebraic map, we have that kernel is ( T − f ) , for instance consider: d (( T − f ) h ( T )) = 0 , (4.11)we will have: h ( T ) + ( T − f ) h ′ ( T ) dT = 0 , (4.12)which implies that the image of ( T − f ) h ( T ) lives in the corresponding quotient A [ T ] / ( T − f ) dT . Therefore we have that the topologized (not complete yet) cotangent complex: [( T − f ) / ( T − f ) / / / / / / A [ T ] / ( T − f ) dT ] , which is quasi-isomorphic to zero. Then we take the corresponding completion with respectto the corresponding topology induced from A we have the desired result. (cid:3) emma 4.4. Let f : ( A, A + ) → ( B, B + ) be any finite étale map. Then we have that B isof topologically finite type.Proof. Since we have that B is affinoid over A by proposition 3.8. (cid:3) Lemma 4.5.
Let f : ( A, A + ) → ( B, B + ) be any finite étale map. Then we have that B isaffinoid over A in the sense definition 3.4.Proof. See the proof of proposition 3.8. (cid:3)
Lemma 4.6.
Let f : ( A, A + ) → ( B, B + ) be any finite étale map. Then we have τ ≤ L B/A, topo (4.13) is quasi-isomorphic to zero.Proof.
This is basically nontrivial due to the fact that we are discussing the correspondingtopological cotangent complex. However, one takes the corresponding finite presentation B = A [ T , ..., T n ] / ( f , ..., f p ) (note that in fact that we have that B is finite over A ) sincewe are considering a finite étale map, which will realize the corresponding desired algebraiccotangent complex: [ I/I / / / / / / Ω B/A ] . For the topological situation we have that B = A { T , ..., T n } / ( f , ..., f p ) by taking the cor-responding completion. Again note that this means that actually the corresponding A -algebra B is still finite over A (since that is the very assumption). Therefore we couldhave the chance to right B as just ⊕ i Ae i this is basically inducing the same differentialmodule L j ( L i Ae i ) dT j in both the corresponding topological setting and the algebraic set-ting. Then one could get the corresponding desired topological cotangent complex which isquasi-isomorphic to zero. (cid:3) Then we consider the corresponding local composition:
Lemma 4.7.
Let f : ( A, A + ) → ( B, B + ) be any naive étale map. Then we have that B isof topologically finite type.Proof. Since we have that B is affinoid over A by proposition 3.8. (cid:3) Lemma 4.8.
Let f : ( A, A + ) → ( B, B + ) be any naive étale morphism. Then we have τ ≤ L B/A, topo (4.14) is quasi-isomorphic to zero locally with respect to the corresponding rational localization.roof.
Locally we have that any naive étale morphism takes the corresponding truncatedcotangent complex to be trivialized, by the corresponding composition properties of thecotangent complex [SP, Tag 08PN]. (cid:3)
In order to globalize the picture one has to work harder. First we have the following:
Proposition 4.9.
Let f : ( A, A + ) → ( B, B + ) be any naive étale morphism. Then f isaffinoid, of topologically finite type.Proof. This is by proposition 3.8 for the affinoidness, which implies the corresponding secondproperty. (cid:3)
Definition 4.10.
We now define localized intrinsic étale morphism to be a morphism f : ( A, A + ) → ( B, B + ) which is affinoid with strongly sheafy target, and locally (with respectto the rational localization) the corresponding truncated topological cotangent complex isquasi-isomorphic to zero.We now consider the corresponding intrinsic étale morphisms of the corresponding specialadic spaces after [HK]: Setting 4.11.
We now consider the three special adic spaces after [HK], they are the corre-sponding strongly sheafy adic spaces, the corresponding sousperfectoid adic spaces and thecorresponding diamantine adic spaces. We will use the notations
T, S, D to denote them ingeneral respectively.The corresponding categories of strongly sheafy adic spaces, sousperfectoid spaces anddiamantine adic spaces are nice enough since at least we have well-defined notion of naiveétale morphisms (which is certainly the correct one) and furthermore well-defined étale sites.
Definition 4.12.
For strongly sheafy adic spaces, a morphism T → T is called localizedintrinsic étale if locally on T this is localized intrinsic étale, namely for any neighbourhood U ⊂ T we have that the morphism ( O T ( U ′ ) , O + T ( U ′ )) → ( O T ( U ) , O + T ( U )) is localizedintrinsic étale. Definition 4.13.
For sousperfectoid adic spaces, a morphism S → S is called localizedintrinsic étale if locally on S this is localized intrinsic étale, namely for any neighbourhood U ⊂ S we have that the morphism ( O S ( U ′ ) , O + S ( U ′ )) → ( O S ( U ) , O + S ( U )) is localizedintrinsic étale. However this could actually be globalized easily. efinition 4.14.
For diamantine adic spaces, a morphism D → D is called localizedintrinsic étale if locally on D this is localized intrinsic étale, namely for any neighbourhood U ⊂ D we have that the morphism ( O D ( U ′ ) , O + D ( U ′ )) → ( O D ( U ) , O + D ( U )) is localizedintrinsic étale. 5. Properties
We now study the corresponding properties of the corresponding localized intrinsic étalemorphisms, following [EGAIV4] and [Hu1]. We now assume the corresponding analyticityof the adic rings.
Conjecture 5.1.
Any localized intrinsic étale morphism of strongly sheafy adic spaces islocally a composition of rational localization and finite étale morphism.
Here is the special situation.
Proposition 5.2.
As in [Hu1] , namely in the strongly noetherian situation we have theconjecture holds.Proof.
This is because in that setting our definition in the intrinsic setting coincides with themore algebraic one in [Hu1]. And note that in this setting the affinoidness of the morphismreduces to just being admitting surjections from Tate algebra over the source. (cid:3)
If this is true then we have:
Corollary 5.3.
Any localized intrinsic étale morphism of sousperfectoid adic spaces is locallya composition of rational localization and finite étale morphism. Any localized intrinsic étalemorphism of diamantine adic spaces is locally a composition of rational localization and finiteétale morphism.
Proposition 5.4.
Compositions of localized intrinsic étale morphisms of strongly sheafy adicspaces are again localized intrinsic étale morphism.Proof.
Locally it is the corresponding compositions of topologically finite type morphism,and locally it is the corresponding compositions of the corresponding affinoid morphisms, andlocally it is the corresponding compositions of morphisms giving rise to the quasi-isomorphicto zero truncated cotangent complex. (cid:3) orollary 5.5.
Compositions of localized intrinsic étale morphisms of sousperfectoid adicspaces are again localized intrinsic étale morphism. Compositions of localized intrinsic étalemorphisms of diamantine adic spaces are again localized intrinsic étale morphism.
Proposition 5.6.
The localized intrinsic étaleness of any morphism T → T of stronglysheafy adic rings is preserved under the base change along any morphism of T → T .Proof. The base change of any morphism of topologically finite type is again of topologicallyfinite type. The affinoidness of morphism is also preserved under any base change morphism.Finally for the cotangent complex locally, we definitely have the corresponding result aswell. (cid:3)
Proposition 5.7.
The étale property of a morphism between strongly sheafy rings could bedetected locally at each point.Proof.
Straightforward. (cid:3)
We now consider some functoriality issue in our current situation. Now we consider thecorresponding localized intrinsic étale morphisms under the construction of Witt vectors.Now let: A → B (5.1)be a general morphism in positive characteristic. Therefore we can take the correspondingWitt vector construction: W ( A ♭ ) → W ( B ♭ ) , (5.2)where we assume that A ♭ → B ♭ is localized intrinsic étale. Here we take the completion ifneeded along the corresponding Fontainisation. Proposition 5.8.
The map W ( A ♭ ) → W ( B ♭ ) , (5.3) is affinoid if the kernel is closed .Proof. We only need to check this locally, locally we have that there is a lifting: W ( A ♭ ) { T , ... } → W ( B ♭ ) → (5.4) This is again due to the very subtle point around the sheafiness such as in [Ked1, Theorem 1.4.20]. rom the corresponding surjection: A ♭ { T , ... } → B ♭ → . (5.5)And what we have is that this map on the Witt vector level is also realizing the target as astably-pseudocoherent module over the source since we have that the target is sheafy ([Ked1,Theorem 1.4.20]). (cid:3) Proposition 5.9.
Same holds for the construction of integral Robba ring e R r ∗ and Robba ring e R [ s,r ] ∗ with respect to closed intervals as in [KL1] and [KL2] . Étale-Like Morphisms of ∞ -Banach Rings and the ∞ -Analytic Stacks Approach through De Rham Stacks.
We now extend the corresponding discus-sion to the E ∞ objects in [BBBK, Remark 3.16] by using the ideas as in [R]. Recallfrom [BBBK, Theorem 3.14] we have the corresponding categories SimpInd m (BanSets H ) and SimpInd(BanSets H ) which are the corresponding categories of the corresponding simpli-cial sets over the corresponding inductive categories of the corresponding Banach sets oversome Banach ring H . Theorem 6.1. (Bambozzi-Ben-Bassat-Kremnizer)
The corresponding categories
SimpInd m (BanSets H ) and SimpInd(BanSets H ) admit symmetric monoidal model categoricalstructure. Same holds for SimpInd m (NrSets H ) and SimpInd(NrSets H ) . Corollary 6.2.
The corresponding categories
SimpInd m (BanSets H ) and SimpInd(BanSets H ) admit presentations as ( ∞ , -categories. Same holds for SimpInd m (NrSets H ) and SimpInd(NrSets H ) . Then recall from [BBBK, Remark 3.16] we have the corresponding ring objects in the ∞ -categories above: sComm(SimpInd m (BanSets H )) , (6.1) sComm(SimpInd(BanSets H )) . (6.2)and sComm(SimpInd m (NrSets H )) , (6.3) sComm(SimpInd(NrSets H )) . (6.4) It is safer to assume the open mapping properties on homotopy groups. ow we use general notation A to denote any object in these categories, regarding as ageneral E ∞ -ring. We consider the general morphism A → B in the first two categories inthe following discussion. Definition 6.3.
For any general morphism A → B , we call this affinoid if we have that that π ( B ) is affinoid over π ( A ) , namely we have that there is a surjection map π ( A ) h X , ..., X d i → π ( B ) . And moreover we assume that π ( B ) ⊗ π ( A ) π n ( A ) ∼ → π n ( B ) , for any n . Remark 6.4.
Kedlaya’s theorem [Ked1, Theorem 1.4.20] is actually expected to hold in moregeneral setting, at least in the situation where the definition of the affinoid morphisms couldbe made independent from the corresponding stably-pseudocoherence for open mapping rings(note that we are working over analytic fields). However in the previous definition, we havebeen not really exact. To be really accurate in the characterization of some desired notionof the affinoidness we think that one has to add certain ∞ -sheafiness (which certainly holdsin [BK]). To be more precise for any general morphism A → B in [BK] (namely in currentsituation one considers the corresponding Banach algebras over the analytic fields in oursituation), we call this affinoid if we have that that π ( B ) is affinoid over π ( A ) , namely wehave that there is a surjection map π ( A ) h X , ..., X d i → π ( B ) . And moreover we assumethat π ( B ) ⊗ π ( A ) π n ( A ) ∼ → π n ( B ) , for any n . In this situation we have the nice sheafiness(up to higher homotopy). Again similar discussion could be made in the context of [CS].Note that [Ked1, Theorem 1.4.20] literally says that the sheafiness is equivalent (in somenice sense but in more flexible derived sense) to the stably-pseudocoherence. Definition 6.5.
For any general morphism A → B , we call this localized intrinsic étale ifwe have that that π ( B ) is localized intrinsic étale over π ( A ) , namely we have that thereis a surjection map π ( A ) h X , ..., X d i → π ( B ) and we have that locally the correspondingtruncated topological cotangent complex is basically quasi-isomorphic to zero. And moreoverwe assume that π ( B ) ⊗ π ( A ) π n ( A ) ∼ → π n ( B ) , for any n .We now use the corresponding X = Spec A to denote the corresponding ∞ -stack in theopposite categories with respect to the ring A . We now define the corresponding de Rhamstack attached to X as in [R, Remark 1.2]: Definition 6.6.
We now define: X dR ( R ) := lim −→ I X ( π ( R ) /I ) (6.5)or any R in sComm(SimpInd m (BanSets H )) , (6.6) sComm(SimpInd(BanSets H )) . (6.7)And the injective limit is taking throughout all nilpotent ideals of π ( R ) . Remark 6.7.
As in [R, Definition 1.1, Remark 1.2], one can actually define the correspond-ing de Rham and crystalline spaces for any functor from sComm(SimpInd m (BanSets H ))) and sComm(SimpInd(BanSets H ))) to s Sets . This means we do not have to consider ( ∞ , -sheaves satisfying certain ∞ -descent conditions. Definition 6.8.
The corresponding formal completion of any morphism X = Spec B → Y =Spec A : Y X, dR (6.8)is defined to be: Y X, dR ( R ) := lim −→ I X ( π ( R ) /I ) × Y ( π ( R ) /I ) Y ( R ) , (6.9)for any R in sComm(SimpInd m (BanSets Q p )) , (6.10) sComm(SimpInd(BanSets Q p )) . (6.11)And the injective limit is taking throughout all nilpotent ideals of π ( R ) . Remark 6.9.
Certainly it is actually not clear how really we should deal with the corre-sponding ideals here, namely we are not for sure if we need to consider closed ideals. Butfor simplicial noetherian rings we really have some nice definitions, which will certainly betangential to the corresponding Huber’s original consideration.
Definition 6.10.
We now define the corresponding de Rham intrinsic étale morphism to bean affinoid morphism X = Spec B → Y = Spec A which satisfies the condition: π ( X ( R )) ∼ → π ( Y X, dR ( R )) , (6.12)for any E ∞ -object R . Proposition 6.11.
Compositions of de Rham intrinsic étale morphisms are again PD in-trinsic étale morphisms.Proof. This is formal. (cid:3) .2.
Approach through Crystalline Stack and PD-morphisms.
We now use the cor-responding X = Spec A to denote the corresponding ∞ -stack in the opposite categories withrespect to the ring A . We now define the corresponding crystalline stack attached to X asin [R, Definition 1.1]: Definition 6.12.
We now define: X crys ( R ) := lim −→ I,γ X ( π ( R ) /I ) (6.13)for any R in sComm(SimpInd m (BanSets H )) , (6.14) sComm(SimpInd(BanSets H )) . (6.15)And the injective limit is taking throughout all nilpotent ideals of π ( R ) and the correspond-ing PD-structures. Definition 6.13.
The corresponding PD completion of any morphism X = Spec B → Y =Spec A : Y X, crys (6.16)is defined to be: Y X, crys ( R ) := lim −→ I,γ X ( π ( R ) /I ) ⊗ Y ( π ( R ) /I ) Y ( R ) , (6.17)for any R in sComm(SimpInd m (BanSets H )) , (6.18) sComm(SimpInd(BanSets H )) . (6.19)And the injective limit is taking throughout all nilpotent ideals of π ( R ) and all the corre-sponding PD structures. Remark 6.14.
Certainly it is actually not clear how really we should deal with the cor-responding ideals here and the corresponding PD structures, namely we are not for sure ifwe need to consider closed ideals. But for simplicial noetherian rings we really have somenice definitions, which will certainly be tangential to the corresponding Huber’s originalconsideration. efinition 6.15.
We now define the corresponding PD intrinsic étale morphism to be anaffinoid morphism X = Spec B → Y = Spec A which satisfies the condition: π ( X ( R )) ∼ → π ( Y X, crys ( R )) , (6.20)for any E ∞ -object R . Proposition 6.16.
We have that any de Rham intrinsic étale morphism is a PD intrinsicétale morphism.Proof.
This is formal. (cid:3)
Proposition 6.17.
Compositions of PD intrinsic étale morphisms are again PD intrinsicétale morphisms.Proof. This is formal. (cid:3) Lisse-Like and Non-Ramifié-Like Morphisms of ∞ -Banach Rings and the ∞ -Analytic Stacks Approach through De Rham Stacks.
We now define the corresponding lisse-likemorphisms along the idea in the previous section:
Definition 7.1.
For any general morphism A → B , we call this localized intrinsic lisse ifwe have that that π ( B ) is localized intrinsic lisse over π ( A ) , namely we have that thereis a surjection map π ( A ) h X , ..., X d i → π ( B ) and we have that locally the correspondingtruncated topological cotangent complex is basically quasi-isomorphic to Ω π ( B ) /π ( A ) [0] . Andmoreover we assume that π ( B ) ⊗ π ( A ) π n ( A ) ∼ → π n ( B ) , for any n . Definition 7.2.
We now define the corresponding de Rham intrinsic lisse morphism to bean affinoid morphism X = Spec B → Y = Spec A which satisfies the condition: π ( X ( R )) → π ( Y X, dR ( R )) (7.1)being surjective, for any E ∞ -object R . Definition 7.3.
For any general morphism A → B , we call this localized intrinsic non-ramifié if we have that that π ( B ) is localized intrinsic non-ramifié over π ( A ) , namely wehave that there is a surjection map π ( A ) h X , ..., X d i → π ( B ) and we have that locallythe corresponding truncated topological cotangent complex is basically quasi-isomorphic to π ( B ) /π ( A ) [0] which vanishes as well. And moreover we assume that π ( B ) ⊗ π ( A ) π n ( A ) ∼ → π n ( B ) , for any n .7.2. Approach through Crystalline Stack and PD-morphisms.Definition 7.4.
We now define the corresponding PD intrinsic lisse morphism to be anaffinoid morphism X = Spec B → Y = Spec A which satisfies the condition: π ( X ( R )) → π ( Y X, crys ( R )) (7.2)being surjective, for any E ∞ -object R . Proposition 7.5.
We have that any de Rham intrinsic lisse morphism is a PD intrinsiclisse morphism. Perfectization and Fontainisation of ∞ -Analytic Stacks Situation Perfectization, Fontainisation and Crystalline Stacks.
Now we consider the cor-responding perfectoidization of ∞ -analytic stacks after [R] and [Dr1] in the situation where H is assumed to be of characteristic p . Definition 8.1.
For any object in ∞ −
Fun(sCommSimpInd(Ban H ) , s Sets) , denoted by X ,we define the corresponding perfectization X /p ∞ of X to be the corresponding functor suchthat for any R ∈ sCommSimpInd(Ban H ) we have that X /p ∞ ( R ) := X ( R ♭ ) where we definethe corresponding tilting Fontainisation R ♭ of R to be the corresponding derived completionof: lim ←−{ ... Fro −→ R Fro −→ R Fro −→ R } . (8.1)In the situation of the corresponding monomorphically inductive Banach sets, we have theparallel definition: Definition 8.2.
For any object in ∞ −
Fun(sCommSimpInd m (BanSets H ) , s Sets) , denotedby X , we define the corresponding perfectization X /p ∞ of X to be the corresponding functorsuch that for any ring R ∈ sCommSimpInd m (BanSets H ) we have that X /p ∞ ( R ) := X ( R ♭ ) where we define the corresponding tilting Fontainisation R ♭ of R to be the correspondingderived completion of: lim ←−{ ... Fro −→ R Fro −→ R Fro −→ R } . (8.2) For non-ramifié situation one considers injectivity. emark 8.3.
This is very general notion beyond the corresponding ( ∞ , -sheaves satisfyingcertain descent with respect to the derived rational localizations or more general homotopyZariski topology as in [BK] and [BBBK].Now we follow [R, Proposition 5.3] and [Dr1, Section 1.1] to give the following discussionaround the corresponding Witt crystalline Stack: Definition 8.4.
We define the corresponding
Witt Crystalline Stack X W of any X over H/ F p in ∞ − Fun(sCommSimpInd(Ban H ) , s Sets) or ∞ − Fun(sCommSimpInd m (Ban H ) , s Sets) tobe the functor ( W ( π ( X ) ♭ ) π ( X ) , crys ) ∧ p . And we define the corresponding pre-crystals to besheaves of O -modules over this functors when we have that X is an ∞ -analytic stack withreasonable topology. Example 8.5.
For instance if we have that X = Spa h ( R ) coming from the correspondingBambozzi-Kremnizer spectrum of any Banach ring over F p (( t )) as constructed in [BK]. Thenwe have that the corresponding functor is ( W ( π ( X ) ♭ ) π ( X ) , crys ) ∧ p is now admitting structurescoming from the corresponding homotopy Zariski topology from X . Example 8.6.
For instance if we have that X = Spec( F p [[ t ]]) coming from the correspondingobject in the opposite category of F p [[ t ]] . Then we have that the corresponding functor is ( W ( π ( X ) ♭ ) π ( X ) , crys ) ∧ p is now admitting structures coming from the corresponding homotopyZariski topology from X , is just the same as the corresponding one in the algebraic settingconstructed in [R, Proposition 5.3] and [Dr1, Section 1.1].8.2. Perfectization, Fontainisation and Robba Stacks.
Now we contact [KL1] and[KL2] to look at the corresponding Robba Stacks. Now take any X to be any ∞ -analytic stackwhich admits structures of simplicial complete Bornological rings or ind-Fréchet structures,namely we have the corresponding complete bornological topology or ind-Fréchet topologyon π ( X ) . We work over H/ F p as well. Example 8.7.
For instance we take that X = Spa h ( R ) coming from the correspondingBambozzi-Kremnizer spectrum of any Banach ring over F p (( t )) as constructed in [BK]. Definition 8.8.
For any ∞ -analytic stack X as above, we consider the corresponding Wittvector functor W n ( π ( X ) ♭ ) and then consider lim −→ n →∞ W n ( π ( X ) ♭ ) , namely W ( π ( X ) ♭ ) , thenwe consider the ring W ( π ( X ) ♭ )[1 /p ] . Then we can take the corresponding completion withrespect to the Gauss norm k . k π ( X ) , [ s,r ] coming from the corresponding norm on π ( X ) withespect to some interval [ s, r ] ∈ (0 , ∞ ) as in [KL2, Definition 4.1.1]: e Π( X ) [ s,r ] := ( W ( π ( X ) ♭ )[1 /p ]) ∧k . k π X ) , [ s,r ] , Fré . (8.3)Then following [KL2, Definition 4.1.1] we consider the following: e Π( X ) r := lim ←− s> ( W ( π ( X ) ♭ )[1 /p ]) ∧k . k π X ) , [ s,r ] , Fré (8.4)and e Π( X ) := lim −→ r> lim ←− s> ( W ( π ( X ) ♭ )[1 /p ]) ∧k . k π X ) , [ s,r ] , Fré . (8.5)We call these Robba stacks . Example 8.9.
In the situation where X is some ∞ -analytic stack carrying the correspond-ing sheaves of simplicial Banach rings (namely not in general bornological or ind-Fréchet)we have that the finite projective modules over the three Robba stacks (for by enough in-tervals) carrying semilinear Frobenius action which realizes the corresponding isomorphismsby Frobenius pullbacks are equivalent. In the noetherian setting we have the same holds forlocally finite presented sheaves as well. This is the main results of [KL2, Theorem 4.6.1]. cknowledgements. This is our independent work, however we would like to thank Pro-fessor Kedlaya for helpful discussion, in particular the corresponding nontrivial input onour understanding of the corresponding affinoid morphisms, Hansen-Kedlaya and the corre-sponding AWS lecture notes. eferences [HK] Hansen, David, and Kiran S. Kedlaya. "Sheafiness criteria for Huber rings." (2020).[Ked1] Kedlaya, K. "Sheaves, stacks, and shtukas, lecture notes from the 2017 Arizona Winter School:Perfectoid Spaces." Math. Surveys and Monographs 242.[EGAIV4] Grothendieck, Alexander. "Éléments de géométrie algébrique (rédigés avec la collaboration deJean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, Quatrieme partie."Publications Mathématiques de l’IHÉS 32 (1967): 5-361.[SP] Stacks Project Authors. "Stacks project." (2019).[Hu1] Huber, Roland. Étale cohomology of rigid analytic varieties and adic spaces. Vol. 30. Springer, 2013.[BBBK] Bambozzi, Federico, Oren Ben-Bassat, and Kobi Kremnizer. "Analytic geometry over F ∼ p -adic derived de Rham cohomology." arXiv preprint arXiv:1204.6560 (2012).[III1] Illusie, Luc. Complexe cotangent et déformations I. Vol. 239. Springer, 2006.III2] Illusie, L. "Complexe cotangent et déformations II, SLN, 283 (1972)." Zbl0224 13014.[LL] Li, Shizhang, and Tong Liu. "Comparison of prismatic cohomology and derived de Rham cohomology."arXiv preprint arXiv:2012.14064 (2020).[SGAIV] Artin, Michael, Alexandre Grothendieck, and Jean-Louis Verdier. "Théorie des topos et cohomolo-gie étale des schémas, LNM 269, 270, 305." (1972).[Sch1] Scholze, Peter. " pp