aa r X i v : . [ m a t h . N T ] F e b PERIOD RINGS WITH BIG COEFFICIENTS AND APPLICATIONS III
XIN TONG
Abstract.
We continue our study on the corresponding period rings with big coefficients,with the corresponding application in mind on relative p -adic Hodge theory and noncom-mutative analytic geometry. In this article, we extend the discussion of the correspondingnoncommutative descent over étale topology to the corresponding noncommutative descentover pro-étale topology in both Tate and analytic setting. Contents
1. Introduction 21.1. Introduction and Main Results 21.2. Future Work 32. Descent over Analytic Adic Banach Rings in the Rational Setting 32.1. Noncommutative Functional-Analytic Pseudocoherence over Pro-Étale Topology 33. Descent over Analytic Huber Rings in the Integral Setting 63.1. Noncommutative Topological Pseudocoherence over Étale Topology 63.2. Noncommutative Topological Pseudocoherence over Pro-Étale Topology 94. Descent in General Setting for Analytic Huber Pairs 124.1. Étale Topology 124.2. Pro-étale Topology 145. Descent in General Setting for Analytic Adic Banach Rings 175.1. Étale Topology 175.2. Pro-étale Topology 195.3. Quasi-Stein Spaces 216. Applications 236.1. Application to Noetherian Cases 236.2. Application to Descent over Noncommutative ∞ -Analytic Prestacks afterBambozzi-Ben-Bassat-Kremnizer 25 Version: Feb 19 2021. Keywords and Phrases: Noncommutative Deformation, Descent. cknowledgements 29References 291. Introduction
Introduction and Main Results.
In our previous work on the corresponding periodrings with big coefficients and applications, we essentially studied many very general defor-mations of the corresponding pseudocoherent sheaves over adic Banach rings in the sense of[KL1] and [KL2]. We studied the corresponding glueing in the fashion of [KL1] and [KL2]carrying sufficiently large coefficients. Although we have some conditions on both the corre-sponding adic spaces we are considering and the corresponding coefficients, but the descentresults on their own are already general enough to tackle some specific situations in theHodge-Iwasawa theoretic consideration and noncommutative analytic geometry such as in[TX3] and [TX4].First in our current scope of discussion we will first consider the corresponding extensionof our paper [TX2] to the corresponding pro-étale topology setting. In the Tate setting, thiswill follow the corresponding unrelative situation in [KL2] while in the analytic situation thecorresponding will following the corresponding unrelative situation in [Ked1]:
Theorem 1.1.
Over sheafy Tate adic Banach rings or analytic Huber rings, the correspond-ing descent of stably pseudocoherent modules (with respect to the underlying spaces) carryingthe corresponding noncommutative coefficients holds over étale sites and pro-étale sites.
We then consider some application to the corresponding ’complete’ noetherian situation(this will mean that we consider the noetherian deformation of noetherian rings):
Theorem 1.2.
Over noetherian sheafy Tate adic Banach rings or analytic Huber rings, thecorresponding descent of finitely presented modules (with respect to the underlying spaces)carrying the corresponding noncommutative coefficients (such that the products of rings arenoetherian as well) holds over étale sites.
We also discussed the corresponding possible application to the descent in certain casesover the corresponding ∞ -analytic stacks after Bambozzi-Ben-Bassat-Kremnizer [BBBK]and Bambozzi-Kremnizer [BK], where the latter is really more related to the adic geometrywe considered after [KL1] and [KL2]. Our approach certainly is definitely parallel to [KL1]and [KL2], which is not actually parallel to more ∞ -method considered by Ben-Bassat-Kremnizer in [BBK]..2. Future Work.
We expect more interesting applications. For instance partial noether-ian spaces could admit some desired descent carrying general noncommutative coefficients(certainly not need to be noetherian coefficients), such as different types of eigenvarieties orShimura varieties.2.
Descent over Analytic Adic Banach Rings in the Rational Setting
Noncommutative Functional-Analytic Pseudocoherence over Pro-Étale Topol-ogy.
Now we consider the corresponding discussion of the corresponding glueing deformedpseudocoherent sheaves over étale topology which generalize the corresponding discussion in[Ked1].
Setting 2.1.
Let ( W, W + ) be Tate adic Banach uniform ring which is defined over Q p . Andwe assume that the ring V over Q p is a Banach ring over the Q p . Assume that the ring W is sheafy. Remark 2.2.
The corresponding glueing of the corresponding pseudocoherent sheaves overthe corresponding pro-étale site does not need the corresponding notions on the correspond-ing stability beyond the corresponding étale-stably pseudocoherence.
Lemma 2.3. (After Kedlaya-Liu [KL2, Proposition 3.4.3] ) Suppose that we are taking ( W, W + ) to be Fontaine perfectoid (note that we are considering the corresponding contextof Tate adic Banach situation). Then we have that over the corresponding étale site, thecorresponding group H i (Spa( W, W + ) ét , O b ⊗ V ) vanishes for each i > , while we have thatthe corresponding group H (Spa( W, W + ) ét , O b ⊗ V ) is just isomorphic W b ⊗ V .Proof. See [KL2, Proposition 3.4.3], where the corresponding [KL2, Corollary 3.3.20] applies. (cid:3)
Now we proceed to consider the corresponding pro-étale topology. First we recall thefollowing result from [KL2, Lemma 3.4.4] which does hold in our situation since we did notchange much on the corresponding underlying adic spaces.
Lemma 2.4. (Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring W as above which isfurthermore assumed to be Fontaine perfectoid in the sense of [KL2, Definition 3.3.1] , andconsider any direct system of faithfully finite étale morphisms as: W / / / / / / W / / / / / / W / / / / / / ..., here A is just the corresponding base A . Then the corresponding completion of the infinitelevel could be decomposed as: W ⊕ dM ∞ k =0 W k /W k − . (2.1) As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above withthe corresponding seminorm spectral. And for the corresponding quotient we endow with thecorresponding quotient norm.
Then we consider the corresponding deformation by the ring V over Q p : Lemma 2.5. (After Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring A as above whichis furthermore assumed to be Fontaine perfectoid in the sense of [KL2, Definition 3.3.1] , andconsider any direct system of faithfully finite étale morphisms as: W ,V / / / / / / W ,V / / / / / / W ,V / / / / / / ..., where W is just the corresponding base W . Then the corresponding completion of the infinitelevel could be decomposed as: W ,V ⊕ dM ∞ k =0 W k,V /W k − ,V . (2.2) Corollary 2.6. (After Kedlaya-Liu [KL2, Corollary 3.4.5] ) Starting with an adic Banachring which is as in [KL2, Corollary 3.4.5] assumed to be Fontaine perfectoid in the sense of [KL2, Definition 3.3.1] . And as in the previous two lemmas we consider an admissible infinitedirect system: W ,V / / / / / / W ,V / / / / / / W ,V / / / / / / ..., whose infinite level will be assumed to take the corresponding spectral seminorm as in the [KL2, Corollary 3.4.5] . Then carrying the corresponding coefficient V , we have in suchsituation the corresponding 2-pseudoflatness of the corresponding embedding map: W ,V → W ∞ ,V . (2.3) Here W ∞ ,V denotes the corresponding completion of the limit lim −→ k →∞ W k,V .Proof. See [KL2, Corollary 3.4.5]. (cid:3) ow recall that from [KL2] since we do not have to modify the corresponding underlyingspatial context, so we will also only have to consider the corresponding stability with respectto étale topology even although we are considering the corresponding profinite étale site inour current section. We first generalize the corresponding Tate acyclicity in [KL2] to thecorresponding V -relative situation in our current situation: Proposition 2.7. (After Kedlaya-Liu [KL2, Theorem 3.4.6] ) Now we consider the cor-responding pro-étale site of X which we denote it by X pét . We assume we have a stablebasis H of the corresponding perfectoid subdomains for this profinite étale site where eachmorphism therein will be étale pseudoflat. We consider as in [KL2, Theorem 3.4.6] a cor-responding module over W b ⊗ V which is assumed to be étale-stably pseudocoherent. Thenconsider the corresponding presheaf e G attached to this étale-stably pseudocoherent modulewith respect to in our situation the corresponding chosen well-defined basis H , we have thatthe corresponding sheaf over some element Y ∈ H (that is to say over O Y pét b ⊗ V ) is acyclicand is acyclic with respect to some Čeck covering coming from elements in H .Proof. The corresponding proof could be made as in the corresponding nonrelative settingas in [KL2, Theorem 3.4.6]. As in [KL2, Theorem 3.4.6] the corresponding first step is tocheck the statement is true with respect to some specific basis consisting of all the perfec-toid subdomains which are faithfully finite étale over some specific perfectoid subdomain Y (along towers). Then the corresponding statement could be reduced to those with respectto covering which is simple Laurent rational and the corresponding covering in the senseof the previous corollary. Then actually in this first scope of consideration, the statementholds by previous corollary and [TX2, Theorem 2.12]. Then in general for some general Y we consider the corresponding covering of Y consisting of elements satisfying the previoussituation, then apply the previous argument to the corresponding basis in current situationsatisfying the previous situation Then we consider the corresponding sheafification of thefollowing chosen short exact sequence: / / / / / / G ′′ / / / / / / G ′ / / / / / / G / / / / / / , with G ′ finite projective. Then the idea is to consider the corresponding diagram factoringthrough the corresponding pro-étale sheafification of this short exact sequence down to thecorresponding étale site. Then the five lemma will finish the corresponding proof as oneonsiders the following commutative diagram: G ′′ V Y / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G ′ V Y / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) G V Y / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) , / / / / / / Γ( Y ét , f G ′′ ) / / / / / / Γ( Y ét , f G ′ ) / / / / / / Γ( Y ét , e G ) / / / / / / . Here the corresponding ring V Y is the adic Banach ring such that Y = Spa( V Y , V + Y ) . (cid:3) Definition 2.8. (After Kedlaya-Liu [KL2, Definition 3.4.7] ) Consider the pro-étale siteof X attached to the adic Banach ring ( W, W + ) . We will define a sheaf of module G over b O pét b ⊗ V to be V -pseudocoherent if locally we can define this as a sheaf attached to a V -étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to consider thecorresponding notion of V -profinite-étale-stably pseudocoherent module. Proposition 2.9. (After Kedlaya-Liu [KL2, Theorem 3.4.8] ) Taking the correspondingglobal section will realize the corresponding equivalence between the following two categories.The first one is the corresponding one of all the V -pseudocoherent sheaves over b O pét b ⊗ V , whilethe second one is the corresponding one of all the V -étale-stably pseudocoherent modules over W b ⊗ V .Proof. As in [KL2, Theorem 3.4.8], the corresponding [KL1, Proposition 9.2.6] applies inthe way that the corresponding conditions of [KL1, Proposition 9.2.6] hold in our currentsituation. (cid:3)
Obviously we have the following analog of [KL2, Corollary 3.4.9]:
Corollary 2.10. (After Kedlaya-Liu [KL2, Corollary 3.4.9] ) The following two categoriesare equivalent. The first is the corresponding category of all V -pseudocoherent sheaves over b O pét b ⊗ V . The second is the corresponding category of all V -pseudocoherent sheaves over O ét b ⊗ V . The corresponding functor is the corresponding pullback along the correspondingmorphism of sites X pét → X ét . Descent over Analytic Huber Rings in the Integral Setting
Noncommutative Topological Pseudocoherence over Étale Topology.
Now weconsider the corresponding discussion of the corresponding glueing deformed pseudocoherentheaves over étale topology which generalize the corresponding discussion in [Ked1]. And wewill consider the corresponding situation as assumed in the following setting:
Setting 3.1.
Let ( W, W + ) be analytic Huber uniform pair which is defined over Z p . Andwe assume that the ring V over Z p is a topological ring (complete) and splitting over the Z p .We now fix a corresponding stable basis H for the corresponding étale site of the adic space Spa(
W, W + ) , locally consisting of the corresponding compositions of rational localizationsand the corresponding finite étale morphisms. And as in [Ked1, Hypothesis 1.10.3] we needto assume that the corresponding basis is made up of the corresponding adic spectrum ofsheafy rings. Assume the corresponding sheafiness of the Huber ring W . Definition 3.2. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) We define a V -étale stablypseudocoherent module over the corresponding Huber ring W with respect to the correspond-ing basis H chosen above for the corresponding étale site of the analytic adic space X . Wedefine a module over W b ⊗ V is a V -étale stably pseudocoherent module if it is algebraicallypseudocoherent (namely formed by the corresponding possibly infinite length resolution offinitely generated and projective modules) and at least complete with respect to the corre-sponding natural topology and also required to be also complete with respect to the naturaltopology along some base change with respect to any morphism in H . Definition 3.3. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) Along the previous defini-tion we have the corresponding notion of V -étale-pseudoflat left (or right respectively) mod-ules with respect to the corresponding chosen basis H . A such module is defined to be atopological module G over W b ⊗ V complete with respect to the natural topology and for anyright (or left respectively) V -étale stably pseudocoherent module, they will jointly give the Tor vanishing. Lemma 3.4. (After Kedlaya-Liu [KL2, Lemma 2.5.10] ) One could find another basis H ′ which is in our situation contained in the corresponding original basis H such that anymorphism in H could be V -étale-pseudoflat with respect H ′ or just H itself.Proof. The derivation of such new basis is actually along the same way as in [KL2, Lemma2.5.10] since the corresponding compositions of rational localizations and the finite étalesare actually satisfying the corresponding conditions in the statement by [TX2, Theorem2.12] over the analytic topology. In general one just show any general morphism will alsodecompose in the same way, which one can proves certainly as in [KL2, Lemma 2.5.10],where one instead consider in the current context the corresponding basis spreading resultin [Ked1, Lemma 1.10.4]. (cid:3) roposition 3.5. (After Kedlaya-Liu [KL2, Theorem 2.5.11] ) Consider the site X ét andconsider the basis H . Take any V -étale stably pseudocoherent module M over W b ⊗ V . Con-sider the corresponding presheaf by taking the inverse limit throughout all the correspondingbase change along morphisms in H . Then we have the corresponding acyclicity of the presheafover any element in H and any covering of this element by elements in H .Proof. As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21]. (cid:3)
Proposition 3.6. (After Kedlaya-Liu [KL2, Lemma 2.5.13] ) The corresponding glueingof V -étale-stably pseudocoherent modules holds in this current situation over W along binarymorphisms (namely along binary rational decomposition).Proof. We adapt the argument of [KL2, Lemma 2.5.13] to our current situation. Take thecorresponding map to be W → W L W , and we denote the two spaces associated W and W by Y and Y . Establish a corresponding descent datum of V -étale-stably pseudocoherentmodules along this decomposition of Spa(
W, W + ) . One can realize this by acyclicity a sheaf G over the whole space Spa(
W, W + ) Then we have for each k = 1 , that the correspondingidentification of H ( Y k, ét , G ) with H ( Y k , G ) and the vanishing of the i -th cohomology forhigher i . By the corresponding results we have in the corresponding rational localizationsituation we vanishing of the corresponding H i (Spa( W, W + ) , G ) , i ≥ . Now consider afinite free covering C of G and take the kernel K to form the corresponding exact sequence: / / / / / / K / / / / / / C / / / / / / G / / / / / / , which will give rise to the following short exact sequence: / / / / / / K (Spa( W, W + )) / / / / / / C (Spa( W, W + )) / / / / / / G (Spa( W, W + )) / / / / / / . To finish we have also to show the corresponding resulting global section is actually V -étale-stably pseudocoherent. For this, we refer the readers to [TX2, Lemma 2.22]. (cid:3) Proposition 3.7. (After Kedlaya-Liu [KL2, Theorem 2.5.14] ) Taking the correspondingglobal section will realize the equivalence between the following two categories. The first isthe one of all the corresponding V -pseudocoherent sheaves over O ét b ⊗ V . The second is theone of all the corresponding V -étale-stably pseudocoherent modules.Proof. This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in ourcurrent context. The refinement comes from the acyclicity, the transitivity is straightforward,he corresponding binary rational localization situation is the previous proposition, while thecorresponding last condition will basically comes from the corresponding f.p. descent [SGAI,Chapitre VIII]. (cid:3)
Noncommutative Topological Pseudocoherence over Pro-Étale Topology.Setting 3.8.
Let ( W, W + ) be analytic Huber ring which is defined over Z p . And we assumethat the ring V over Z p is a complete topological ring over the Z p which is completely exact.Assume that the ring W is sheafy. Remark 3.9.
The corresponding glueing of the corresponding pseudocoherent sheaves overthe corresponding pro-étale site does not need the corresponding notions on the correspond-ing stability beyond the corresponding étale-stably pseudocoherence.
Lemma 3.10. (After Kedlaya-Liu [KL2, Proposition 3.4.3] ) Suppose that we are taking ( W, W + ) to be perfectoid now in the sense of [Ked1, Definition 2.1.1] (note that we areconsidering the corresponding context of analytic Huber pair situation). Then we have thatover the corresponding étale site, the corresponding group H i (Spa( W, W + ) ét , O b ⊗ V ) vanishesfor each i > , while we have that the corresponding group H (Spa( W, W + ) ét , O b ⊗ V ) is justisomorphic W b ⊗ V .Proof. See [KL2, Proposition 3.4.3], where the corresponding [KL2, Corollary 3.3.20] applies. (cid:3)
Now we proceed to consider the corresponding pro-étale topology. First we recall thefollowing result from [KL2, Lemma 3.4.4] which does hold in our situation since we did notchange much on the corresponding underlying adic spaces.
Lemma 3.11. (Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring W as above which isfurthermore assumed to be perfectoid in the sense of instead [Ked1, Definition 2.1.1] , andconsider any direct system of faithfully finite étale morphisms as: W / / / / / / W / / / / / / W / / / / / / ..., where A is just the corresponding base A . Then the corresponding completion of the infinitelevel could be decomposed as: W ⊕ dM ∞ k =0 W k /W k − . (3.1) s in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above withthe corresponding seminorm spectral. And for the corresponding quotient we endow with thecorresponding quotient norm. Then we consider the corresponding deformation by the ring V over Z p : Lemma 3.12. (After Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring A as abovewhich is furthermore assumed to be perfectoid in the sense of [Ked1, Definition 2.1.1] , andconsider any direct system of faithfully finite étale morphisms as: W ,V / / / / / / W ,V / / / / / / W ,V / / / / / / ..., where W is just the corresponding base W . Then the corresponding completion of the infinitelevel could be decomposed as: W ,V ⊕ dM ∞ k =0 W k,V /W k − ,V . (3.2) Corollary 3.13. (After Kedlaya-Liu [KL2, Corollary 3.4.5] ) Starting with an analyticHuber ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in the sense of [Ked1, Definition 2.1.1] . And as in the previous two lemmas we consider an admissibleinfinite direct system: W ,V / / / / / / W ,V / / / / / / W ,V / / / / / / ..., whose infinite level will be assumed to take the corresponding spectral seminorm as in the [KL2, Corollary 3.4.5] . Then carrying the corresponding coefficient V , we have in suchsituation the corresponding 2-pseudoflatness of the corresponding embedding map: W ,V → W ∞ ,V . (3.3) Here W ∞ ,V denotes the corresponding completion of the limit lim −→ k →∞ W k,V .Proof. See [KL2, Corollary 3.4.5]. (cid:3)
Now recall that from [KL2] since we do not have to modify the corresponding underlyingspatial context, so we will also only have to consider the corresponding stability with respectto étale topology even although we are considering the corresponding profinite étale site inour current section. We first generalize the corresponding Tate acyclicity in [KL2] to thecorresponding V -relative situation in our current situation: roposition 3.14. (After Kedlaya-Liu [KL2, Theorem 3.4.6] ) Now we consider the cor-responding pro-étale site of X which we denote it by X pét . Then we assume we have a stablebasis H of the corresponding perfectoid subdomains for this profinite étale site where eachmorphism therein will be étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] acorresponding module over W b ⊗ V which is assumed to be étale-stably pseudocoherent. Thenconsider the corresponding presheaf e G attached to this étale-stably pseudocoherent with re-spect to in our situation the corresponding chosen well-defined basis H , we have that thecorresponding sheaf over some element Y ∈ H (that is to say over O Y pét b ⊗ V ) is acyclic andis acyclic with respect to some Čeck covering coming from elements in H .Proof. See proposition 2.7. (cid:3)
Definition 3.15. (After Kedlaya-Liu [KL2, Definition 3.4.7] ) Consider the pro-étale siteof X attached to the adic Banach ring ( W, W + ) . We will define a sheaf of module G over b O pét b ⊗ V to be V -pseudocoherent if locally we can define this as a sheaf attached to a V -étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to consider thecorresponding notion of V -profinite-étale-stably pseudocoherent module. Proposition 3.16. (After Kedlaya-Liu [KL2, Theorem 3.4.8] ) Taking the correspondingglobal section will realize the corresponding equivalence between the following two categories.The first one is the corresponding one of all the V -pseudocoherent sheaves over b O pét b ⊗ V , whilethe second one is the corresponding one of all the V -étale-stably pseudocoherent module over W b ⊗ V .Proof. As in [KL2, Theorem 3.4.8], the corresponding [KL1, Proposition 9.2.6] applies inthe way that the corresponding conditions of [KL1, Proposition 9.2.6] hold in our currentsituation. (cid:3)
Obviously we have the following analog of [KL2, Corollary 3.4.9]:
Corollary 3.17. (After Kedlaya-Liu [KL2, Corollary 3.4.9] ) The following two categoriesare equivalent. The first is the corresponding category of all V -pseudocoherent sheaves over b O pét b ⊗ V . The second is the corresponding category of all V -pseudocoherent sheaves over O ét b ⊗ V . The corresponding functor is the corresponding pullback along the correspondingmorphism of sites X pét → X ét . . Descent in General Setting for Analytic Huber Pairs
We now consider the situation where there is no condition on the base (except that we willassume certainly the corresponding sheafiness). We will discuss along [Ked1] the correspond-ing descent in analytic and étale topology, as well as the corresponding quasi-Stein spaces.The current situation will not assume we are working over Z p . Remark 4.1.
The corresponding discussion in this section and in the following section isactually an exercise proposed by Kedlaya included in [Ked1], where the corresponding étalesituation essentially is proposed by Kedlaya in [Ked1, Discussion after Lemma 1.10.4] whilewe give the corresponding exposition in both étale and pro-étale situations closely after [KL1]and [KL2].4.1.
Étale Topology.Setting 4.2.
Let ( W, W + ) be analytic Huber uniform pair. We now fix a correspondingstable basis H for the corresponding étale site of the adic space Spa(
W, W + ) , locally consist-ing of the corresponding compositions of rational localizations and the corresponding finiteétale morphisms. And as in [Ked1, Hypothesis 1.10.3] we need to assume that the corre-sponding basis is made up of the corresponding adic spectrum of sheafy rings. Assume thecorresponding sheafiness of the Huber ring W . Remark 4.3.
One should treat our discussion in this section as essentially some detailizationof some exercise predicted in [Ked1].
Definition 4.4. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) We define the étale stablypseudocoherent module over the corresponding Huber ring W with respect to the corre-sponding basis H chosen above for the corresponding étale site of the analytic adic space X .We define a module over W to be an étale stably pseudocoherent module if it is algebraicallypseudocoherent (namely formed by the corresponding possibly infinite length resolution offinitely generated and projective modules) and at least complete with respect to the corre-sponding natural topology and also required to be also complete with respect to the naturaltopology along some base change with respect to any morphism in H . Definition 4.5. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) Along the previous defini-tion we have the corresponding notion of étale-pseudoflat left (or right respectively) moduleswith respect to the corresponding chosen basis H . A such module is defined to be a topologi-cal module G over W complete with respect to the natural topology and for any right (or leftrespectively) étale stably pseudocoherent module, they will jointly give the Tor vanishing. emark 4.6. Although in this definition we considered the corresponding left or rightmodules, but essentially speaking this is no different from the usual situation since we arein the corresponding essential site theoretic situation.
Lemma 4.7. (After Kedlaya-Liu [KL2, Lemma 2.5.10] ) One could find another basis H ′ which is in our situation contained in the corresponding original basis H such that anymorphism in H could be étale-pseudoflat with respect H ′ or just H itself.Proof. The derivation of such new basis is actually along the same way as in [KL2, Lemma2.5.10] since the corresponding compositions of rational localizations and the finite étalesare actually satisfying the corresponding conditions in the statement by [TX2, Theorem2.12] over the analytic topology. In general one just show any general morphism will alsodecompose in the same way, which one can proves certainly as in [KL2, Lemma 2.5.10],where one instead consider in the current context the corresponding basis spreading resultin [Ked1, Lemma 1.10.4]. (cid:3)
Proposition 4.8. (After Kedlaya-Liu [KL2, Theorem 2.5.11] ) Consider the site X ét andconsider the basis H . Take any étale stably pseudocoherent module M over W . Considerthe corresponding presheaf by taking the inverse limit throughout all the corresponding basechange along morphisms in H . Then we have the corresponding acyclicity of the presheafover any element in H and any covering of this element by elements in H .Proof. As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21]. (cid:3)
Proposition 4.9. (After Kedlaya-Liu [KL2, Lemma 2.5.13] ) The corresponding glueingof étale-stably pseudocoherent modules holds in this current situation over W along binarymorphisms (namely along binary rational decomposition).Proof. We adapt the argument of [KL2, Lemma 2.5.13] to our current situation. Take thecorresponding map to be W → W L W , and we denote the two spaces associated W and W by Y and Y . Establish a corresponding descent datum of V -étale-stably pseudocoherentmodules along this decomposition of Spa(
W, W + ) . One can realize this by acyclicity a sheaf G over the whole space Spa(
W, W + ) Then we have for each k = 1 , that the correspondingidentification of H ( Y k, ét , G ) with H ( Y k , G ) and the vanishing of the i -th cohomology forhigher i . By the corresponding results we have in the corresponding rational localizationsituation we have vanishing of the corresponding H i (Spa( W, W + ) , G ) , i ≥ . Now consider afinite free covering C of G and take the kernel K to form the corresponding exact sequence: / / / / / / K / / / / / / C / / / / / / G / / / / / / , hich will give rise to the following short exact sequence: / / / / / / K (Spa( W, W + )) / / / / / / C (Spa( W, W + )) / / / / / / G (Spa( W, W + )) / / / / / / . To finish we have also to show the corresponding resulting global section is actually étale-stably pseudocoherent. For this, we refer the readers to [TX2, Lemma 2.22]. (cid:3)
Proposition 4.10. (Kedlaya [Ked1, Below Lemma 1.10.4] , After Kedlaya-Liu [KL2, Theorem 2.5.14] ) Taking the corresponding global section will realize the equivalence between the following twocategories. The first is the one of all the corresponding pseudocoherent sheaves over O ét .The second is the one of all the corresponding étale-stably pseudocoherent modules.Proof. This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in ourcurrent context. The refinement comes from the acyclicity, the transitivity is straightforward,the corresponding binary rational localization situation is the previous proposition, while thecorresponding last condition will basically come from the corresponding f.p. descent [SGAI,Chapitre VIII]. (cid:3)
Pro-étale Topology.Setting 4.11.
Let ( W, W + ) be an analytic Huber ring. Assume that the ring W is sheafy. Remark 4.12.
Although it seems that the current discussion is basically outside the cor-responding deformed setting, but the foundation here may be generalized to the deformedsetting by taking the corresponding completed tensor product over F as in [BBBK]. Remark 4.13.
The corresponding glueing of the corresponding pseudocoherent sheaves overthe corresponding pro-étale site does not need the corresponding notions on the correspond-ing stability beyond the corresponding étale-stably pseudocoherence.
Lemma 4.14. (After Kedlaya-Liu [KL2, Proposition 3.4.3] ) Suppose that we are taking ( W, W + ) to be perfectoid now in the sense of [Ked1, Definition 2.1.1] (note that we areconsidering the corresponding context of analytic Huber pair situation). Then we have thatover the corresponding étale site, the corresponding group H i (Spa( W, W + ) ét , O ) vanishesfor each i > , while we have that the corresponding group H (Spa( W, W + ) ét , O ) is justisomorphic to W .Proof. See [KL2, Proposition 3.4.3], where the corresponding [KL2, Corollary 3.3.20] applies. (cid:3) ow we proceed to consider the corresponding pro-étale topology. First we recall thefollowing result from [KL2, Lemma 3.4.4] which does hold in our situation since we did notchange much on the corresponding underlying adic spaces.
Lemma 4.15. (Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring W as above which isfurthermore assumed to be perfectoid in the sense of instead [Ked1, Definition 2.1.1] , andconsider any direct system of faithfully finite étale morphisms as: W / / / / / / W / / / / / / W / / / / / / ..., where W is just the corresponding base W . Then the corresponding completion of the infinitelevel could be decomposed as: W ⊕ dM ∞ k =0 W k /W k − . (4.1) As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above withthe corresponding seminorm spectral. And for the corresponding quotient we endow with thecorresponding quotient norm.
Corollary 4.16. (After Kedlaya-Liu [KL2, Corollary 3.4.5] ) Starting with an analyticHuber ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in the sense of [Ked1, Definition 2.1.1] . And as in the previous two lemmas we consider an admissibleinfinite direct system: W / / / / / / W / / / / / / W / / / / / / ..., whose infinite level will be assumed to take the corresponding spectral seminorm as in the [KL2, Corollary 3.4.5] . Then we have in such situation the corresponding 2-pseudoflatnessof the corresponding embedding map: W → W ∞ . (4.2) Here W ∞ denotes the corresponding completion of the limit lim −→ k →∞ W k .Proof. See [KL2, Corollary 3.4.5]. (cid:3)
Now recall that from [KL2] since we do not have to modify the corresponding underlyingspatial context, so we will also only have to consider the corresponding stability with respectto étale topology even although we are considering the corresponding profinite étale site inour current section. We first generalize the corresponding Tate acyclicity in [KL2] to theorresponding our current situation.
Proposition 4.17. (After Kedlaya-Liu [KL2, Theorem 3.4.6] ) Now we consider the cor-responding pro-étale site of X which we denote it by X pét . Then we assume we have a stablebasis H of the corresponding perfectoid subdomains for this profinite étale site where eachmorphism therein will be étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] acorresponding module over W which is assumed to be étale-stably pseudocoherent. Then con-sider the corresponding presheaf e G attached to this étale-stably pseudocoherent with respectto in our situation the corresponding chosen well-defined basis H , we have that the corre-sponding sheaf over some element Y ∈ H (that is to say over O Y pét ) is acyclic and is acyclicwith respect to some Čeck covering coming from elements in H .Proof. See proposition 2.7. (cid:3)
Definition 4.18. (After Kedlaya-Liu [KL2, Definition 3.4.7] ) Consider the pro-étale siteof X attached to the analytic Huber pair ( W, W + ) . We will define a sheaf of module G over b O pét to be pseudocoherent if locally we can define this as a sheaf attached to an étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to consider thecorresponding notion of profinite-étale-stably pseudocoherent modules. Proposition 4.19. (Kedlaya [Ked1, Section 3.8] , after Kedlaya-Liu [KL2, Theorem 3.4.8] ) Taking the corresponding global section will realize the corresponding equivalence between thefollowing two categories. The first one is the corresponding one of all the pseudocoherentsheaves over b O pét , while the second one is the corresponding one of all the étale-stably pseu-docoherent modules over W .Proof. As in [KL2, Theorem 3.4.8], the corresponding [KL1, Proposition 9.2.6] applies inthe way that the corresponding conditions of [KL1, Proposition 9.2.6] hold in our currentsituation. (cid:3)
Obviously we have the following analog of [KL2, Corollary 3.4.9]:
Corollary 4.20. (After Kedlaya-Liu [KL2, Corollary 3.4.9] ) The following two categoriesare equivalent. The first is the corresponding category of all pseudocoherent sheaves over b O pét . The second is the corresponding category of all pseudocoherent sheaves over O ét . Thecorresponding functor is the corresponding pullback along the corresponding morphism ofsites X pét → X ét . . Descent in General Setting for Analytic Adic Banach Rings
We now translate the results in previous section to the corresponding analytic adic Banachcontext. Again everything is essentially as proposed in [Ked1, Below Lemma 1.10.4, also seeSection 3.8].5.1.
Étale Topology.Setting 5.1.
Let ( W, W + ) be analytic adic Banach ring. We now fix a corresponding stablebasis H for the corresponding étale site of the adic space Spa(
W, W + ) , locally consisting ofthe corresponding compositions of rational localizations and the corresponding finite étalemorphisms. And as in [Ked1, Hypothesis 1.10.3] we need to assume that the correspondingbasis is made up of the corresponding adic spectrum of sheafy rings. Assume the correspond-ing sheafiness of the Huber ring W . Definition 5.2. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) We define the étale-stablypseudocoherent module over the corresponding adic Banach W with respect to the corre-sponding basis H chosen above for the corresponding étale site of the analytic adic space X .We define a module over W to be an étale stably pseudocoherent module if it is algebraicallypseudocoherent (namely formed by the corresponding possibly infinite length resolution offinitely generated and projective modules) and at least complete with respect to the corre-sponding natural topology and also required to be also complete with respect to the naturaltopology along some base change with respect to any morphism in H . Definition 5.3. (After Kedlaya-Liu [KL2, Definition 2.5.9] ) Along the previous defini-tion we have the corresponding notion of étale-pseudoflat left (or right respectively) moduleswith respect to the corresponding chosen basis H . A such module is defined to be a topologi-cal module G over W complete with respect to the natural topology and for any right (or leftrespectively) étale stably pseudocoherent module, they will jointly give the Tor vanishing. Lemma 5.4. (After Kedlaya-Liu [KL2, Lemma 2.5.10] ) One could find another basis H ′ which is in our situation contained in the corresponding original basis H such that anymorphism in H could be étale-pseudoflat with respect H ′ or just H itself.Proof. See lemma 4.7. (cid:3)
Proposition 5.5. (After Kedlaya-Liu [KL2, Theorem 2.5.11] ) In our current analyticadic Banach situation, consider the site X ét and consider the basis H . Take any étale sta-bly pseudocoherent module M over W . Consider the corresponding presheaf by taking thenverse limit throughout all the corresponding base change along morphisms in H . Then wehave the corresponding acyclicity of the presheaf over any element in H and any covering ofthis element by elements in H .Proof. As in [KL2, Theorem 2.5.11] apply the corresponding [KL1, Proposition 8.2.21]. (cid:3)
Proposition 5.6. (After Kedlaya-Liu [KL2, Lemma 2.5.13] ) In our current analytic adicBanach situation, the corresponding glueing of étale-stably pseudocoherent modules holdsin this current situation over W along binary morphisms (namely along binary rationaldecomposition).Proof. We adapt the argument of [KL2, Lemma 2.5.13] to our current situation over analyticadic Banach rings. Take the corresponding map to be W → W L W , and we denotethe two spaces associated W and W by Y and Y . Establish a corresponding descentdatum of V -étale-stably pseudocoherent modules along this decomposition of Spa(
W, W + ) .One can realize this by acyclicity a sheaf G over the whole space Spa(
W, W + ) Then wehave for each k = 1 , that the corresponding identification of H ( Y k, ét , G ) with H ( Y k , G ) and the vanishing of the i -th cohomology for higher i . By the corresponding results wehave in the corresponding rational localization situation we vanishing of the corresponding H i (Spa( W, W + ) , G ) , i ≥ . Now consider a finite free covering C of G and take the kernel K to form the corresponding exact sequence: / / / / / / K / / / / / / C / / / / / / G / / / / / / , which will give rise to the following short exact sequence: / / / / / / K (Spa( W, W + )) / / / / / / C (Spa( W, W + )) / / / / / / G (Spa( W, W + )) / / / / / / . To finish we have also to show the corresponding resulting global section is actually étale-stably pseudocoherent. For this, we refer the readers to [TX2, Lemma 2.22]. (cid:3)
Proposition 5.7. (After Kedlaya-Liu [KL2, Theorem 2.5.14] ) In our current analyticadic Banach situation, taking the corresponding global section will realize the equivalencebetween the following two categories. The first is the one of all the corresponding pseu-docoherent sheaves over O ét . The second is the one of all the corresponding étale-stablypseudocoherent modules.Proof. This is by considering and applying the corresponding [Ked1, Lemma 1.10.4] in ourcurrent context. The refinement comes from the acyclicity, the transitivity is straightforward,he corresponding binary rational localization situation is the previous proposition, while thecorresponding last condition will basically comes from the corresponding f.p. descent [SGAI,Chapitre VIII]. (cid:3)
Pro-étale Topology.Setting 5.8.
Let ( W, W + ) be an analytic adic Banach ring. Assume that the ring W issheafy. Remark 5.9.
As in the corresponding analytic Huber ring situation, the corresponding glue-ing of the corresponding pseudocoherent sheaves over the corresponding pro-étale site doesnot need the corresponding notions on the corresponding stability beyond the correspondingétale-stably pseudocoherence.We translate the corresponding notions of perfectoid rings to the current context from[Ked1, Definition 2.1.1]:
Definition 5.10. (After Kedlaya [Ked1, Definition 2.1.1] ) Consider a general analyticadic Banach ring ( W, W + ) , we will call it perfectoid if there exists a definition ideal d ⊂ W + such that d p contains p and we have the corresponding Frobenius map from W + /d to W + /d p is required to be surjective. Lemma 5.11. (After Kedlaya-Liu [KL2, Proposition 3.4.3] ) Suppose that we are taking ( W, W + ) to be perfectoid now in the sense of definition 5.10 (note that we are consideringthe corresponding context of analytic adic Banach situation). Then we have that over thecorresponding étale site, the corresponding group H i (Spa( W, W + ) ét , O ) vanishes for each i > , while we have that the corresponding group H (Spa( W, W + ) ét , O ) is just isomorphic W .Proof. See [KL2, Proposition 3.4.3], where the corresponding [KL2, Corollary 3.3.20] applies. (cid:3)
Now we proceed to consider the corresponding pro-étale topology. First we recall thefollowing result from [KL2, Lemma 3.4.4].
Lemma 5.12. (Kedlaya-Liu [KL2, Lemma 3.4.4] ) Consider the ring W as above which isfurthermore assumed to be perfectoid in the sense of instead definition 5.10, and considerny direct system of faithfully finite étale morphisms as: W / / / / / / W / / / / / / W / / / / / / ..., where W is just the corresponding base W . Then the corresponding completion of the infinitelevel could be decomposed as: W ⊕ dM ∞ k =0 W k /W k − . (5.1) As in [KL2, Lemma 3.4.4] we endow the corresponding completion mentioned above withthe corresponding seminorm spectral. And for the corresponding quotient we endow with thecorresponding quotient norm.
Corollary 5.13. (After Kedlaya-Liu [KL2, Corollary 3.4.5] ) Starting with an analyticadic Banach ring which is as in [KL2, Corollary 3.4.5] assumed to be perfectoid in the senseof definition 5.10. And as in the previous two lemmas we consider an admissible infinitedirect system: W / / / / / / W / / / / / / W / / / / / / ..., whose infinite level will be assumed to take the corresponding spectral seminorm as in the [KL2, Corollary 3.4.5] . Then we have in such situation the corresponding 2-pseudoflatnessof the corresponding embedding map: W → W ∞ . (5.2) Here W ∞ denotes the corresponding completion of the limit lim −→ k →∞ W k .Proof. See [KL2, Corollary 3.4.5]. (cid:3)
Now recall that from [KL2] since we do not have modified the corresponding underlyingspatial context, so we will also only have to consider the corresponding stability with respectto étale topology even although we are considering the corresponding profinite étale site inour current section. We first generalize the corresponding Tate acyclicity in [KL2] to thecorresponding our current situation:
Proposition 5.14. (After Kedlaya-Liu [KL2, Theorem 3.4.6] ) Now we consider the cor-responding pro-étale site of X which we denote it by X pét . Then we assume we have a stablebasis H of the corresponding perfectoid subdomains for this profinite étale site where eachmorphism therein will be étale pseudoflat. Then we consider as in [KL2, Theorem 3.4.6] aorresponding module over W which is assumed to be étale-stably pseudocoherent. Then con-sider the corresponding presheaf e G attached to this étale-stably pseudocoherent with respectto in our situation the corresponding chosen well-defined basis H , we have that the corre-sponding sheaf over some element Y ∈ H (that is to say over O Y pét ) is acyclic and is acyclicwith respect to some Čeck covering coming from elements in H .Proof. See proposition 2.7. (cid:3)
Definition 5.15. (After Kedlaya-Liu [KL2, Definition 3.4.7] ) Consider the pro-étale siteof X attached to the adic Banach ring ( W, W + ) . We will define a sheaf of module G over b O pét to be pseudocoherent if locally we can define this as a sheaf attached to an étale-stably pseudocoherent module. As in [KL2, Definition 3.4.7], we do not have to consider thecorresponding notion of profinite-étale-stably pseudocoherent module. Proposition 5.16. (After Kedlaya-Liu [KL2, Theorem 3.4.8] ) Taking the correspondingglobal section will realize the corresponding equivalence between the following two categories.The first one is the corresponding one of all the pseudocoherent sheaves over b O pét , while thesecond one is the corresponding one of all the étale-stably pseudocoherent modules over W .Proof. As in [KL2, Theorem 3.4.8], the corresponding [KL1, Proposition 9.2.6] applies inthe way that the corresponding conditions of [KL1, Proposition 9.2.6] hold in our currentsituation. (cid:3)
Obviously we have the following analog of [KL2, Corollary 3.4.9]:
Corollary 5.17. (After Kedlaya-Liu [KL2, Corollary 3.4.9] ) The following two categoriesare equivalent. The first is the corresponding category of all pseudocoherent sheaves over b O pét . The second is the corresponding category of all pseudocoherent sheaves over O ét . Thecorresponding functor is the corresponding pullback along the corresponding morphism ofsites X pét → X ét . Quasi-Stein Spaces.
We now generalize our discussion in [TX2] around pseudocoher-ent sheaves over quasi-Stein spaces to the general base situation without the correspondingrestrictive assumption that we are working over Z p . Setting 5.18.
Assume that we are working over some quasi-Stein space X which could bewritten as the corresponding direct limit of analytic adic Banach affinoids: X := lim −→ i X i such that X i could be written as the corresponding analytic affinoid Spa( W i , W + i ) . emma 5.19. (After Kedlaya-Liu [KL2, Lemma 2.6.3] ) Consider a compatible family ofBanach modules over the projective system O X i ( X i ) . Suppose that the corresponding tran-sition map p i : O X i ( X i ) b ⊗ O Xi +1 ( X i +1 ) M i +1 → M i is surjective. Then we have: I. The cor-responding global section of M throughout the limit O ( X ) = lim ←− i O X i ( X i ) is dense in eachsection M i for any i ≥ ; II. The corresponding vanishing of R lim ←− i →∞ holds in our situa-tion. (Parallel statement in analytic Huber ring situation holds as well.)Proof. This is basically the general version of the corresponding result in [KL2, Lemma 2.6.3].One could prove this in the same way. (cid:3)
Lemma 5.20. (After Kedlaya-Liu [KL2, Corollary 2.6.4] ) Consider a compatible familyof Banach modules over the projective system O X i ( X i ) . Suppose that each member in thefamily is stably-pseudocoherent. And suppose that the corresponding transition map p i : O X i ( X i ) b ⊗ O Xi +1 ( X i +1 ) M i +1 → M i is strictly isomorphism. Then we have in our situation thecorresponding projection from the global section M = lim ←− i M i to each M i (for each i ≥ ) isthen isomorphism in our current situation as in [KL2, Corollary 2.6.4] . (Parallel statementin analytic Huber ring situation holds as well.)Proof. This is basically the general version of the corresponding result in [KL2, Corollary2.6.4]. One could prove this in the same way. (cid:3)
Proposition 5.21. (After Kedlaya-Liu [KL2, Theorem 2.6.5] ) With the correspondingnotations as above, we have that the global section of any pseudocoherent sheaf M over X will be dense in the section over any quasicompact. (Parallel statement in analytic Huberring situation holds as well.)Proof. See [KL2, Theorem 2.6.5]. (cid:3)
Proposition 5.22. (After Kedlaya-Liu [KL2, Theorem 2.6.5] ) With the correspondingnotations as above, we have that the global section of any pseudocoherent sheaf M over X will finitely generate each fiber M x over O X,x for any x ∈ X . (Parallel statement inanalytic Huber ring situation holds as well.)Proof. See [KL2, Theorem 2.6.5]. (cid:3)
Proposition 5.23. (After Kedlaya-Liu [KL2, Theorem 2.6.5] ) With the correspondingnotations as above, we have that the corresponding sheaf cohomology of any pseudocoherentsheaf M over X admits vanishing for degree bigger than 0. (Parallel statement in analyticHuber ring situation holds as well.)roof. See [KL2, Theorem 2.6.5]. (cid:3)
Proposition 5.24. (After Kedlaya-Liu [KL2, Corollary 2.6.6] ) For any pseudocoherentsheaf M over the space X , we have in our current situation that the exact functor from thecorresponding category of all the pseudocoherent sheaves to the corresponding one of all thecorresponding O X ( X ) -modules through the corresponding taking the global section. (Parallelstatement in analytic Huber ring situation holds as well.) Proposition 5.25. (After Kedlaya-Liu [KL2, Corollary 2.6.8] ) For any pseudocoherentsheaf M admitting a structure of vector bundles over the space X , we have in our currentsituation that the corresponding sufficient and necessary condition for the global section tobe finite projective is exactly that the global section of M is finitely generated. (Parallelstatement in analytic Huber ring situation holds as well.)Proof. See [KL2, Corollary 2.6.8], it is straightforward to exact a global splitting from thelocal ones over quasi-compacts. (cid:3)
Proposition 5.26. (After Kedlaya-Liu [KL2, Proposition 2.6.17] ) Assume that the space X is basically m -uniform in the sense of [KL2] . Then we have the corresponding finitenessof the global section is equivalent to to the corresponding uniform finiteness through out all X i , i = 0 , , , ... . (Parallel statement in analytic Huber ring situation holds as well.)Proof. See the proof given in [TX2, Proposition 2.6.17]. (cid:3) Applications
Application to Noetherian Cases.
The corresponding noetherian situation is ex-pected to be in some sense better than just being pseudoflat with respect to the correspondingrational localization.
Setting 6.1.
We now work with the corresponding analytic adic Banach rings over Q p or F p (( t )) as the corresponding base spaces. The deformation will happen along some Banachring over Q p or F p (( t )) . And moreover we have to assume that the corresponding noetheri-anness preserves under the corresponding deformation over some ring V . Namely we have toassume that W b ⊗ ∗ V is always noetherian, for instance this could be achieved when we havethat V over Q p comes from truncations of distribution algebra over some p -adic Lie groupsby considering some further p -adic microlocal analysis. emma 6.2. With the notations in the [KL2, Lemma 2.4.10] , we have that the correspondingmorphisms W b ⊗ V → B b ⊗ V , W b ⊗ V → B b ⊗ V and W b ⊗ V → B b ⊗ V are 2-pseudoflat. Andfurthermore in our current context they are actually flat.Proof. In our current situation, the corresponding 2-pseudoflatness is achieved since thisis already true in the corresponding more general situation. However for finite presentedmodules (we do not have to consider the topological issues since we are in the noetheriansituation) this is already flat which implies for finitely generated modules (we do not haveto consider the topological issues since we are in the noetherian situation) this is already flatas well. Then for any module which could be written as injective limit of finite ones, theresult holds. (cid:3)
Corollary 6.3.
The corresponding rational localization with respect to the adic Banach ring ( W, W + ) is flat (namely for all the corresponding finitely presented module over W b ⊗ V ). Proposition 6.4.
Then we consider the corresponding presheafification of any finitely gen-erated module M over W b ⊗ V , to be more precise over Spa(
W, W + ) we will define the cor-responding presheaf f M by taking the inverse limit of the base changes of M throughout allthe rational localizations of W . Then we have that the Tate glueing property holds in ourcurrent situation for such presheaf f M .Proof. This reduces to the previous lemma by using [KL1, Proposition 2.4.20, Proposition2.4.21]. (cid:3)
The corresponding glueing finitely generated modules is also achieved in current noetheriansituation. First we consider the corresponding result:
Proposition 6.5.
The descent along the morphism W → B L B is effective as long asone focuses on the category of all the finitely presented modules over W b ⊗ V .Proof. This is the corresponding noetherian implication of the corresponding result in [TX2,Lemma 2.14]. (cid:3)
Proposition 6.6.
The corresponding functor of taking the corresponding global section willgive rise to the corresponding equivalence between the corresponding V -coherent sheaves andthe V finitely presented modules.Proof. This is also the corresponding noetherian implication of the corresponding result in[TX2, Theorem 2.15]. (cid:3) hen fixing a stable basis H for the étale site of the space Spa(
W, W + ) , then we have theresults over the étale site as well: Proposition 6.7.
We consider the corresponding presheafification of any finitely generatedmodule M over W b ⊗ V , to be more precise over Spa(
W, W + ) we will define the correspondingpresheaf f M by taking the inverse limit of the base changes of M throughout all the memberin the basis H . Then we have that the Tate glueing property holds in our current situationfor such presheaf f M . Proposition 6.8.
The descent (in the étale topology) along the morphism W → B L B is effective as long as one focuses on the category of all the finitely presented modules over W b ⊗ V .Proof. This is the corresponding noetherian implication of the corresponding result in [TX2,Lemma 2.22]. (cid:3)
Proposition 6.9.
The corresponding functor (in the étale topology) of taking the correspond-ing global section will give rise to the corresponding equivalence between the corresponding V -coherent sheaves and the V finitely presented modules.Proof. This is also the corresponding noetherian implication of the corresponding result in[TX2, Theorem 2.23]. (cid:3)
Application to Descent over Noncommutative ∞ -Analytic Prestacks afterBambozzi-Ben-Bassat-Kremnizer. We now contact with the corresponding derived an-alytic spaces from [BBBK]. Recall that we have the categories
Simp(Ind m (BanSets E )) and Simp(Ind(BanSets E )) of the corresponding simplicial sets in the corresponding inductivecategory of Banach sets over E = Q p , F p (( t )) . Theorem 6.10. (Bambozzi-Ben-Bassat-Kremnizer)
The categories
Simp(Ind m (BanSets E )) and Simp(Ind(BanSets E )) admit symmetric monoidal model categorical structure. Therefore based on this nice structure [BBBK] defined the corresponding ∞ -categories of E ∞ -rings: sComm(Simp(Ind m (BanSets E ))) (6.1) sComm(Simp(Ind(BanSets E ))) . (6.2)e now consider the corresponding some object A which is a corresponding ∞ -locally con-vex ring in the categories above and we consider the corresponding object in the correspond-ing opposite category we call that Spec A . And we consider the corresponding homotopyZariski topology, which allows us to talk about the corresponding ∞ -analytic stacks.Now recall that a connective E -ring is called noetherian if we have that π ( A ) is noetherianand π n ( A ) is basically finitely generated over π ( A ) . Remark 6.11.
The corresponding Koszul simplicial Banach ring considered in [BK] is ac-tually concentrated in the nonpositive degrees , in the cohomological convention, namely H − n ( . ) = 0 for n < . But as in any abstract homotopy theory we consider the correspond-ing conventional transformation: π n := H − n , n ≥ . (6.3)Now we consider an ∞ -analytic stack X as considered in [BBBK]. And we consider thecorresponding ringed site attached to X under the corresponding homotopy Zariski topology,denoted by ( X, O X ) . Example 6.12.
The examples which are very interesting to us are the corresponding ∞ -Huber spectra constructed from any Banach rings over E from [BK] by using Koszul complex.The corresponding classical sheafiness issue could be really forgotten.Now we consider the corresponding E -ring A in the corresponding categories Simp(Ind m (BanSets E )) and Simp(Ind(BanSets E )) . We make the following assumption: Assumption 6.13.
We assume that all the ring objects below are noetherian, as E -rings.Suppose now we have four E Banach rings
A, A , A , A in Simp(Ind m (BanSets E )) and Simp(Ind(BanSets E )) respectively. And we assume that we have the following short strictlyexact sequence in the sense of a glueing square: / / / / / / π ( A ) / / / / / / π ( A ) ⊕ π ( A ) / / / / / / π ( A ) / / / / / / , and we assume that the image of π ( A i ) in π ( A ) is dense for each i = 1 , . And we assumethe corresponding morphism π ( A i ) → π ( A ) is flat for i = 1 , . And we assume that A, A , A , A form the corresponding derived glueing sequence. Thanks for Federico Bambozzi for reminding us of this. Namely all the homotopy groups have to be Banach instead of Bornological or ind-Banach. xample 6.14.
One can construct the following example. First consider any glueing se-quence of Banach rings (not required to be sheafy or commutative) over Q p : / / / / / / Π / / / / / / Π ⊕ Π / / / / / / Π / / / / / / . This is very natural in the corresponding situation for example where we deform from acorresponding nice short exact sequence of commutative sheafy rings of the same type, but inany rate we do not require the corresponding commutativity or the correponding sheafyness.Then take any derived global section of some
Spa h ( R ) from [BK], denoted by R h . Over Q p ,we have the following short strictly exact sequence: / / / / / / Π b ⊗ Q p π ( R h ) / / / / / / Π b ⊗ Q p π ( R h ) ⊕ Π b ⊗ Q p π ( R h ) / / / / / / Π b ⊗ Q p π ( R h ) / / / / / / , but since the corresponding ring Π , Π , Π , Π will actually realize the situation where Π ∗ b ⊗ π ( R h ) ∼ → π (Π ∗ b ⊗ L R h ) , we will have the desired situation as long as restrict to nowthe noetherian situation. Conjecture 6.15.
The map A → Q i =1 , A i is an effective descent morphism with respect tothe finitely presented left module spectra (meaning we have finitely presented π ). Proposition 6.16.
The operation of taking equalizer along the corresponding map Q i =1 , A i → A preserves the property of being finitely presented.Proof. The situation where we deform from the corresponding E ∞ by some noncommutativedeformation could be implied by [TX2, Lemma 2.14]. While in general this follows from[TX4, Lemma 6.83]. Note that we achieve the finiteness for each π n ( M ) just by applicationof the results we know in the classical deformed situation, which is because the connectinghomomorphism vanishes on each level. (cid:3) What could happen in the commutative setting is actually also interesting. We have thefollowing example in mind:
Example 6.17.
Consider the context of the previous example, but assume that all the ringsinvolved are Banach commutative. First we consider over Q p : / / / / / / Π / / / / / / Π ⊕ Π / / / / / / Π / / / / / / , a strictly exact sequence of sheafy rings. We now deform this along some spectrum Spa h ( R ) in [BK] which will produce a very interesting situation where we can actually obtain desiredsituation for the descent of finitely presented module spectra. And note that in commutativesetting we could also have more geometric contexts to rely on as in [BK] and [BBBK]. Thisay have contact with the corresponding derived Galois deformation theory of [GV] forinstance by considering the corresponding simplicial pro-Artin rings. Remark 6.18.
The corresponding machinery from [CS] should definitely reflect similarthings here even in the E -ring context. Also we would like to mention in the commutativesetting that actually the descent for certain quasi-coherent modules are also considered exten-sively in [BBK]. Strikingly the ideas in [BBK] (although developed in a quite commutativesetting) come from partially work from Kontsevich-Rosenberg [KR] namely essentially thenoncommutative descent. Our feeling is that definitely the descent for certain quasi-coherentmodules in [BBBK] could be established to noncommutative setting in some form both inthe archimedean and nonarchimedean situations. cknowledgements. We would like to thank Professor Kedlaya for helpful discussion onthe corresponding materials in the chapter [Ked1], especially the discussion beyond the bookwhich helps us have the chance to enrich the presentation here.
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