Spreading out the Hodge filtration in non-archimedean geometry
SSPREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY
JORGE ANTÓNIO
Abstract.
The goal of the current text is to study non-archimedean analytic derived de Rham cohomologyby means of formal completions. Our approach is inspired by the deformation to the normal cone providedin [GR17b]. More specifically, given a morphism f : X → Y of (derived) k -analytic spaces we constructthe non-archimedean deformation to the normal cone associated to f . The latter can be thought as an A k -parametrized deformation whose fiber at 1 ∈ A k coincides with the formal completion of f and the fiberat 0 ∈ A k with the (derived) normal cone associated to f . We further show that such deformation can beendowed with a natural filtration which spreads out the usual Hodge filtration on the (completed shifted)analytic tangent bundle to the formal completion. Such filtration agrees with the I -adic filtration in the casewhere f is a locally complete intersection morphism between (derived) k -affinoid spaces. Along the way wedevelop the theory of (ind-inf)- k -analytic spaces or in other words k -analytic formal moduli problems. Contents
1. Introduction 21.1. Background 21.2. Main results 21.3. Relation with other works and future questions 41.4. Organization of the paper 51.5. Notations and Conventions 61.6. Acknowledgments 72. Analytic formal moduli problems 72.1. Preliminaries 72.2. Non-archimedean differential geometry 92.3. Analytic formal moduli problems under a base 152.4. Analytic formal moduli problems over a base 242.5. Non-archimedean nil-descent for almost perfect complexes 292.6. Non-archimedean formal groupoids 322.7. The affinoid case 353. Non-archimedean Deformation to the normal bundle 393.1. General construction in the algebraic case 393.2. The construction of the deformation in the affinoid case 463.3. Gluing the Deformation 543.4. The Hodge filtration 56References 62 a r X i v : . [ m a t h . AG ] M a y JORGE ANTÓNIO Introduction
Background.
Let k be a field of characteristic zero and X a variety over k . Following Illusie we canassociate to X its derived de Rham cohomology as follows: we consider the (Hodge completed) derived de Rhamalgebra dR X/k := Sym( L X/k [ − ∧ where ( − ) ∧ stands for the completion of the usual symmetric algebraSym( L X/Y [ − ∈ CAlg k , with respect to its natural Hodge filtration. Furthermore, we can identify the graded pieces of the latter withexterior powers ∧ i L X/Y [ − i ] , for i ≥ , c.f. [Ill72, Vol. II, §VIII]. The Hodge completed derived de Rham algebra is thus a natural generalization ofthe usual de Rham complex for singular varieties over k , where we replace the sheaf of differentialsΩ X/k , by the (algebraic) cotangent complex L X/k and force a spectral sequence of the form Hodge-to-de Rham toconverge.On the other hand, Hartshorne introduced in [Har75] the algebraic de Rham cohomology for singular varieties
X/k in terms of formal completions: suppose we are given a closed immersion i : X , → Y , where Y is a smoothalgebraic variety over k . Then one can form the formal completion , Y ∧ X , of Y at X along i . Inspired by thetheory of tubular neighborhoods, Hartshorne defined an algebraic cohomology theory by taking global sectionsof Y ∧ X Γ( Y ∧ X , O Y ∧ X ) . Moreover, the cohomology groups so obtained are independent of the choice of the closed embedding i : X → Y .In [Bha12], Bhatt proved a general comparison between the two mentioned notions. More precisely, the authorproves a natural multiplicative equivalence of complexesdR X/k ’ Γ( Y ∧ X , O Y ∧ X ) , c.f. [Bha12, Proposition 4.16]. In particular, the results of Bhatt imply that the algebraic de Rham cohomologyis equipped with a Hodge filtration , which is finer than the standard infinitesimal Hodge filtration and theDeligne-Hodge filtration. Futhermore, when f : Spec A → Spec B is a closed immersion of affine (derived)schemes, the Hodge filtration is finer than the I -adic filtration on the completion Y ∧ X ’ Spf( B ∧ I ) , where I := fib( B → A ). In the particular case where f is a locally complete intersection morphism both the I -adic filtration and the Hodge filtration coincide (c.f. [Bha12, Remark 4.13]).1.2. Main results.
The main goal of the present text is to prove a non-archimedean analogue of the aboveresults. Namely, let k be a non-archimedean field of characteristic 0, we prove the following: Theorem 1.1.
Let f : X → Y be a morphism of (locally geometric) derived k -analytic stacks. Consider the Hodge completed derived de Rham complex associated to f , defined as dR an X/Y := Sym an ( L an X/Y [ − ∧ , in CAlg k . Then one has a natural multiplicative equivalence dR an X/Y ’ Γ( Y ∧ X , O Y ∧ X ) , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 3 of derived k -algebras, where Y ∧ X denotes the derived k -analytic stack corresponding to the formal completion of X in Y along the morphism f . Unfortunately, the methods of B. Bhatt are algebraic in nature and cannot be directly extrapolated to thenon-archimedean setting. Instead, we follow closely the approach of Gaitsgory-Rozemblyum given in [GR17b,§9]. The main idea consists of defining the non-archimedean deformation to the normal cone as a derived k -analytic stack D an X/Y ∈ dAnSt k , mapping naturally to the k -analytic affine line A k . Denote by p : D an X/Y → A k the natural morphism. ThenTheorem 1.1 follows immediately from the following more precise result (see Theorem 3.42): Theorem 1.2.
Let f : X → Y be a morphism of locally geometric derived k -analytic stacks. Then thenon-archimedean deformation to the normal cone satisfies the following assertions:(1) The fiber of p : D an X/Y → A k at { } ⊆ A k identifies naturally with the completion of the shifted analytictangent bundle T an X/Y [1] ∧ , completed along the zero section s : X → T an X/Y [1] . Moreover for λ = 0 we have a natural identification ( D an X/Y ) λ ’ Y ∧ X , where Y ∧ X denotes the formal completion of Y at X along f .(2) There exists a natural sequence of morphisms admitting a deformation theory X × A k = X (0) → X (1) , → · · · , → X ( n ) , → · · · → Y × A k , such that the colimit of derived k -analytic stacks identifies naturally colim n ≥ X ( n ) ’ D an X/Y . In particular, the latter induces the usual Hodge filtration on (global sections of) T an X/Y [1] ∧ and itinduces a further filtration on (global sections of) the formal completion Y ∧ X , which we shall also referto as the Hodge filtration on the formal completion Y ∧ X ;(3) When f : X → Y is a locally complete intersection morphism of derived k -affinoid spaces, the Hodgefiltration on Y ∧ X identifies with the I -adic filtration on global sections. More precisely, let A := Γ( X, O alg X ) and B := Γ( Y, O alg Y ) , denote the corresponding derived global sections derived k -algebras. Then the global sections of Y ∧ X agree with the completion B ∧ I ∈ CAlg k , I := fib( B → A ) , and the Hodge filtration on B ∧ I coincides with the usual I -adic filtration. For applications, it is desirable to establish Theorem 1.2 in such generality. As an example, it is interesting tohave at our disposal a natural Hodge filtration for the de Rham cohomology of the moduli of ‘ -adic continuousrepresentations of an étale fundamental group, see [Ant17] and [Ant19b]. The latter is a locally geometricderived k -analytic stack. Notwithstanding, the reader can safely assume that f is simply a morphism of(derived) k -analytic spaces.In order to construct the deformation to the normal cone D an X/Y we shall need two main ingredients. Namely,a general theory of k -analytic formal moduli problems and the relative analytification functor introduced in[HP18] (see also [PY20, §6]). So far, a general theory of analytic formal moduli problems was lacking in the JORGE ANTÓNIO literature. For this reason, we devote the whole §2 to develop such framework. Let X ∈ dAn k be a (derived) k -analytic space. Denote by AnNil cl X/ the ∞ -category of closed embeddings g : X → S, where S ∈ dAn k , such that g red : X red → S red is an isomorphism of k -analytic spaces. We shall define a k -analytic formal moduli problem under X (resp. over X ) as a functor Y : (AnNil cl X/ ) op → S , (resp. Y : (AnNil cl /X ) op → S )satisfying some additional requirements, see Definition 2.26 (resp. Definition 2.43). It turns out that analyticformal moduli problems can be described in very explicit terms as follows: Theorem 1.3.
Let X ∈ dAn k denote a derived k -analytic space. Then the following assertions hold:(1) The ∞ -category AnNil cl X//Y (resp. (AnNil cl /X ) /Y ) is filtered and we have a natural equivalence colim S ∈ AnNil cl X//Y S ’ Y resp. colim X ∈ (AnNil cl /X ) /Y S ’ Y ; (2) The natural morphism f : X → Y (resp. f : Y → X ) admits a relative (pro-)cotangent complex L X/Y ∈ Pro(Coh + ( X )) , which completely controls the deformation theory of the morphism f ;(3) In the case where X is a derived k -affinoid space, the ∞ -category of analytific formal moduli problems AnFMP X/ is presentable and monadic over the usual ∞ -category of ind-coherent sheaves on A := Γ( X, O alg X ) . We refer the reader to Corollary 2.47, Proposition 2.30, Corollary 2.55 and Proposition 2.81, for a detailedtreatment of Theorem 1.3.Granting a general deformation theory we are able to construct the deformation D an X/Y as a k -analytic formalmoduli problem under X . Furthermore, we are able to extrapolate the main formal properties of the deriveddeformation to the normal cone in the algebraic setting, proved in [GR17b, §9]. The latter is done via a carefulanalysis of the behaviour of the relative analytification functor, denoted ( − ) an Y , and by means of Noetherianapproximation.In particular, when f : X → Y is a morphism of derived k -affinoid spaces, we have a natural identification D an X/Y ’ ( D X alg /Y alg ) an Y , where X alg := Spec A and Y alg := Spec B , with A and B are as in Theorem 1.2.Along the way, we establish a precise comparison between the works of [Bha12] and [GR17b, §9], which weconsider to be of relevance for experts in derived algebraic geometry. Moreover, we shall also explicit mentionthat the techniques employed in the current text, namely Noetherian approximation, allows to construct thedeformation to the normal cone and its Hodge filtration as in [GR17b, §9] outside the the almost of finitepresentation scope, provided that we work under Noetherian assumptions. The latter might be helpful even inpurely algebraic contexts.1.3. Relation with other works and future questions.
As previously explained, the content of this text isbuild upon two major prior works, namely [GR17b, §9] and [Bha12]. It follows essentially from Proposition 3.8that Bhatt’s construction of the Hodge filtration on algebraic de Rham cohomology with the approach ofGaitsgory-Rozemblyum (c.f. [GR17b, §9]). Such comparison result is not completely obvious. Indeed, in[Bha12] the author produces the Hodge filtration in the general case, by left Kan extending the adic filtrationfrom the case of local complete intersection morphisms, whereas the construction of [GR17b, §9] is completelycanonical. Therefore, there are two main advantages provided by our comparison result: first it proves the
PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 5 canonicity of Bhatt’s approach and secondly it enlightens many important aspects of [GR17b, §9], which werenot made completely explicitly.In [GK + analytic de Rham cohomology theory for a very large class of rigid k -analytic spaces. Let X be a rigid k -analytic space, Grosse-Klönne’s construction is related to over-convergent global sections of formal completions along closed immersion i : X , → Y , where Y is a smooth rigid k -analyticspace. Furthermore, the author proves finiteness of over-convergent analytic de Rham cohomology under certainfiniteness conditions on X .We expect that our derived de Rham cohomology theory, dR an X/k , can be compared to the constructionin [GK + k -analytic spaces, similar to the algebraic case (see[Bha12, §1, Corollary]). When k is a field of equi-characteristic zero, we expect that a non-smooth extension ofthe main construction of [AT19], in the non-archimedean setting, could be related to the formal completion Y ∧ X .More interestingly, would be to study the p -adic setting. Even though the author currently does not have anidea of how to accomplish this, it might be useful to extrapolate certain results in [Bra18] to the singular case,via derived geometry.Another future goal closely related to this work is the study of a rigid cohomology theory for singularalgebraic varieties and possibly algebraic stacks over perfect fields of positive characteristic. Our hope is thattechniques from derived geometry can be useful to reduce questions about the rigid cohomology of singularalgebraic varieties to the free case, via simplicial polynomial resolutions of general k -algebras.Finally, we expect that the results in this text provide a first step in a consistent study of D X -modulesand curved modules outside the smooth case, in the non-archimedean setting. In order to develop such aframework, one has to set the main ingredients of [BZN12] in non-archimedean geometry. One such ingredientis the deformation to the normal cone done in the current text. Another important ingredient is a k -analyticHKR theorem, which is a work in progress with M. Porta and F. Petit (see the author’s thesis [Ant19a] for asketch of proof of the latter statement).1.4. Organization of the paper.
In §2, we develop a theory of formal moduli problems in the non-archimedeansetting. We devote 2.1 to a brief review of the main notions in non-archimedean geometry that will be importantfor us. We then proceed to §2.2 where a careful study of the structure of nil-isomorphisms is performed. Thenotion of nil-isomorphism is basic building block of the theory of analytic formal moduli problems under (resp. over ) a given base. The latter notions are introduced and study at length in §2.3 (resp. §2.4). In §2.4, weestablish pseudo -nil-descent for pro-almost perfect complexes or in other words pro-coherent complexes. Givena morphism X → Y which exhibits X as an analytic formal moduli problem, we prove that the ∞ -category ofPro(Coh + ( Y )) does satisfy descent along the usual ∗ -upper functoriality, that it is, it can be realized as thetotalization of Pro(Coh + ( X • )) , where X • denotes the Čech nerve associated to X → Y . The main novelty compared with the main result in[GR17a, §7.2] is that we use the entire Coh + and the ∗ -upper functoriality. Our nil-descent statement is basedin the work of Halpern-Leistner and Preygel, c.f. see [HLP14, Theorem 3.3.1]. Section 2.6 is devoted to thestudy of the notion of formal groupoids over a base X . We will prove the existence of a natural equivalencebetween the ∞ -categories of analytic formal moduli problems under X and analytic formal groupoids over X JORGE ANTÓNIO (see Theorem 2.65). The content of this equivalence can be interpreted as: every analytic formal moduli problemunder X arises as a classifying formal derived k -analytic stack associated to an analytic formal groupoid (whichcorresponds to the associated Čech nerve). This equivalence of ∞ -categories is based on the pseudo-nil-descentresults, proved earlier. Notice that Theorem 2.65, is a direct non-archimedean analogue of [GR17b, §5, Theorem2.3.2]. In §2.6, we study the tangent complex associate to an analytic formal moduli problem under X . Inparticular, we prove that such a functor is monadic, see Proposition 2.81.The main bulk of the text lies in §3, where we introduce the deformation to the normal cone, D X/Y , in thenon-archimedean setting. In §3.1 we review the algebraic situation. We shall summarize the main algebraicformal properties of D X/Y and then proceed to give a simplification of the formalism developped in [GR17b,§9] in the case of closed immersions of derived schemes, see Proposition 3.8. This will enable us to prove acomparison statement between Bhatt’s construction of the Hodge filtration in algebraic de Rham cohomology,c.f. [Bha12] and the work of Gaitsgory-Rozemblyum.In §3.2 we introduce the non-archimedean deformation in the local case, via the relative analyticificationfunctor and Noetherian approximation. We devote §3.3 to gluing the deformation to the class of morphismsbetween locally geometric derived k -analytic stacks. In particular, our results hold for any derived k -analyticspace. We then introduce the construction of the non-archimedean Hodge filtration in the local, in §3.4. This ispossible by extending the usual Hodge filtration of Gaitsgory-Rozemblyum to the non-archimedean settingas follows: we first extrapolate the main results in [GR17b, §9.5] to the Noetherian setting not necessarily ofalmost of finite presentation (which was the only case treated in [GR17b, §9]) and then to the non-archimedeansetting via the relative analytification functor. Finally, in §3.4 we glue the non-archimedean Hodge filtrationfor general morphisms between locally geometric derived k -analytic stacks.1.5. Notations and Conventions.
In this text, k denotes a non-archimedean field of characteristic zero. Welet dAff k and dAff laft k denote the ∞ -categories of affine derived schemes (resp. affine derived schemes almost offinite presentation). We shall denote by dSt k the ∞ -category of derived k -stacks. We shall further denote bydSt laft k ⊆ dSt k the full subcategory spanned by derived k -stacks locally almost of finite presentation. Givenderived k -stacks X, Y ∈ dSt k we shall denote byMap( X, Y ) ∈ dSt k , the corresponding mapping stack. If X and Y live over A k , we then shall denote byMap / A k ( X, Y ) ∈ (dSt k ) / A k . We denote dAnSt k the ∞ -category of derived k -analytic stacks introduced in [PY18b]. We shall denote bydAn k and dAfd k the ∞ -categories of derived k -analytic spaces (resp., derived k -affinoid spaces). We will furtherdenote by An k the usual ordinary category of (discrete) k -analytic spaces as in [Ber93]. We shall instead usebold letters for the corresponding notions in the non-archimedean setting. Therefore, we shall denote by Map ( X, Y ) ∈ dAnSt k , the corresponding k -analytic mapping stack, whenever X, Y ∈ dAnSt k . Let C be a stable ∞ -category. We shalldenote by C fil the associated ∞ -category of filtered objects in C . Similarly, we denote by C gr the ∞ -category ofgraded objects in C . We shall employ the notations C gr , ≥ and C gr , = n , the respective full subcategories of positively indexed graded objects in C and objects concentrated in a singledegree n . Notice that the latter ∞ -category is equivalent to C itself. PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 7
Acknowledgments.
The author would like to express his deep gratitude to Mauro Porta for the encour-agement to pursue this project and for many commentaries and suggestions during the realization of the presentwork. The author would also like to thank Marco Robalo for a clarifying remark.2.
Analytic formal moduli problems
Preliminaries.
Recall the notions of k -affinoid/analytic spaces introduced in [PY18b, Definition 7.3 andDefinition 2.5.], respectively. Definition 2.1.
Let f : X → Y be a morphism in the ∞ -category dAn k . We shall say that f is an affinemorphism if for every morphism Z → Y in dAn k such that Z is a derived k -affinoid space, the pullback Z := Z × X Y ∈ dAn k , is a derived k -affinoid space. Definition 2.2.
Let f : X → Y be a morphism in dAn k . We shall say that f is an admissible open immersion if the induced morphism on 0-th truncationst ( f ) : t ( X ) → t ( Y ) , is an admissible open immersion in the sense of [Ber93, §1.3]. Definition 2.3.
Let f : X → Y be a morphism in An k . We shall say that f is a finite morphism if f is affineand for every admissible open covering a j ∈ J V j → Y, the base change morphism U j := V j × Y Y → V j , exihibits the k -affinoid algebra B j := Γ( U j , O U j ) as a finite A = Γ( V j , O V j )-module. Given f : X → Y in the ∞ -category dAn k , we say that f is finite if its truncation t ( f ) is a finite morphism. Definition 2.4.
Let X ∈ An k denote an (ordinary) k -analytic space. We denote by J X ⊆ O X , the nil-radicalideal of O X , that is the sheaf ideal spanned by nilpotent sections on X . We denote by X red the k -analytic spaceobtained as the pair ( X, O X / J X ), which we shall refer to as the underlying reduced k -analytic space associatedto X . Remark . Let X ∈ An k , then the underlying reduced X red ∈ An k is a reduced k -analytic space, byconstruction. Notation 2.6.
We denote by An red k ⊆ An k the full subcategory spanned by reduced k -analytic spaces. Wefurther denote by ( − ) red : dAn k → An red k , the functor obtained by the association Z ∈ An k Z red ∈ An red k . We now extend the construction ( − ) red : An k → An red k to the ∞ -category of derived k -analytic spaces: Definition 2.7.
Consider the functor dAn k → An red k defined as the composite( − ) red : dAn k t ( − ) −−−→ An k ( − ) red −−−−→ An red k . Given X ∈ dAn k a derived k -analytic space we denote by X red ∈ An red k the underlying reduced k -analytic space associated with X . JORGE ANTÓNIO
Lemma 2.8.
Let f : X → Y be an admissible open immersion of derived k -analytic spaces. Then f red : X red → Y red is an admissible open immersion, as well.Proof. By the definitions, it is clear that the truncationt ( f ) : t ( X ) → t ( Y ) , is an admissible open immersion of ordinary k -analytic spaces. In the case of ordinary k -analytic spaces thequestion is local and we reduce ourselves to the affinoid case. Then it is clear that localization in the ordinarycategory Afd k commutes with quotients (as the former is defined via a colimit). (cid:3) Definition 2.9.
In [PY17, Definition 5.41] the authors introduced the notion of a square-zero extensionbetween derived k -analytic spaces. In particular, given a morphism f : Z → Z in dAn k , we shall say that f has the structure of a square-zero extension if f exhibits Z as a square-zero extension of Z . Remark . Let X ∈ An k . Let J ⊆ O X be an ideal satisfying J = 0. Consider the fiber sequence J → O X → O X / J , in the ∞ -category Coh + ( X ). We have an induced natural fiber sequence of the form L an O X → L an O X / J → L an( O X / J ) / O X , and we have a further identification τ ≤ ( L ( O X / J ) / O X ) ’ J [1]. For this reason, we obtain a well defined morphism d : L O X / J → J [1] , in the derived ∞ -category Mod O X / J . This derivation classifies a square-zero extension of O X / J by J [1] whichcan be identified with the object O X itself. In particular, we deduce that X is a square-zero extension of theordinary k -analytic space ( X, O X / J ). Remark . Recall the ∞ -category T op R ( T an ( k )) of T an ( k )-structured ∞ -topoidefined in [PY18b, Definition2.4], where T an ( k ) denotes the k -analytic pre-geometry (see for instance [PY18b, Construction 2.2]). Let O ∈ Str loc T an ( k ) ( X ) be a local T an ( k )-structure on X (c.f. [PY18b, Definition 2.4]). Since the pregeometry T an ( k )is compatible with n -truncations, c.f. [PY18b, Theorem 3.23], it follows that π ( O ) ∈ Str loc T an ( k ) ( X ), as well.Moreover, if ( X , O ) ∈ T op R ( T an ( k )) is a T an ( k )-structured ∞ -topos, we define its underlying reduced T an ( k ) -structured ∞ -topos as the T an ( k )-structured space( X , π ( O ) / J ) ∈ T op R ( T an ( k )) , where J ⊆ π ( O ) denotes the ideal sheaf spanned by nilpotent sections on π ( O ). Moreover, the quotient π ( O ) / J is considered in the ∞ -category of local structures on X π ( O ) / J ∈ Str loc T an ( k ) ( X ) . Recall the notion of the underlying algebra functor ( − ) alg : Str loc T an ( k ) ( X ) → CAlg k ( X ) introduced in [PY18b,Lemma 3.13]. Lemma 2.12.
Let Z := ( Z , O Z ) ∈ T op R ( T an ( k )) denote a T an ( k ) -structure ∞ -topos such that π ( O alg Z ) is aNoetherian derived k -algebra on Z . Suppose that the reduction Z red is equivalent to a derived k -affinoid space.Then the truncation t ( Z ) is isomorphic to an ordinary k -affinoid space. If we assume further that for every i > , the homotopy sheaves π i ( O Z ) are coherent π ( O Z ) -modules, then Z itself is equivalent to a derived k -affinoid space. PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 9
Proof.
The second claim of the Lemma follows readily from the first assertion together with the definitions.We are thus reduced to prove that t ( Z ) is isomorphic to an ordinary k -affinoid space. Let J ⊆ π ( O Z ), denotethe coherent ideal sheaf associated to the closed immersion Z red , → Z . Notice that the ideal J agrees with thenilradical ideal of π ( O Z ). Thanks to our assumption that π ( O Z ) is a Noetherian derived k -algebra on Z , itfollows that there exists a sufficiently large integer n ≥ J n = 0 . Arguing by induction, we can suppose that n = 2, that is to say that J = 0 . In particular, Remark 2.10 implies that the the natural morphism Z red → Z has the structure of a square-zeroextension. The assertion now follows from [PY17, Proposition 6.1] and its proof. (cid:3) Remark . We observe that the converse of Lemma 2.12 holds true. Indeed, the natural morphism Z red → Z is a closed immersion. In particular, if Z ∈ dAfd k we deduce readily that Z red ∈ Afd k , as well. Lemma 2.14.
Let f : X → Y be an affine morphism in dAn k . Suppose we are given an admissible opencovering g : a j ∈ J U j → Y, where for each j ∈ J , U j ∈ dAfd k . For each j ∈ J , let V j := U j × X Y ∈ dAfd k , then ‘ j ∈ J V j → Y is an admissible open covering by derived k -affinoid spaces.Proof. It is clear from our assumption that f is an affine morphism that for every index j ∈ J , the objects V j ∈ dAfd k . The claim of the Lemma follows immediately from the observation that both the classes ofeffective epimorphisms of ∞ -topoi and admissible open immersions of derived k -analytic spaces are stable underpullbacks, cf. [Lur09, Proposition 6.2.3.15] and [PY17, Corollary 5.11, Proposition 5.12], respectively. (cid:3) Non-archimedean differential geometry.
In this §, we introduce the notion of a nil-isomorphismbetween derived k -analytic spaces and study certain important features of such class of morphisms. The resultscontained in this paragraph will be of crucial importance to the study of analytic formal moduli problems. Definition 2.15.
Let f : X → Y be a morphism in dAn k . We say that f is a nil-isomorphism if f is almostof finite presentation and furthermore f red : X red → Y red is an isomorphism of ordinary k -analytic spaces. Wewill denote by AnNil X/ the full subcategory of (dAn k ) X/ spanned by nil-isomorphisms X → Y . Lemma 2.16.
Let f : X → Y be a nil-isomorphism of derived k -analytic spaces. Then:(1) Given any morphism Z → Y in dAn k , the base change morphism Z × X Y → Z, is an nil-isomoprhism, as well;(2) f is an affine morphism;(3) f is a finite morphism.Proof. To prove (1), it suffices to prove that the functor (-) red : dAn k → An red k commutes with finite limits.The truncation functor t : dAn k → An k , commutes with finite limits, c.f. [PY18b, Proposition 6.2 (5)]. It suffices then to prove that the usual underlyingreduced functor ( − ) red : An k → An red k , commutes with finite limits. By construction, the latter assertion is equivalent to the claim that the usualcomplete tensor product of ordinary k -affinoid algebras commutes with the operation of taking the quotient bythe Jacobson radical, which is immediate.We now prove (2). Let Z → Y be an admissible open immersion such that Z is a derived k -affinoid space.We claim that the pullback Z × X Y is again a derived k -affinoid space. Thanks to Lemma 2.12 we reduceourselves to prove that ( Z × X Y ) red is equivalent to an ordinary k -affinoid space. Thanks to (1), we deducethat the induced morphism ( Z × X Y ) red → Z red , is an isomorphism of ordinary k -analytic spaces. In particular, ( Z × X Y ) red is a k -affinoid space. The resultnow follows by invoking Lemma 2.12 again.To prove (3), we shall show that the induced morphism on the underlying 0-th truncations t ( X ) → t ( Y ) isa finite morphism of ordinary k -affinoid spaces. But this follows immediately from the fact that both t ( X )and t ( Y ) can be obtained from the reduced X red by means of a finite sequence of square-zero extensions as inRemark 2.10. (cid:3) Definition 2.17.
A morphism X → Y be a morphism in dAn k is called a nil-embedding if the inducedmorphism of ordinary k -analytic spaces t ( X ) → t ( Y ) is a closed immersion with nilpotent ideal of definition. Proposition 2.18.
Let f : X → Y be a nil-embedding of derived k -analytic spaces. Then there exists a sequenceof morphisms X = X , → X , → · · · , → X n = X , → X . . . X n , → · · · , → Y, such that for each ≤ i ≤ n and j ≥ , the morphisms X i , → X i +10 and X j → X j +1 have the structure ofsquare-zero extensions. Furthermore, the induced morphisms t ≤ j ( X j ) → t ≤ j ( Y ) are equivalences of derived k -analytic spaces, for every j ≥ .Proof. The proof follows the same scheme of reasoning as [GR17b, Proposition 5.5.3]. For the sake ofcompleteness we present the complete argument here. Consider the induced morphism on the underlyingtruncations t ( f ) : t ( X ) → t ( Y ) . By construction, there exists a sufficiently large integer n ≥ J n +1 = 0 , where J ⊆ π ( O Y ) denotes the ideal associated to the nil-embedding t ( f ). Therefore, we can factor the latteras a finite sequence of square-zero extensions of ordinary k -analytic spacest ( X ) , → X ord , , → · · · , → X ord ,n = t ( Y ) , as in the proof of Lemma 2.12. For each 0 ≤ i ≤ n , we set X i := X G t ( X ) X ord ,i . By construction, we have that the natural morphism t ( X n ) → t ( Y ) is an isomorphism of ordinary k -analyticspaces. We now argue by induction on the Postnikov tower associated to the morphism f : X → Y . Suppose PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 11 that for a certain integer i ≥
0, we have constructed a derived k -analytic space X i together with morphisms g i : X → X i and h i : X i → Y such that f ’ h i ◦ g i and the induced morphismt ≤ i ( X i ) → t ≤ i ( Y )is an equivalence of derived k -analytic spaces. We shall proceed as follows: by the assumption that h i is( i + 1)-connective, we deduce from [PY17, Proposition 5.34] the existence of a natural equivalence τ ≤ i ( L an X i /Y ) ’ , in Mod O Xi . Consider the natural fiber sequence h ∗ i L an Y → L an X i → L an X i /Y , in Mod O Xi . The natural morphism L an X i /Y → π i +1 ( L an X i /Y )[ i + 1] , induces a morphism L an X i → π i +1 ( L an X i /Y )[ i + 1], such that the composite h ∗ i L an Y → L an X i → π i +1 ( L an X i /Y )[ i + 1] , (2.1)is null-homotopic, in Mod O Xi . By the universal property of the relative analytic cotangent complex, (2.1)produces a square-zero extension X i → X i +1 , together with a morphism h i +1 : X i +1 → Y , factoring h i : X i → Y . We are reduced to show that the morphism O Y → h i +1 , ∗ ( O X i +1 ) , is ( i + 2)-connective. Consider the commutative diagram h i, ∗ ( π i +1 ( L an X i /Y ))[ i + 1] h i +1 , ∗ ( O X i +1 ) h i, ∗ ( O X i ) I O Y h ∗ ( O X i ) J J s i = , (2.2)in Mod O Y , where both the vertical and horizontal composites are fiber sequences. By our inductive hypothesis, I is ( i + 1)-connective. Moreover, thanks to [PY17, Proposition 5.34] we can identify the natural morphism s i : I → h i, ∗ ( π i +1 ( L an X i /Y ))[ i + 1]with the natural morphism I → τ ≤ i +1 ( I ). We deduce that the fiber of the morphism s i must be necessarily( i + 2)-connective. The latter observation combined with the structure of (2.2) implies that h i +1 : X i +1 → Y induces an equivalence of derived k -analytic spacest ≤ i +1 ( X i +1 ) → t ≤ i +1 ( Y ) , as desired. (cid:3) Corollary 2.19.
Let X ∈ dAn k . Then the following assertions hold:(1) The natural morphism X red → X, in dAn k , can be approximated by successive square-zero extensions; (2) For each n ≥ , the natural morphism X red → t ≤ n ( X ) , can be approximated by a finite number of square-zero extensions.Proof. Both the assertions of the Corollary follow readily from Proposition 2.18 by observing that the canonicalmorphisms X red → X and X red → t ≤ n ( X ) have the structure of nil-embeddings and that in the latter case thefiniteness assumption on the Postkinov tower forces the finiteness of the approximation sequence. (cid:3) Lemma 2.20.
Let f : X → Y be a finite morphism of derived k -affinoid spaces. Let Z → Y be an admissibleopen immersion and denote by A := Γ( X, O alg X ) , B := Γ( Y, O alg Y ) , C := Γ( Z, O alg Z ) , the corresponding derived k -algebras of derived global sections. Consider the base change Z := Z × Y X ∈ dAfd k . Then one has a natural equivalence Γ( Z , O alg Z ) ’ A ⊗ B C, in the ∞ -category CAlg k .Proof. The Lemma is a direct consequence of [PY18b, Proposition 3.17] (iii). (cid:3)
Lemma 2.21.
Let f : X → Y be a finite morphism of derived k -affinoid spaces and g : Z → Y an admissibleopen immersion in dAfd k . Form the pullback diagram Z XZ Y, g f fg in the ∞ -category dAfd k . Then the commutative diagram Coh + ( Y ) Coh + ( X )Coh + ( Z ) Coh + ( Z ) , f ∗ g ∗ ( g ) ∗ ( f ) ∗ is right adjointable. In other words, the Beck-Chevalley natural transformation α : g ∗ ◦ f ∗ → ( f ) ∗ ◦ g is an equivalence of functors.Proof. Since f is assumed to be a finite morphism of derived k -affinoid spaces, it follows from the derived Tateaciclicity theorem, c.f. [PY18a, Theorem 3.1] that the right adjoint f ∗ : Coh + ( X ) → Coh + ( Y ) is well defined.The assertion of the Lemma is now an immediate consequence of Lemma 2.20 together with [Lur18, Proposition2.5.4.5] and the derived Tate aciclicity theorem. (cid:3) Proposition 2.22.
Let f : S → S be a nil-isomorphism between derived k -analytic spaces. Then the pullbackfunctor f ∗ : Coh + ( S ) → Coh + ( S ) , admits a well defined right adjoint, f ∗ . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 13
Proof.
Since f : S → S is a nil-isomorphism, we conclude from Lemma 2.16 that f is an affine morphismbetween derived k -analytic spaces. By admissible descent of Coh + , cf. [AP19, Theorem 3.7], together withLemma 2.14 and Lemma 2.21 we reduce the statement of the Lemma to the case where both S and S are equivalent to derived k -affinoid spaces. In this case, the result follows by our assumptions on f andLemma 2.16. (cid:3) Proposition 2.23.
Let f : X → Y be a nil-embedding of derived k -analytic spaces. Let g : X → Z be a finitemorphism in dAn k . The the diagram X YZ fg admits a pushout in dAn k , denoted Z . Moreover, the natural morphism Z → Z is also a nil-embedding.Proof. The ∞ -category of T an ( k )-structured ∞ -topoi T op R ( T an ( k )) is a presentable ∞ -category. Consider thepushout diagram X YZ Z , fg in the ∞ -category T op R ( T an ( k )). By construction, the underlying ∞ -topos of Z can be computed as thepushout in the ∞ -category T op R of the induced diagram on the underlying ∞ -topoi of X , Z and Y . Moreover,since g is a nil-isomorphism it induces an equivalence on underlying ∞ -topoi of both X and Y . It follows thatthe induced morphism Z → Z in T op R ( T an ( k )) induces an equivalence on the underlying ∞ -topoi. Moreover,it follows essentially by construction that we have a natural equivalence O Z ’ g ∗ ( O Y ) × g ∗ ( O X ) O Z ∈ Str loc T an ( k ) ( Z ) . As the morphism g ∗ ( O Y ) → g ∗ ( O X ) is an effective epimorphism and effective epimorphisms are preserved underfiber products in an ∞ -topos, c.f. [Lur09, Proposition 6.2.3.15], it follows that the natural morphism O Z → O Z , is an effective epimorphism, as well. Consider now the commutative diagram of fiber sequences J O Z O Z J g ∗ ( O Y ) g ∗ ( O X ) , in the stable ∞ -category Mod O Z . Since the right commutative square is a pullback square it follows that themorphism J → J , is an equivalence. In particular, π ( J ) is a finitely generated nilpotent ideal of π ( O alg Z ). Indeed, finitelygeneration follows from our assumption that g is a finite morphism. Thanks to Lemma 2.12, it follows thatt ( Z ) is an ordinary k -analytic space and the morphism t ( Z ) → t ( Z ) is a nil-embedding. We are thus reduced to show that for every i >
0, the homotopy sheaf π i ( O Z ) ∈ Coh + (t ( Z )). But this follows immediatelyfrom the existence of a fiber sequence O Z → g ∗ ( O Y ) ⊕ O Z → g ∗ ( O X ) , in the ∞ -category Mod O Z together with the fact that g ∗ ( O Y ) and g ∗ ( O Z ) have coherent homotopy sheaves, byour assumption that g is a finite morphism combined with Lemma 2.16. (cid:3) Corollary 2.24.
Let f : S → S be a square-zero extension and g : S → T a nil-isomorphism in dAn k . Supposewe are given a pushout diagram S S T T f , in dAn k . Then the induced morphism T → T is a square-zero extension.Proof. Since g is a nil-isomorphism of derived k -analytic spaces, Proposition 2.22 implies that the pullbackfunctor g ∗ : Coh + ( T ) → Coh + ( S ) admits a well defined right adjoint g ∗ : Coh + ( S ) → Coh + ( T ) . Let F ∈ Coh + ( S ) ≥ and d : L an S → F [1] be a derivation associated with the morphism f : S → S . Considernow the natural composite d : L an T → g ∗ ( L an S ) g ∗ ( d ) −−−→ g ∗ ( F )[1] , in the ∞ -category Coh + ( T ). By the universal property of the relative analytic cotangent complex, we deducethe existence of a square-zero extension T → T , in the ∞ -category dAn k . Let X ∈ dAn k together with morphisms S → X and T → X compatible with both f and g . By the universal property of the relative analytic cotangent complex, the morphism S → X inducesa uniquely defined (up to a contractible indeterminacy space) morphism L an S/X → F [1] , in Coh + ( S ), such that the compositve L an S → L an S/X → F [1] agrees with d . By applying the right adjoint g ∗ above we obtain a commutative diagram L an T L an T/X g ∗ ( L an S ) g ∗ ( L an S/X ) g ∗ ( F )[1] , can d g ∗ (can) in the ∞ -category Coh + ( T ). From this, we conclude again by the universal property of the relative analyticcotangent complex the existence of a uniquely defined natural morphism T → X extending both T → X and S → X and compatible with the restriction to S . The latter assertion is equivalent to state that thecommutative square S S T T , is a pushout diagram in dAn k . The proof is thus concluded. (cid:3) PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 15
Construction . Let n ≥
0. We have a well defined functorAnNil t ≤ n ( X ) / → AnNil X/ , given by the formula (t ≤ n ( X ) → S ) ∈ AnNil t ≤ n ( X ) / ( X → S G t ≤ n ( X ) X ) . Given any functor F : AnNil X/ → S , we define F ≤ n : AnNil t ≤ n ( X ) / → S as the functor given on objects by theassociation (t ≤ n ( X ) → S ) ∈ AnNil t ≤ n ( X ) / F ≤ n ( S G t ≤ n ( X ) X ) ∈ S . It follows from the construction that, for each n ≥ S ∈ AnNil X/ , we have a natural morphism F ( S ) → F ≤ n (t ≤ n ( S )) , in the ∞ -category S .2.3. Analytic formal moduli problems under a base.
We study the ∞ -category of k -analytic formalmoduli problems under a base X ∈ dAn k and explore certain important features of such. The results presentedhere will prove to be crucial for the study of the deformation to the normal cone in the k -analytic setting,which we treat in the next section. We start with the following central definition: Definition 2.26. An analytic formal moduli problem under X corresponds to the datum of a functor F : (AnNil X/ ) op → S , satisfying the following two conditions:(1) F ( X ) ’ ∗ in S ;(2) Given any S ∈ AnNil op X/ , the natural morphism F ( S ) → lim n ≥ F ≤ n (t ≤ n ( S )) , is an equivalence in S (see Construction 2.25). In other words, Y is convergent ;(3) Given any pushout diagram S S T T , f in the ∞ -category AnNil X/ , such that f has the structure of a square-zero extension, the inducedmorphism F ( T ) → F ( T ) × F ( S ) F ( S ) , is an equivalence in S .We shall denote by AnFMP X/ the full subcategory of Fun((AnNil X/ ) op , S ) spanned by analytic formal moduliproblems under X . Remark . In the previous definition we remark that the functor F ≤ n : AnNil opt ≤ n ( X ) / → S , is itself an analytic formal moduli problem under t ≤ n ( X ).We shall give some important examples of formal moduli problems under X : Example 2.28. (1) Let X ∈ dAn k . As in the algebraic case, we can consider the de Rham pre-stackassociated to X , X dR : dAfd op k → S , determined by the formula X dR ( Z ) := X ( Z red ) , Z ∈ dAfd k . We have a natural morphism X → X dR induced from the natural morphism Z red → Z . The restrictionof X dR to Fun(AnNil op X/ , S ) via the functor h ∗ : dAnSt k → AnFMP X/ , induced by the inclusion h : AnNil X/ ⊆ dAn k , is an analytic formal moduli problem under X . Moreover,it follows from the construction of X dR that the object h ∗ ( X dR ) is equivalent to a final object inAnFMP X/ .(2) Let f : X → Y be a morphism in the ∞ -category dAn k . We define the formal completion of X in Y along f as the pullback Y ∧ X := Y × Y dR X dR ∈ dAnSt k . By construction we have a natural factorization X → Y ∧ X → Y in dAnSt k , and moreover the restrictionof X → Y ∧ X to the ∞ -category Fun(AnNil op X/ , S ) along the natural functor h ∗ : (dAnSt k ) X/ → AnFMP X/ , exhibits Y ∧ X as a formal moduli problem under X .(3) Let f : X → Y be a closed immersion in the ∞ -category dAn k . Consider the shifted tangent bundleassociated to f together with the zero section X T an X/Y [1] X. s p The completion T an X/Y [1] ∧ ∈ AnFMP X/ along s will play an important role in what follows. Notation 2.29.
We set AnNil cl X/ ⊆ AnNil X/ to be the full subcategory spanned by those objects correspondingto nil-embeddings X → S, in dAn k . Proposition 2.30.
Let Y ∈ AnNil X/ . The following assertions hold:(1) The inclusion functor AnNil cl X//Y , → AnNil
X//Y , is cofinal.(2) The natural morphism colim Z ∈ AnNil cl X//Y Z → Y, is an equivalence in Fun(AnNil op X/ , S ) .(3) The ∞ -category AnNil cl X//Y is filtered.
PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 17
Proof.
We start by proving claim (1). Let n ≥
0, and consider the usual restriction along the natural morphism X red → t ≤ n ( X ) functor res ≤ n : AnNil t ≤ n ( X ) / → AnNil X red / . Such functor admits a well defined left adjoint push ≤ n : AnNil X red / → AnNil t ≤ n ( X ) / , which is determined by the formula( X red → T ) ∈ AnNil X red / (t ≤ n ( X ) → T ) ∈ AnNil X/ , where we have set T := t ≤ n ( X ) G X red T ∈ AnNil X/ . (2.3)We claim that T ∈ AnNil t ≤ n ( X ) / belongs to the full subcategory AnNil clt ≤ n ( X ) / ⊆ AnNil t ≤ n ( X ) / . Indeed, sincethe structural morphism X red → T , is necessarily a nil-embedding we deduce the claim from Proposition 2.23.We shall denote by res ≤ n ! ( Y ) : AnNil op X red / → S , the left Kan extension of Y along the functor res ≤ n above. By the colimit formula for left Kan extensions, c.f.[Lur09, Lemma 4.3.2.13], it follows that res ≤ n ! ( Y ) is given by the formula( X red → T ) ∈ AnNil X red / Y ≤ n ( T ) ∈ S , where T is as in (2.3). We thus have a diagram of functors res ≤ n : AnNil t ≤ n ( X ) //Y ≤ n (cid:29) AnNil X red // res ≤ n ! ( Y ) : push ≤ n , where res ≤ n is given on objects by the formula(t ≤ n ( X ) → S → Y ≤ n ) ∈ AnNil t ≤ n ( X ) //Y ≤ n ( X red → S → res ≤ n ! ( Y )) ∈ AnNil X red // res ≤ n ! ( Y ) and the functor push ≤ n is given by the association( X red → T → Y ≤ n ) ∈ AnNil X red //Y ≤ n (t ≤ n ( X ) → T t X red t ≤ n ( X ) → Y ≤ n ) ∈ AnNil t ≤ n ( X ) //Y ≤ n . We claim that the pair ( res ≤ n , push ≤ n ) forms an adjunction. Indeed, a morphism( X red → S → res ≤ n ! ( Y )) → res ≤ n ! (t ≤ n ( X ) → T → Y ≤ n ) , corresponds to a commutative diagram X red S res ≤ n ! ( Y ) X red T res ≤ n ! ( Y ) , = =8 JORGE ANTÓNIO in the ∞ -category Fun(AnNil op X red / , S ). The latter datum is equivalent to the datum of a commutative diagramt ≤ n ( X ) X red S S Y ≤ n X red T T Y ≤ n = . (2.4)Since the morphism t ≤ n ( X ) → T factors through the structural mapt ≤ n ( X ) → T, we deduce that the datum of (2.4) is equivalent to the datum of a commutative diagramt ≤ n ( X ) S Y ≤ n t ≤ n ( X ) T Y ≤ n , = which corresponds to a uniquelly well defined morphism push ≤ n ( X red → S → res ≤ n ! ( Y )) → (t ≤ n ( Y ) → T → Y ≤ n ) , in the ∞ -category AnNil t ≤ n ( X ) //Y ≤ n . We further observe that for every n ≥ m ≥
0, the objects res ≤ n ! ( Y ) and res ≤ m ! ( Y ) , are equivalent as functors AnNil op X red / → S , we shall denote this functor simply by res ! ( Y ).Passing to the limit over n ≥ X red // res ! ( Y ) lim n ≥ AnNil t ≤ n ( X ) //Y ≤ n AnNil
X//Y . The horizontal morphism is cofinal since it fits into an adjunction, by our previous considerations. Thanks to[Lur09, Corollary 4.1.1.9] in order to show that the natural morphismAnNil X red // res ! ( Y ) → AnNil
X//Y , is cofinal, it suffices to prove that AnNil X//Y → lim n ≥ AnNil t ≤ n ( X ) //Y ≤ n is itself cofinal. But the latter isan immediate consequence of the fact that derived k -analytic spaces are nilcomplete, c.f. [PY17, Lemma7.7], combined with assumption (3) in Definition 2.26. Assertion (1) of the Proposition now follows from theobservation that the functor AnNil X red // res ! ( Y ) → AnNil
X//Y , factors through the full subcategory AnNil cl X//Y ⊆ AnNil
X//Y .Claim (2) follows immediately from (1) combined with Yoneda Lemma. To prove (3) we shall make use of[Lur09, Lemma 5.3.1.12]. Let F : ∂ ∆ n → AnNil cl X//Y . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 19
For each [ m ] ∈ ∆ n , denote by S m := F ([ m ]) in AnNil cl X//Y . The pushout S n G X S n − , exists in AnNil cl X/ . We wish to show that S n F X S n − admits a morphism S n G X S n − → Y, compatible with the diagram F . In order to prove the latter assertion, we observe that Proposition 2.18 canfilter the diagram F by diagrams F i → F such that X → F is formed by square-zero extensions and so areeach transition morphisms F i → F i +1 . This implies that for every j ≥
0, we can find morphismst ≤ j ( S n ) G t ≤ j ( X ) t ≤ j ( S n − → Y ≤ j , which are compatible for varying j ≥
0. Moreover, the fact that Y satisfies condition (2) in Definition 2.26implies that we can find a well defined morphism S n G X S n − → Y, which is compatible with F , as desired. (cid:3) As a Corollary we deduce the following important result:
Corollary 2.31.
Let X ∈ dAn k , then the ∞ -category AnFMP X/ is presentable. In particular, the latteradmits all small colimits.Proof. Consider the fully faithful functorAnNil cl X/ , → AnNil X/ → AnFMP X/ . It follows from the definitions that the ∞ -category AnFMP X/ admits filtered colimits. In particular, we obtaina well defined functor F : Ind(AnNil cl X/ ) → AnFMP X/ , which is further fully faithful, since every the image of every ( X → S ) ∈ AnNil X/ is compact in AnFMP X/ . Itfollows from Proposition 2.30 that F itself is essentially surjective. Since the ∞ -category AnNil cl X/ is essentiallysmall we are reduced to show that AnFMP X/ admits all small colimits. We already know that AnFMP X/ admits filtered colimits. We shall prove that AnFMP X/ admits finite colimits as well. As a consequenceof Proposition 2.23, we deduce that the ∞ -category AnNil cl X/ admits all finite colimits. It is now clear thatAnFMP X/ admits finite colimits as well, and the proof is concluded. (cid:3) Definition 2.32.
Let Y ∈ AnFMP X/ denote an analytic formal moduli problem under X . The relativepro-analytic cotangent complex of Y under X is defined as the pro-object L an X/Y := { L an X/Z } Z ∈ AnNil cl X//Y ∈ Pro(Coh + ( X )) , where, for each Z ∈ AnNil cl X//Y L an X/Z ∈ Coh + ( X ) , denotes the usual relative analytic cotangent complex associated to the structural morphism X → Z inAnNil cl X//Y . Remark . Let Y ∈ AnFMP X/ . For a general Z ∈ dAn k , there exists a natural morphism L an X → L an X/Z , in the ∞ -category Coh + ( X ). Passing to the limit over Z ∈ AnNil cl X//Z , we obtain a natural map L an X → L an X/Y , in Pro(Coh + ( X )), as well.The following result justifies our choice of terminology for the object L an X/Y ∈ Pro(Coh + ( X )): Lemma 2.34.
Let Y ∈ AnFMP X/ . Let X , → S be a square-zero extension associated to an analytic derivation d : L an S → F [1] , where F ∈ Coh + ( X ) ≥ . Then there exists a natural morphism Map
AnFMP X/ ( S, Y ) → Map
Pro(Coh + ( X )) ( L an X/Y , F ) × Map
Coh+( X ) ( L an , F ) { d } which is furthermore an equivalence in the ∞ -category S .Proof. Thanks to Proposition 2.30 combined with the Yoneda Lemma we can identify the space of liftings ofthe map X → Y along X → S with the mapping spaceMap AnFMP X/ ( S, Y ) ’ colim Z ∈ AnNil
X//Y
Map
AnNil X/ ( S, Z ) . Fix Z ∈ AnNil cl X//Y . Then we have a natural identification of mapping spacesMap
AnNil X/ ( S, Z ) ’ Map (dAn k ) X/ ( S, Z ) (2.5) ’ Map
Coh + ( X ) ( L an X/Z , F ) × Map
Coh+( X ) ( L an X , F ) { d } , (2.6)see [PY17, §5.4] for a justification of the latter assertion. Passing to the colimit over Z ∈ AnNil cl X//Y , weconclude thanks to the formula for mapping spaces in pro- ∞ -categories that we have a natural equivalenceMap AnFMP X/ ( S, Y ) ’ Map
Pro(Coh + ( X )) ( L an X/Y , F ) × Map
Coh+( X ) ( L an , F ) { d } , as desired. (cid:3) Construction . Let f : Y → Z denote a morphism in AnFMP X/ . Then, for every S ∈ AnNil cl X//Y , theinduced morphism S → Z, in AnFMP X/ factors necessarily through some S ∈ AnNil cl X//Z . For this reason, we obtain a natural morphism L an X/S → L an X/S , in the ∞ -category Coh + ( X ). Passing to the limit over S ∈ AnNil cl X//Y we obtain a canonically definedmorphism θ ( f ) : L an X/Z → L an X/Y , in Pro(Coh + ( X )). Moreover, this association is functorial and thus we obtain a well defined functor L an X/ • : AnFMP X/ → Pro(Coh + ( X )) , given by the formula ( X → Y ) ∈ AnFMP X/ L an X/Y ∈ Pro(Coh + ( X )) . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 21
Proposition 2.36.
Let X ∈ dAn k be a derived k -analytic space. Then the functor L an X/ • : AnFMP X/ → Pro(Coh + ( X )) , obtained via ?? ?? Let f : Y → Z be a morphism in AnFMP X/ . Thanks to Proposition 2.30 we are reduced to show thatgiven any S ∈ AnNil cl X//Z , the structural morphism g S : X → S admits a unique extension S → Y which factors the structural morphism X → Z . Thanks to Proposition 2.18 we can reduce ourselves to the case where X → S has the structure of asquare-zero extension. In this case, the result follows from Lemma 2.34 combined with our hypothesis. (cid:3) Our goal now is to give an alternative description of analytic formal moduli problems under X ∈ dAn k , interms of derived k -analytic stacks: Construction . Consider the ∞ -category of derived k -analytic stacks, dAnSt k . We have a natural functor h : AnNil X/ → dAn k , → dAnSt k . Therefore, given any derived k -analytic stack Y equipped with a morphism X → Y , one can consider itsrestriction to the ∞ -category AnNil X/ : Y ◦ h : AnNil op X/ → S . We have thus a natural restriction functor h ∗ : dAnSt k → Fun(AnNil op X/ , S ) . On the other hand, Proposition 2.30 allows us to define a natural functor F : AnNil op X/ → dAnSt k via the formula ( X → Y ) ∈ AnFMP X/ colim S ∈ AnNil cl X//Y S, the colimit being computed in the ∞ -category dAnSt k . The latter agrees with the left Kan extension of thefunctor h : AnNil X/ → dAnSt k , along the natural inclusion functor AnNil X/ , → AnFMP X/ . In particular, any analytic formal moduli under X when regarded as a derived k -analytic stack can be realized as an ind - inf -object, i.e. it can be written as afiltered colimit of nil-embeddings X → Z . We refer the reader to [GR17b, §1] for a precise meaning of thelatter notion in the algebraic setting. Definition 2.38.
Let f : X → Y be a morphism in the ∞ -category dAnSt k . We shall say that f has a deformation theory if it satisfies the following conditions:(1) Both X and Y is nilcomplete , c.f. [PY17, Definition 7.4];(2) Y is infinitesimally cartesian if it satisfies [PY17, Definition 7.3];(3) The morphism f : X → Y admits a relative analytic pro-cotangent complex , i.e., if it satisfies [PY17, Defi-nition 7.6] under the weaker assumption that the corresponding derivation functor is pro-corepresentable. Proposition 2.39.
Let Y ∈ (dAnSt k ) X/ . Assume further that Y admits a deformation theory. Then Y isequivalent to an analytic formal moduli problem under X . Proof.
We must prove that given a pushout diagram
S S T T fg in the ∞ -category AnNil X/ , where f has the structure of a square-zero extension, then the natural morphism Y ( T ) → Y ( T ) × Y ( S ) Y ( S ) , is an equivalence in the ∞ -category S . Suppose further that S , → S is associated to some analytic derivation d : L an S → F [1] , for some F ∈ Coh + ( S ) ≥ . Thanks to Corollary 2.24 we deduce that the induced morphism T → T admits astructure of a square-zero extension, as well. Then, by our assumptions that Y is infinitesimally cartesian andit admits a relative pro-cotangent complex, we obtain a chain of natural equivalences of the form Y ( T ) ’ G f : T → Y Map T/ ( T , Y ) ’ G f : T → Y Map
Pro(Coh + ( T )) L an T / ( L an T/Y , g ∗ ( F )[1]) ’ G f : T → Y Map
Pro(Coh + ( S )) g ∗ L an T / ( g ∗ L an T/Y , F [1]) ’ G f : T → Y Map
Pro(Coh + ( S )) L an S / ( L an S/Y , F [1]) ’ G f : T → Y Map S/ ( S , Y ) ’ Y ( T ) × Y ( S ) Y ( S ) , where only the fourth equivalence requires an additional justification, namely: it follows from the existence of acommutative diagram between fiber sequences g ∗ f ∗ L an Y g ∗ L an T g ∗ L an T/Y ( f ◦ g ) ∗ L an Y L an S L an S/Y , = in the ∞ -category Pro(Coh + ( S )) combined with the fact that the derivation d T : L an T → g ∗ ( F )[1] is inducedfrom d : L an S → F [1] , as in the proof of Corollary 2.24. The result now follows. (cid:3) Proposition 2.40.
Let Z ∈ (dAnSt k ) X/ such that the structural morphism X → Z is a nil-isomorphism andassume that Z admits a deformation theory. Then the natural morphism colim S ∈ AnNil cl X//Z S → Z, in the ∞ -category (dAnSt k ) /X , is an equivalence. PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 23
Proof.
We shall prove that for every derived k -analytic space T ∈ dAn k , any diagram of the form X ZT fg h , factors through an object X → S → Z, in AnNil cl X//Z . Consider the commutative diagram X red T red X T Z, (2.7)in the ∞ -category dAnSt k . Since Z red ’ X red we obtain that T red → Z factors necessarily through the colimitcolim S ∈ AnNil cl X//Z S ∈ AnPreStk . Assume first that T is bounded, i.e. T ∈ dAn < ∞ k . Then we can construct T out of T red via a finite sequence ofsquare-zero extensions, as in Proposition 2.18. Therefore, in order to construct a factorization T → S → Z, in AnPreStk X/ , we reduce ourselves to the case where the morphism T red → T is itself a square-zero extension.In this case, let d : L an T red → F [1] , where F ∈ Coh + ( T red ) ≥ be the associated derivation. The existence of the diagram (2.7) implies that we havea commutative diagram of the form g ∗ L an T red L an X red L an X red /T red f ∗ L an Z L an X red L an X red /Z , in the ∞ -category Pro(Coh + ( X )). For this reason, the limit-colimit formula for mapping spaces in pro- ∞ -categories implies that the natural morphism L an X red /Z → L an X red /T red , in the ∞ -category Pro(Coh + ( X )) factors necessarily via a morphism of the form L an X red /S → L an X red /T red , for some suitable S ∈ AnNil cl X//Z . Thus the existence problem T red TX red = S red S Z, admits a solution T → S → Z , as desired. If T ∈ dAn k is a general derived k -analytic space, we reduceourselves to the bounded case using the fact that Z is nilcomplete. (cid:3) Corollary 2.41.
The functor F : AnFMP X/ → (dAnSt k ) X/ , is fully faithful. Moreover, its essential image coincides with those Z ∈ (dAnSt k ) X/ which admit deformationtheory.Proof. Let (dAnSt k ) def X/ ⊆ (dAnSt k ) X/ denote the full subcategory spanned by derived k -analytic stacks under X admitting a deformation theory. It is clear that the natural functor F : AnFMP X/ → (dAnSt k ) X/ , factors through the full subcategory (dAnSt k ) def X/ . Moreover, the restriction functorres : (dAnSt k ) def X/ → Fun(AnNil op X/ , S ) , factors through AnFMP X/ ⊆ Fun(AnNil op X/ , S ). Moreover, Proposition 2.40 implies that the restriction functorthat F and res are mutually inverse functors, proving the claim. (cid:3) Analytic formal moduli problems over a base.
Let X ∈ dAn k denote a derived k -analytic space.In [PY20, Definition 6.11] the authors introduced the ∞ -category of analytic formal moduli problems over X ,which we shall review: Notation 2.42.
Let X ∈ dAn k . We shall denote by AnNil /X the full subcategory of (dAn k ) /X spanned bynil-isomorphisms Z → X, in the ∞ -category dAn k . Definition 2.43.
We denote by AnFMP /X ⊆ Fun(AnNil op /X , S ) the full subcategory spanned by those functors Y : AnNil op /X → S satisfying the following:(1) Y ( X red ) ’ ∗ ;(2) The natural morphism Y ( S ) → lim n ≥ Y (t ≤ n ( S )) , induced by the natural inclusion morphisms t ≤ n ( S ) → S for each n ≥
0, is an equivalence in S ;(3) For each pushout diagram S S T T , fg in AnNil /X , for which f has the structure of a square-zero extension, the canonical morphism Y ( T ) → Y ( T ) × Y ( S ) Y ( S ) , is an equivalence in S . Definition 2.44.
We shall denote by AnNil cl /X ⊆ AnNil /X the faithful subcategory in wich we allow morphisms i : S → S , where i is a nil-embedding in dAn k .We start with the analogue of Proposition 2.30 in the setting of analytic formal moduli problems over X : Proposition 2.45.
Let Y ∈ AnFMP /X . The following assertions hold: PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 25 (1) The inclusion functor (AnNil cl /X ) /Y → (AnNil /X ) /Y , is cofinal.(2) The natural morphism colim Z ∈ (AnNil cl /X ) /Y Z → Y, is an equivalence in the ∞ -category AnFMP /X .(3) The ∞ -category AnNil cl /X is filtered.We shall refer to objects in AnFMP /X as formal moduli problems over X .Proof. We first prove assertion (1). Let S → Z be a morphism in (AnNil cl /X ) /Y . Consider the pushout diagram S red SZ Z , (2.8)in the ∞ -category AnNil /X whose existence is guaranteed by Proposition 2.23. Since the upper horizontalmorphism in (2.8) is a nil-embedding combined with the fact that Y is itself convergent, we can reduce ourselvesvia Proposition 2.18 to the case where the latter is an actual square-zero extension. Since Y is assumed to bean analytic formal moduli problem over X we then deduce that the canonical morphism Y ( Z ) → Y ( Z ) × Y ( S red ) Y ( S ) ’ Y ( Z ) × Y ( S ) , is an equivalence (we implicitly used above the fact that S red ’ X red ). As a consequence the object ( Z → X )in AnNil /X admits an induced morphism Z → Y making the required diagram commute. Thanks toProposition 2.23 we deduce that both S → Z and Z → Z are nil-embeddings. Therefore, we can factor thediagram S ZY via a closed nil-isomorphism Z → Z . As a consequence, we deduce readily that the inclusion functor(AnNil cl /X ) /Y → (AnNil /X ) /Y , is cofinal. Assertion (2) is now an immediate consequence of (1). We now prove (3). Let θ : K → (AnNil cl /X ) /Y , be a functor where K is a finite ∞ -category. We must show that θ can be extended to a functor θ (cid:66) : K (cid:66) → (AnNil cl /X ) /Y . Thanks to Proposition 2.18 we are allowed to reduce ourselves to the case where morphisms indexed by K aresquare-zero extensions. The result now follows from the fact that Y being an analytic moduli problem sendsfinite colimits along square-zero extensions to finite limits. (cid:3) Lemma 2.46.
Let X ∈ dAn k . Given any Y ∈ AnFMP X/ , then for each i = 0 , the i -th projection morphism p i : X × Y X → X, computed in the ∞ -category dAnSt k lies in the essential image of AnFMP /X via the canonical functor AnFMP /X → (dAnSt k ) /X .Proof. Consider the pullback diagram X × Y X XX Y, p p computed in the ∞ -category dAnSt k . Thanks to Proposition 2.30 together with the fact that fiber productscommute with filtered colimis in the ∞ -category dAnSt k , we deduce that X × Y X ’ colim Z ∈ AnNil cl X//Y X × Z X, in dAnSt k . It is clear that ( p i : X × Z X → X ) lies in the essential image of AnFMP /X , for each Z ∈ (AnNil cl /X ) /Y and i = 0 ,
1. Thus also the filtered colimit( p i : X × Y X → X ) ∈ AnFMP /X , for i = 0 , , as desired. (cid:3) Just as in the previous section we deduce that every analytic formal moduli problem over X admits thestructure of an ind - inf -object in AnPreStk k : Corollary 2.47.
Let Y ∈ (dAnSt k ) /X . Then Y is equivalent to an analytic formal moduli problem over X ifand only if there exists a presentation Y ’ colim i ∈ I Z i , where I is a filtered ∞ -category and for every i → j in I , the induced morphism Z i → Z j , is a closed embedding of derived k -affinoid spaces that are nil-isomorphic to X .Proof. It follows immediately from Proposition 2.45 (2). (cid:3)
Definition 2.48.
Let Y ∈ AnFMP /X . We define the ∞ -category of coherent modules on Y , denoted Coh + ( Y ),as the limit Coh + ( Y ) := lim Z ∈ (dAn k ) /Y Coh + ( Z ) , computed in the ∞ -category Cat st ∞ . We define the ∞ -category of pseudo-pro-coherent modules on Y , denotedPro ps (Coh + ( Y )), as Pro ps (Coh + ( Y )) := lim Z ∈ (AnNil cl /X ) /Y Pro(Coh + ( Z )) , where the limit is computed in the ∞ -category Cat st ∞ . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 27
Definition 2.49.
Let Y ∈ AnFMP /X , Z ∈ dAfd k and let F ∈ Coh + ( Z ) ≥ . Suppose furthermore that we aregiven a morphism f : Z → Y . We define the tangent space of Y at f twisted by F as the fiber T an Y,Z, F ,f := fib f (cid:0) Y ( Z [ F ]) → Y ( Z ) (cid:1) ∈ S . Whenever the morphism f is clear from the context, we shall drop the subscript f above and denote the tangentspace simply by T an Y,Z, F . Remark . Let Y ∈ AnFMP /X . The equivalence of ind-objects Y ’ colim S ∈ (AnNil cl /X ) /Y S, in the ∞ -category dAnSt k , implies that, for any Z ∈ dAfd k , one has an equivalence of mapping spacesMap dAnSt k ( Z, Y ) ’ colim S ∈ (AnNil cl /X ) /Y Map
AnPreStk ( Z, S ) . For this reason, given any morphism f : Z → Y and any F ∈ Coh + ( Z ) ≥ , we can identify the tangent space T an Y,Z, F with the filtered colimit of spaces T an Y,Z, F ’ colim S ∈ (AnNil cl /X ) Z//Y fib f (cid:0) S ( Z [ F ]) → S ( Z ) (cid:1) (2.9) ’ colim S ∈ (AnNil cl /X ) Z//Y T an S,Z, F (2.10) ’ colim S ∈ (AnNil cl /X ) Z//Y
Map
Coh + ( Z ) ( f ∗ S,Z ( L an S ) , F ) , (2.11)where f S,Z : Z → S is a transition morphism, in (dAn k ) /X , factoring f : Z → Y such that( S → X ) ∈ AnNil cl /X . The final equivalence in (2.9), follows from [PY17, Lemma 7.7]. Therefore, we deduce that the analytic formalmoduli problem Y ∈ AnFMP /X admits an absolute pro-cotangent complex given as L an Y := { f ∗ S,Z ( L an S ) } Z,S ∈ (AnNil cl /X ) /Y ∈ Pro ps (Coh + ( Y )) . Corollary 2.51.
Let Y ∈ AnFMP /X . Then its absolute cotangent complex L an Y classifies analytic deformationson Y . More precisely, given any morphism Z → Y where Z ∈ dAfd k and F ∈ Coh + ( Z ) ≥ one has a naturalequivalence of mapping spaces T an Y,Z, F ’ Map
Pro(Coh + ( Y )) ( L an Y , F ) . Proof.
It follows immediately from the natural equivalences displayed in (2.9) combined with the description ofmapping spaces in ∞ -categories of pro-objects. (cid:3) We now introduce the notion of square-zero extensions of analytic formal moduli problems over X : Construction . Let ( f : Y → X ) ∈ AnFMP /X . Let d : L an Y → F [1] be an analytic derivation inPro(Coh + ( Y )), where F ∈ Coh + ( Y ) ≥ , such that F ’ f ∗ ( F ) , for some suitable object F ∈ Coh + ( X ) ≥ . Thanks to Remark 2.50 one has the following sequence of naturalequivalences of mapping spacesMap Pro(Coh + ( Y )) ( L an Y , F [1]) ’ lim S ∈ (AnNil cl /X ) /Y colim S ∈ (AnNil cl /X ) S//Y
Map
Pro(Coh + ( S )) ( f ∗ S,S ( L an S ) , g ∗ S ( F )[1]) (2.12) ’ lim S ∈ (AnNil cl /X ) /Y colim S ∈ (AnNil cl /X ) S//Y
Map
Pro(Coh + ( S )) ( L an S , ( f S,S ) ∗ g ∗ S ( F )[1]) , (2.13)where g S : S → X denotes the structural morphism in AnNil cl /X and f S,S : S → S a given transition morphismin the ∞ -category (AnNil cl /X ) /Y . For this reason, we can form the filtered colimit Y := colim S ∈ (AnNil cl /X ) /Y colim S ∈ (AnNil cl /X ) S//Y S ∈ dAnSt k , where S → S denotes the square-zero extension induced from d together with (2.12). By construction, one hasa natural morphism Y , → Y in the ∞ -category (dAnSt k ) /X . Moreover, thanks to Proposition 2.39 it followsthat Y ∈ AnFMP /X . Definition 2.53.
Let Y ∈ AnFMP /X . Suppose we are given an analytic derivation d : L an Y → F [1] , in Pro(Coh + ( Y )) where F ∈ Coh + ( Y ) ≥ is such that F ’ f ∗ ( F ), for some F ∈ Coh + ( X ) ≥ . We shall saythat the induced morphism h : Y → Y , defined in Construction 2.52, is a square-zero extension of Y associated to the analytic derivation d . Corollary 2.54.
Let Y ∈ AnFMP /X . Let h : X , → S denote a square-zero extension in dAn k . Then the spaceof cartesian squares Y Y X S, h f gh such that h : Y → Y is a square-zero extension and g : Y → S exhibits the former as an analytic formalmoduli problem over S is naturally equivalent to the space of factorizations f ∗ L an X → L an Y → f ∗ ( F )[1] , in Pro ps (Coh + ( Y )) , of the analytic derivation d : L an X → F [1] associated to the morphism h above.Proof. By the universal property of filtered colimits together with the fact that these preserve fiber productswe reduce the statement to the case where Y ∈ AnNil /X and thus Y ∈ AnNil /S , in which case the statementfollows immediately by the universal property of the relative analytic cotangent complex. (cid:3) Corollary 2.55.
Let f : Z → X be a morphism in the ∞ -category dAn k . Suppose we are given analytic formalmoduli problems f : Y → X and g : e Z → Z together with a commutative diagram e Z YZ X, sf PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 29 in the ∞ -category dAnSt k . Let d : L an Z → F [1] , where F ∈ Coh + ( Z ) ≥ , be an analytic derivation correspondingto a square-zero extension morphism Z → Z in the ∞ -category dAn k . Assume that d induces a square-zeroextension e d : L an e Z → F [1] and let h : e Z , → e Z be the induced square-zero extension in dAnSt k such that we havea cartesian diagram e Z e Z Z Z in the ∞ -category dAnSt k . Then the space of factorizations s : e Z → e Z → Y, is naturally equivalent to the space of factorizations e d : L an e Z → L an e Z/Y → F [1] , in the ∞ -category Pro(Coh + ( e Z )) .Proof. The statement holds true in the case where e Z ∈ AnNil /Z and Y ∈ AnNil /X , by the universal propertyof the relative cotangent complex. The general case is reduced to the previous one by a standard argumentwith ind-objects in dAnSt k . (cid:3) Non-archimedean nil-descent for almost perfect complexes.
In this §, we prove that the ∞ -category Coh + ( X ), for X ∈ dAn k satisfies nil-descent with respect to morphims Y → X , which exhibit theformer as an analytic formal moduli problem over X . Proposition 2.56.
Let f : Y → X , where X ∈ dAn k and Y ∈ AnFMP /X . Consider the Čech nerve Y • : ∆ op → dAnSt k associated to f . Then the natural functor f ∗• : Coh + ( X ) → lim ∆ (Coh + ( Y • )) , is an equivalence of ∞ -categories.Proof. Consider the natural equivalence of derived k -analytic stacks Y ’ colim Z ∈ (AnNil cl /X ) /Y Z. Then, by definition one has a natural equivalenceCoh + ( Y ) ’ lim Z ∈ (AnNil cl /X ) /Y Coh + ( Z ) , of ∞ -categories. In particular, since totalizations commute with cofiltered limits in Cat ∞ , it follows that wecan suppose from the beginning that Y ’ Z for some Z ∈ AnNil /X . In this case, the morphism f : Y → X isaffine. In particular, the fact that Coh + ( − ) satisfies descent along admissible open immersions, combined withLemma 2.14 we further reduce ourselves to the case where both X and Y are derived k -affinoid spaces. Inthis case, by Tate acyclicity theorem it follows that letting A := Γ( X, O alg X ) and B := Γ( Y, O alg Y ) , the pullbackfunctor f ∗ can be identified with the usual base change functorCoh + ( A ) → Coh + ( B ) . In this case, it follows that B is nil-isomophic to A . Moreover, since the latter are noetherian derived k -algebrasthe statement of the proposition follows due to [HLP14, Theorem 3.3.1]. (cid:3) We now deduce pseudo-pro-nil-descent for moprhisms of the form Y → X , which exhibit Y as an analyticformal moduli problem over X : Corollary 2.57.
Let X ∈ dAn k and f : Y → X a morphism in dAnSt k which exhibits Y as an analytic formalmoduli problem over X . Then the natural functor f ∗• : Pro(Coh + ( X )) → lim ∆ (Pro ps Coh + ( Y • /X )) , is fully faithful, where Y • denotes the Čech nerve associated to the morphism f . Moreover, the essential imageof the functor f ∗• identifies canonically with the full subcategory lim ∆ Pro ps (Coh + ( X )) ⊆ lim ∆ Pro ps (Coh + ( Y • /X )) , spanned by those { F i, [ n ] } i ∈ I op , [ n ] ∈ lim ∆ (Pro ps (Coh + ( Y • /X ))) , for some filtered ∞ -category I , which belong tothe essential image of the natural functor lim ∆ Fun( I op , Coh + ( Y • /X )) → lim ∆ Pro ps (Coh + ( Y • /X )) . Proof.
By the very definition of the ∞ -category Pro ps (Coh + ( Y )), we reduce ourselves as in Proposition 2.56 tothe case where Y = S , for some S ∈ AnNil /X . In this case, it follows readily from Proposition 2.56 that thenatural functor f ∗• : Pro(Coh + ( X )) → lim ∆ Pro ps (Coh + ( S • /X )) , is fully faithful. We now proceed to prove the second claim of the corollary. Notice that, Proposition 2.22implies that there exists a well defined right adjoint f ∗ : Coh + ( S ) → Coh + ( X ) , to the usual pullback functor f ∗ : Coh + ( X ) → Coh + ( S ). We can extend the right adjoint f ∗ to a well definedfunctor f ∗ : Pro(Coh + ( S )) → Pro(Coh + ( X )) , which commutes with cofiltered limits. For this reason, we have a well defined functor f • , ∗ : lim ∆ (Pro(Coh + ( S • /X ))) → Pro(Coh + ( X )) , which further commutes with cofiltered limits. Denote by D := lim ∆ (Pro(Coh + ( S • /X ))). We claim that f • , ∗ is a right adjoint to f ∗• above. Indeed, given any { F i } i ∈ I op ∈ Pro(Coh + ( X )) and { G j, [ n ] } j ∈ J op[ n ] , [ n ] ∈ ∆ op ∈ lim ∆ (Pro(Coh + ( S • /X ))), we computeMap D ( f ∗• ( { F i } i ∈ I op ) , { G j, [ n ] } j ∈ J op[ n ] , [ n ] ∈ ∆ op ) ’ lim [ n ] ∈ ∆ Map
Pro ps (Coh + ( S [ n ] )) (cid:18) { f • [ n ] ( F i ) } i ∈ I op , { G i, [ n ] } i ∈ I op[ n ] (cid:19) lim [ n ] ∈ ∆ op lim j ∈ J op[ n ] colim i ∈ I Map
Coh + ( S [ n ] ) ( f ∗ [ n ] ( F i ) , G i, [ n ] ) ’ lim [ n ] ∈ ∆ lim j ∈ J op[ n ] colim i ∈ I Map
Coh + ( X ) ( F i , f [ n ] , ∗ ( G i, [ n ] ))lim [ n ] ∈ ∆ op Map
Pro(Coh + ( X )) ( { F i } i ∈ I op , { f [ n ] , ∗ ( G i, [ n ] ) } i ∈ I op[ n ] ) ’ Map
Pro(Coh + ( X )) ( { F i } i ∈ I op , lim [ n ] ∈ ∆ { f [ n ] , ∗ ( G i, [ n ] ) } i ∈ I op[ n ] ) , as desired. It is clear that the functor f ∗• above factors through the full subcategory lim ∆ (Pro ps (Coh + ( S • /X ))) ⊆ lim ∆ (Pro(Coh + ( S • /X ))) . For this reason, the pair ( f ∗• , f • , ∗ ) restricts to a well defined adjunction f ∗• : Pro(Coh + ( X )) (cid:29) lim ∆ (Pro(Coh + ( S • /X ))) : f • , ∗ . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 31
In order to conclude, we will show that the functor f • , ∗ : lim ∆ (Pro(Coh + ( S • /X ))) → Pro(Coh + ( X )) , is conservative. Since both the ∞ -categories Pro(Coh + ( X )) and lim ∆ (Pro ps (Coh + ( Y S • /X ))) are stable, weare reduced to prove that given any { G i, [ n ] } i ∈ I op ∈ lim ∆ (Pro ps (Coh + ( S • /X ))) , such that lim [ n ] ∈ ∆ f • , ∗ ( { G i, [ n ] } i ∈ I op ) ’ , (2.14)we necessarily have { G i, [ n ] } i ∈ I op ’ , in lim ∆ (Pro ps (Coh + ( S • /X ))). Assume then (2.14). Under our hypothesis, for each index i ∈ I , the object { G i, [ n ] } [ n ] ∈ ∆ satisfies descent datum and thanks to Proposition 2.56 it produces a uniquely well defined object G i ∈ Pro(Coh + ( X )) , such that for every [ n ] ∈ ∆ , one has a natural equivalence of the form f ∗ [ n ] ( G i ) ’ G i, [ n ] ∈ Coh + ( S [ n ] ) . We deduce then that f ∗• ( { G i } i ∈ I op ) ’ { G i, [ n ] } i ∈ I op , [ n ] ’ , in lim ∆ Pro ps (Coh + ( S • /X )), as desired. (cid:3) We now use the pseudo-pro-nil-descent for Pro(Coh + ( X )) to compute relative analytic cotangent complexesof analytic formal moduli problems over X : Corollary 2.58.
Let f : Z → X be a morphism in dAn k . Suppose we are given a pullback square e Z YZ X, h in the ∞ -category dAnSt k , where ( Y → X ) ∈ AnFMP /X and ( g : e Z → Z ) ∈ AnFMP /Z . Then the lax-object { L an e Z [ n ] /Y [ n ] } ∈ lim lax , ∆ (Pro ps (Coh + ( e Z • /Z ))) , defines a cartesian section { L an e Z [ n ] /Y [ n ] } ∈ lim ∆ Pro(Coh + (( e Z ) • /Z )) , which belongs to the essential image of the natural functor g ∗• : Pro(Coh + ( Z )) → lim ∆ Pro(Coh + ( e Z • /Z )) . Proof.
We first show that the object { L an( e Z ) [ n ] /Y [ n ] } ∈ lim lax , ∆ Pro ps (Coh + (( e Z ) • /Z )) , defines a cartesian section in lim ∆ Pro(Coh + (( e Z ) • /Z )) . In order to show this assertion, it is sufficient to prove for every [ n ] ∈ ∆ , that we have a natural equivalence p ∗ i,n,n +1 ( L an e Z [ n ] /Y [ n ] ) ’ L an e Z [ n +1] /Y [ n +1] , in the ∞ -category Pro ps (Coh + (( e Z ) [ n +1] )), for each projection morphism p i,n,n +1 : e Z [ n +1] → e Z [ n ] , in the Čech nerve associated to the morphism g . The latter claim is an immediate consequence of the base changeproperty for the relative analytic cotangent complex whenever Y ∈ AnNil /X (and thus so do e Z ∈ AnNil /Z )),c.f. [PY17, Proposition 5.12]. In the general case where Y ∈ AnFMP /X , we reduce to the previous caseby combining Proposition 2.45 with the observation that filtered colimits commute with finite limits in the ∞ -category dAnSt k .We now prove the second assertion of the Corollary: thanks to the characterization of the essential image ofnatural functor g ∗• : Pro(Coh + ( Z )) → lim ∆ Pro ps (Coh + (( e Z ) • /Z )) , provided in Corollary 2.57, we are reduced to show that for each [ n ] ∈ ∆ , we have a natural equivalence ofpro-objects L an e Z [ n ] /Y [ n ] ’ { L an e S [ n ] /S [ n ] } e S ∈ (AnNil /Z ) / e Z , S ∈ (AnNil /X ) /Y , where ˜ S := S × X Z , for each S ∈ (AnNil cl /X ) /Y .The latter statement follows readily from the first part of theproof combined with a standard inductive argument. (cid:3) Non-archimedean formal groupoids.
Let X ∈ dAn k . We start with the definition of the notion of analytic formal groupoids over X : Definition 2.59.
We denote by AnFGrpd( X ) the full subcategory of the ∞ -category of simplicial objectsFun( ∆ op , AnFMP /X ) , spanned by those objects F : ∆ op → AnFMP /X satisfiying the following requirements:(1) F ([0]) ’ X ;(2) For each n ≥
1, the morphism F ([ n ]) → F ([1]) × F ([0]) · · · × F ([0]) F ([1]) , induced by the morphisms s i : [1] → [ n ] given by (0 , ( i, i + 1), is an equivalence in AnFMP /X .We shall refer to objects in AnFGrpd( X ) as analytic formal groupoids over X . Remark . Note that Proposition 2.45 implies that fiber products exist in AnFMP /X . Therefore, theprevious definition is reasonable. PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 33
Construction . Thanks to Lemma 2.46, there exists a well defined functor Φ : AnFMP X/ → AnFGrpd( X )given by the formula ( X f −→ Y ) ∈ AnFMP X/ Y ∧ X ∈ AnFGrpd( X ) , where we shall denote by Y ∧ X ∈ AnFGrpd( X ) the Čech nerve of the morphism f computed in AnFMP /X . Thelatter defines a formal groupoid over X admitting . . . X × Y X × Y X X × Y X X , as simplicial presentation. We shall denote the later simplicial object simply by X × Y • ∈ AnFGrpd( X ). Remark . Let ( X → Y ) ∈ AnFMP X/ and consider the corresponding analytic formal groupoid Y ∧ X ∈ AnFMP( X ). The diagonal morphism ∆ : X → X × Y X, in AnFMP X/ induces a well defined morphism X → X × Y • , in Fun( ∆ op , AnFMP X/ ), where X × Y • denotes the Čech nerve of the morphism X → Y computed in the ∞ -category AnFMP X/ . Construction . Let G ∈ AnFGrpd( X ). Consider the classifying derived k -analytic stack , B X ( G ) pre ∈ dAnSt k ,obtained as the geometric realization of the simplicial object G , regarded naturally as a functor G : ∆ op → dAnSt k , via the natural composite AnFMP /X → (dAnSt k ) /X → dAnSt k . Given any Z ∈ dAfd k , the space of Z -pointsof B X ( G ) pre , B X ( G ) pre ( Z ) , can be identified with the space whose objects correspond to the datum of:(1) A morphism e Z → X , where e Z ∈ dAnSt k , such that e Z ’ Z × B X ( G ) pre X ;(2) A morphism of groupoid-objects e Z × Z e Z → G , in the ∞ -category dAnSt k .We now define B X ( G ) → B pre X ( G ) as the sub-object spanned by those connected components of B X ( G ) pre corresponding to morphisms e Z → Z in Construction 2.63 (1) which exhibit e Z ∈ AnFMP /Z . Denote bycan : B X ( G ) → B X ( G ) pre , the canonical morphism. It follows from the construction that the natural morphism X → B X ( G ) pre , factors as X → B X ( G ) can −−→ B X ( G ) pre .We are able to prove that the object B X ( G ) admits a deformation theory: Lemma 2.64.
The natural morphism X → B X ( G ) exhibits the latter as an object in the ∞ -category AnFMP X/ of analytic formal moduli problems under X . Proof.
Thanks to Proposition 2.39 it suffices to prove that B X ( G ) is infinitesimally cartesian and it admitsfurthermore a pro-cotagent complex. The fact that B X ( G ) is infinitesimally cartesian follows from the modulardescription of B X ( G ) combined with the fact that G is infinitesimally cartesian, as well. Similarly, B X ( G ) beingnilcomplete follows again from its modular description combined with the fact that analytic formal moduliproblems are nilcomplete.We are thus required to show that B X ( G ) admits a global pro-cotangent complex. Let Z ∈ dAn k and supposewe are given an arbitrary morphism q : Z → B X ( G ) , in the ∞ -category dAnSt k . Thanks to Corollary 2.57 combined with Corollary 2.58 it follows that the object { L an e Z [ n ] / G [ n ] } [ n ] ∈ ∆ ∈ lim ∆ Pro ps (Coh + ( e Z • /Z )) , defines a well defined object ( L an Z/ B X ( G ) ) ∈ Pro(Coh + ( Z )). Moreover, it is clear that there exists a naturalmorphism θ : L an Z → ( L an Z/ B X ( G ) ) , in the ∞ -category Pro(Coh + ( Z )), as this holds in each level of the totalization. Let q ∗ ( L anB X ( G ) ) := fib( θ ) , computed in the ∞ -category Pro(Coh + ( Z )). We claim that q ∗ ( L anB X ( G ) ) identifies with the analytic cotangentcomplex of B X ( G ) at the point q : Z → B X ( G ). Let Z , → Z , denote a square-zero extension which corresponds to a certain analytic derivation d : L an Z → F [1] , for some F ∈ Coh + ( Z ) ≥ . Using Corollary 2.54 we deduce that the space of cartesian squares of the form e Z e Z Z Z where the morphism e Z → e Z is a square-zero extension in the ∞ -category dAnSt k is equivalent to the space offactorizations d : g ∗ L an Z → L an e Z d −→ g ∗ ( F )[1] , in the ∞ -category Pro(Coh + ( e Z )). Apply the same reasoning to the each object in the Čech nerve e Z • → Z. Furthermore, Corollary 2.55 implies that the space of factorizations e Z • → ( e Z ) • → G • , identifies with the space of factorizations d : L an e Z • → L an e Z • / G → g ∗ ( F )[1] , in the ∞ -category Pro(Coh + ( e Z )). Corollary 2.58 implies that the above factorization space can be identifiedwith the space of factorizations d : L an Z → ( L an Z/ B X ( G ) ) → F [1] . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 35
This implies that ( L an Z/ B X ( G ) ) satisfies the universal property of the relative analytic pro-cotangent complexassociated to X → B X ( G ), as desired. (cid:3) Theorem 2.65.
The functor
Φ : AnFMP /X → AnFGrpd( X ) of Construction 2.61 is an equivalence of ∞ -categories.Proof. Let G ∈ AnFGrpd( X ). Thanks to Lemma 2.64, we have a well defined functorB X ( − ) : AnFGrpd( X ) → AnFMP X/ , given on objects by the association G ∈ AnFGrpd( X ) B X ( G ) ∈ AnFMP X/ . Thanks to (1) in Construc-tion 2.63 it follows that one has a canonical equivalence X × B X ( G ) X ’ G , in dAnSt k . This shows that the constructionB X ( G ) : AnFGrpd( X ) → AnFMP X/ , is a right inverse to Φ. As a consequence the functor Φ is essentially surjective. Thanks to Proposition 2.36 weare reduced to show that the canonical morphism Y → B X ( X × Y X ) , induces an equivalence on the associated relative analytic cotangent complexes. The claim is an immediateconsequence of the description of L an X/ B X ( X × Y X ) provided in Lemma 2.64 together with Corollary 2.58. (cid:3) Remark . It follows from Theorem 2.65 that given ( X → Y ) ∈ AnFMP X/ , we can identify the latter withthe geometric realization of the associated Čech nerve, the latter regarded as a simplicial object in AnFMP X/ via the diagonal morphisms.2.7. The affinoid case.
Let X ∈ dAfd k denote a derived k -affinoid space. Thanks to derived Tate aciclycitytheorem, cf. [PY18a, Theorem 3.1] the global sections functor Γ : Coh + ( X ) → Coh + ( A ) , where A := Γ( X, O alg X ) ∈ CAlg k , is an equivalence of ∞ -categories. Since ordinary k -affinoid algebras areNoetherian, we deduce that A ∈ CAlg k is a Noetherian derived k -algebra. Notation 2.67.
Let X ∈ dAfd k and A := Γ( X, O X ). We denote byFMP / Spec A ∈ Cat ∞ , the ∞ -category of algebraic formal moduli problems over Spec A , (c.f. [PY20, Definition 6.11]). Theorem 2.68. ([PY20, Theorem 6.12])
Let X ∈ dAfd k and A := Γ( X, O alg X ) ∈ CAlg k . Then the inducedfunctor ( − ) an : FMP / Spec A → AnFMP /X , is an equivalence of ∞ -categories. As an immediate consequence, we obtain the following result:
Corollary 2.69.
Let X ∈ dAfd k and A := Γ( X, O alg X ) . Then one has an equivalence of ∞ -categories FMP
Spec
A//
Spec A → AnFMP
X//X , of pointed algebraic formal moduli problems over Spec A and pointed analytic formal moduli problems over X ,respectively. Proof.
It is an immediate consequence of Theorem 2.68. Indeed, equivalences of ∞ -categories with final objectsinduce natural equivalences on the associated ∞ -categories of pointed objects. (cid:3) Lemma 2.70.
Consider the natural functor F : AnFGrpd( X ) → AnFMP
X//X , given on objects by the formula G ∈ AnFGrpd( X ) G ([1]) ∈ Ptd(AnFMP /X ) . Then F is conservative andcommutes with sifted colimits.Proof. The proof of [GR17b, Corollary 5.2.2.4] applies in our setting to show that the functor F : AnFGrpd( X ) → Ptd(AnFMP /X ) commutes with sifted colimits. We are reduced to prove that F is also conservative. Let f : G → G be a morphism in AnFGrpd( X ). The equivalence of ∞ -categoriesB X ( • ) : AnFGrpd( X ) → AnFMP X/ , of Theorem 2.65 implies that the morphism f can be obtained as the Čech nerve of a morphism e f : B X ( G ) → B X ( G ) , in AnFMP X/ . For this reason, for every [ n ] ∈ ∆ , the morphism f [ n ] : G ([ n ]) → G ([ n ]) , is obtained as (an iterated) pullback of f [1] : G ([1]) → G ([1]) . Therefore, under the assumption that F ( F ) ’ f [1] is an equivalence in AnFMP X//X we deduce that f itselfmust be an equivalence of analytic formal groupoids, as it is each of its components. (cid:3) Remark . Let ( X f −→ Y ) ∈ AnFMP X/ . Consider the diagram X X × Y XX ∆ p p in the ∞ -category dAnSt k , where ∆ : X → X × Y X denotes the usual diagonal embedding . We then obtain anatural fiber sequence associated to the above diagram of the form∆ ∗ L an X × Y X/X → L an X/X → L an X/X × Y X . (2.15)Notice further that by [PY17, Proposition 5.12] one has an equivalence L an X × Y X/X ’ p ∗ i L an X/Y , for i = 0 ,
1. We further deduce that ∆ ∗ L an X × Y X/X ’ L an X/Y , in the ∞ -category Pro(Coh + ( X )). Moreover, since L an X/X ’
0, we obtain from the fiber sequence (2.15) anatural equivalence L an X/X × Y X ’ L an X/Y [1] , in Pro(Coh + ( X )). Moreover, we can identify the L an X × Y X/X with the pro-object L an X/X × Y X ’ { L an X/X × S X } S ∈ AnNil cl X//Y , where L an X/X × S X ∈ Coh + ( X ) denotes the relative analytic cotangent complex associated to the closed embedding X → X × S X, PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 37 for S ∈ AnNil cl X/Y . Thanks to [PY17, Corollary 5.33] we deduce that L an X/X × S X ’ L A/A ⊗ BS A , in Coh + ( A ), where B S = Γ( S, O alg S ), where we have a natural identification A b ⊗ B S A ’ A ⊗ B S A, since the morphism A → B S is finite. Lemma 2.72.
Let X ∈ dAfd k be a derived k -affinoid space. Then the derived k -algebra of global sections A := Γ( X, O alg X ) , admits a dualizing module (see [Lur12a, Definition 4.2.5] for the definition of the latter notion).Proof. This is an immediate consequence of the following facts:(1) Every regular k -algebra R admits a dualizing module, c.f. [Sta13, Tag 0AWX];(2) Every quotient of an algebra R which admits a dualizing module admits itself a dualizing module, c.f.[Sta13, Tag 0A7I].(3) The ordinary affinoid k -algebra π ( A ) can be realized as a quotient of a Tate algebra on n -generators, k h T , . . . , T n i . The latter being regular, we deduce from the previous items that π ( A ) itself admits adualizing module (cf. [Sta13, Tag 0AWX]).(4) Thanks to [Lur12a, Theorem 4.3.5] it follows that A itself admits a dualizing module.The proof is thus concluded. (cid:3) Lemma 2.73.
Let X ∈ dAfd k be a bounded derived k -affinoid space and let ω A ∈ Mod A denote a dualizingmodule for A := Γ( X, O alg X ) . Then ω A ∈ Coh b ( A ) , where the latter denotes the full subcategory of (bounded) coherent A -modules.Proof. By definition it follows that for every i ∈ Z , π i ( ω A ) is finitely generated as a discrete π ( A )-module. Weare thus reduced to show that ω A is bounded. Thanks to [Lur12a, Theorem 4.2.7] it follows that ω A ’ Map
Mod A ( A, ω A ) , is truncated. Moreover, by definition it follows that ω A ∈ Mod A is of finite injective dimension. In particular,for every j ≥
0, we have that Map
Mod A ( π j ( A ) , ω A ) , is bounded. Under our assumption that X ∈ dAfd k is bounded it follows that A itself is a bounded derived k -algebra and therefore by a standard argument on Postnikov towers, we deduce thatMap Mod A ( A, ω A ) ’ ω A , is bounded as well. (cid:3) Remark . Let X ∈ dAfd k be a derived k -affinoid algebra. If X is not bounded then neither it is A ∈ CAlg k .For this reason, a dualizing module ω A ∈ Mod A is not in general bounded. Nonetheless, the latter is alwaystruncated. Let m ∈ Z be such that ω A ∈ Mod A is m -truncated. Thanks to the proof of [Lur12a, Theorem4.3.5], we deduce that we have a natural equivalence ω A ’ colim n ≥ ω τ ≤ n ( A ) , in the ∞ -category Mod A , where for each n ≥ ω τ ≤ n ( A ) ∈ Mod τ ≤ n ( A ) is a suitable m -truncated dualizingmodule for the n -th truncation τ ≤ n ( A ). Definition 2.75.
Let X ∈ dAfd k we define its ∞ -category of ind-coherent sheaves asIndCoh( X ) := Ind(Coh b ( A )) . Definition 2.76.
Let X ∈ dAfd k be a derived k -affinoid space. Whenever X is bounded we shall denote ω X ∈ IndCoh( X ) the image of a fixed dualizing module for A := Γ( X, O alg X ) under the natural inclusionCoh b ( X ) ⊆ IndCoh( X ) . In the unbounded case, we shall define ω X := colim n ≥ ω t ≤ n ( X ) ∈ IndCoh( X ) , where the { ω t ≤ n ( X ) } n ≥ denotes a compatible sequence of dualizing modules for the consecutive truncations of X . Notation 2.77.
Let X ∈ dAfd k be a derived k -affinoid space together with a dualizing module ω X ∈ IndCoh( X ). We shall denote by D Serre X : Ind(Coh b ( X ) op ) → IndCoh( X ) , the associated Serre duality functor , see [Ant20, §2.4].
Remark . Let ( X → Y ) ∈ AnFMP X/ . The pro-cotangent complex L an X/Y ∈ Pro(Coh + ( X )) can benaturally regarded as an object in the full subcategory Pro(Coh b ( X )) of pro-objects of bounded almost perfect O X -modules, c.f. [Ant20, Lemma 4.6].Notice that we have an equivalence of ∞ -categories Pro(Coh b ( X )) op ’ Ind(Coh b ( X ) op ). We now introducethe central notion of the Serre tangent complex : Definition 2.79 (Serre Tangent complex) . Let Y ∈ AnFMP X/ . We define the relative analytic Serre tangentcomplex of X → Y as the object T an X/Y := D Serre X ( L an X/Y ) ’ colim S ∈ AnNil cl X//Y D Serre X ( L an X/S ) , in IndCoh( X ). Remark . The previous definition depends on the choice of a dualizing module for X . Nonetheless, giventwo different choices, these differ only by an invertible A -module, c.f. [Ant20, Proposition 2.4].The following result will play a major role in the study of the deformation to the normal bundle: Proposition 2.81.
The functor T an X/ • : AnFMP X/ → QCoh( X ) given on objects by the formula ( X → Y ) ∈ AnFMP X/ T an X/Y ∈ IndCoh( X ) , is conservative and commutes with sifted colimits.Proof. Observe that Lemma 2.70 combined with (2.15) imply that it suffices to prove that the right verticalfunctor in the diagram FMP
Spec
A//
Spec A AnFMP
X//X
IndCoh( A ) IndCoh( X ) , ( − ) an X T Spec A/ • T an X/ • ( − ) an (2.16) PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 39 commutes with sifted colimits and is further conservative. The diagram is commutative as a consequenceof [PY20, Lemma 6.9 (2)] combined with the definition of Serre duality in IndCoh( X ). Moreover, [Ant20,Corollary 4.31] implies that T Spec A/ • : FMP Spec
A//
Spec A → IndCoh( A ) , is both conservative and commutes with sifted colimits. Furthermore, the bottom horizontal functor in (2.16)is an equivalence of ∞ -categories, see for instance [PY20, Theorem 4.5]. Thanks to Corollary 2.69, we furtherdeduce that the upper horizontal functor in (2.16) is an equivalence, and the result follows. (cid:3) Non-archimedean Deformation to the normal bundle
In this §, we introduce the construction of the deformation to the normal cone in the setting of derived k -analytic geometry. This construction has already been performed in the literature in the particular case ofthe natural inclusion morphism i : t ( X ) → X, X ∈ dAn k , c.f [PY20]. On the other hand, the algebraic situation is largely understood mainly due to [GR17b].3.1. General construction in the algebraic case.
We start by recalling the definition of the simplicialobject B • scaled ∈ Fun( ∆ op , dSt k ) , introduced in [GR17b, §9.2.2]. Namely, for each [ n ] ∈ ∆ , the object B n scaled is obtained by gluing n + 1 copiesof A k together along 0 ∈ A k . Moreover, the transition morphisms p i,n : B n +1scaled → B n scaled , for i ∈ { , . . . , n } , collapse two given different irreducible components of B n +1scaled into B n scaled . More explicitly, for n = 0, we haveB = A k , and, for n = 1, B = Spec k [ x, y ] / ( x − y ). Construction . Let f : X → Y denote a morphism between locally geometric derived k -stacks in dSt laft k .Consider the formal completion of Y on X along the morphism fY ∧ X ∈ dSt k In [GR17b, §9.3], the authors introduced the parametrized deformation to the normal bundle associated to f : X → Y , as the pullback D • X/Y Y ∧ X × A k Map / A k (B • scaled , X × A k ) Map / A k (B • scaled , Y ∧ X × A k ) , (3.1)where both X × A k and Y ∧ X × A k are considered as constant simplicial objects in the ∞ -category dSt k . As aconsequence of [GR17b, Theorem 9.2.3.4], it follows that each component of the simplicial object D • X/Y admitsa deformation theory relative to X .In particular, the object D • X/Y defines a formal groupoid over the stack X × A k . Moreover, Theorem 5.2.3.4in loc. cit. allows us to associate to D • X/Y a formal moduli problem under X × A k D X/Y ∈ FMP X × A k //Y ∧ X × A k , obtained via the construction provided in §5.2.4 in loc. cit. More explicitly, the object D X/Y ∈ FMP X × A k / iscomputed as the sifted colimit of the simplicial diagram D • X/Y ∈ Fun( ∆ op , FMP X × A k / ) , the colimit being computed in FMP X × A k / . We can furthermore consider D X/Y naturally as an object in(dSt laft k ) X × A k //Y ∧ X × A k . Construction . Let f : X → Y a morphism of locally geometric derived k -stacks in dSt laft k . In [GR17b,§9.2.5] the authors constructed an explicit left-lax action of the monoid-scheme object A k (with respect to multiplication ) on the deformation to the normal bundle D X/Y . The above action implies that the structuresheaf of D X/Y is equipped with a (negatively indexed) filtration.
Notation 3.3.
We shall denote by p : D X/Y → A k the natural composite morphism D X/Y → Y ∧ X × A k → A k , in the ∞ -category dSt laft k .We shall now describe formal geometric properties of the deformation to the normal bundle D X/Y ∈ (dSt laft k ) X × A k //Y ∧ X × A k : Proposition 3.4.
Let f : X → Y denote a morphism of locally geometric derived k -stacks in the ∞ -category dSt laft k . The following assertions hold:(1) The fiber of the morphism p : D X/Y → A k at the fiber { } ⊆ A k identifies naturally with the formalcompletion T X/Y [1] ∧ ∈ FMP X/ , of the shifted tangent bundle T X/Y [1] → X along the zero section s : X → T X/Y [1] . (2) For λ ∈ A k with λ = 0 , the fiber ( D X/Y ) λ canonically identifies with the formal completion Y ∧ X ∈ dSt laft k , of X in Y along the morphism f .(3) (Hodge Filtration) There exists a natural sequence of morphisms X × A k = X (0) → X (1) → · · · → X ( n ) → · · · → Y, admitting a deformation theory, in (dSt laft k ) X × A k //Y × A k . For each n ≥ , the relative tangent complex T X ( n ) /Y ∈ IndCoh( X ( n ) ) , has first non-zero associated graded piece in degree n and the latter identifies naturally with i IndCoh n, ∗ (Sym n +1 ( T X/Y [1])[ − ∈ IndCoh( X ( n ) ) . Dually, the pro-relative cotangent complex L X/Y ∈ Pro(Coh( X ( n ) )) is equipped with a decreasingfiltration, starting in degree n , and the corresponding associated n -th graded piece identifies naturallywith i n, ∗ (Sym n +1 ( L X/Y [ − ∈ Pro(Coh b ( X ( n ) )) . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 41 (4) Assume that f : X → Y is a closed immersion of derived schemes locally almost of finite presentationthen, for each n ≥ , the morphism X ( n ) → X ( n +1) , has the structure of a square-zero extension associated to a canonical morphism d : L X ( n ) → L X ( n ) /Y → i n, ∗ (Sym n +1 ( L X/Y [ − , in the ∞ -category Coh + ( X ( n ) ) , where i n : X × A k → X ( n ) denotes the structural morphism. In particular, if X is a derived scheme almost of finite presentation,then so are the X ( n ) , for each n ≥ .(5) There exists a natural morphism colim n X ( n ) → D X/Y , in the ∞ -category FMP X × A k //Y ∧ X × A k . Moreover, the latter is an equivalence of derived k -stacks almostof finite presentation.Proof. Assertion (1) follows formally from [GR17b, §9, Proposition 2.3.6]. Assertion (2) follows from theobservation that the fiber of the simplicial object D • X/Y at λ = 0 can be identified with the Čech nerve of the(completion) of the morphism f X × Y ∧ X • ∈ Fun( ∆ op , dSt laft k ) . Therefore, its colimit taken in FMP X × A k / agrees with the formal completion Y ∧ X ∈ FMP X × A k / . The first partof assertion (3) follows formally from [GR17b, §9, Theorem 5.1.3]. For each n ≥
0, denote by i n : X × A k → X ( n ) , the structural morphism. The latter is a proper morphism due to [Sta13, Tag 0CYK].The statement concerning the relative cotangent complex follows formally from [GR17b, §9, Theorem 5.1.3]by applying the Serre duality functor and combining [Gai11, Corollary 9.5.9 (b)] with [Gai11, Proposition 3.1.3]and [GR17b, Corollary 1.4.4.2] to produce a natural identification D Serre X (cid:0) ( i IndCoh n, ∗ (Sym( T X/Y [1])[ − (cid:1) ’ i n, ∗ (cid:0) Sym( L X/Y [ − (cid:1) , in the ∞ -category Pro(Coh b ( X )). Assertion (5) is an immediate consequence of [GR17b, §9, Proposition 5.2.2].We shall now deduce claim (4) of the Proposition: it follows from Theorem 9.5.1.3, in loc. cit., that the (Serre)tangent complex T X ( n ) /Y ∈ IndCoh( X ( n ) ) , admits a natural (increasing) filtration whose n -th piece identifies with i IndCoh n, ∗ (Sym n +1 ( T X/Y [1]))[ − ∈ IndCoh( X ( n ) ) . For this reason, we have a structural morphism i IndCoh n, ∗ (Sym n +1 ( T X/Y [1]))[ − → T X ( n ) /Y , in the ∞ -category IndCoh( X ( n ) ). Consider then the natural composite i IndCoh n, ∗ (Sym n +1 ( T X/Y [1])[ − → T X ( n ) /Y → T X ( n ) , in the ∞ -category IndCoh( X ). Using the equivalence of ∞ -categories between Lie algebroids in IndCoh( X )and the ∞ -category FMP X/ , c.f. [GR17b, §8.5], one produces then a canonical morphism X ( n ) → X ( n +1) , in the ∞ -category FMP X × A k / .In the case where f : X → Y is a closed immersion of derived schemes locally almost of finite presentationthe latter construction can be adapted in terms of the (usual) cotangent complex formalism: thanks to [Lur12b,Corollary 8.4.3.2], the relative cotangent complex L X/Y ∈ Coh + ( X ) , is 1-connective. In particular, the shift L X/Y [ −
1] is 0-connective. By applying the usual Serre duality functor(c.f. [Gai11, §9]) we obtain a well defined fiber sequence L X ( n ) → L X ( n ) /Y → i n, ∗ (cid:0) Sym n +1 ( L X/Y [1])[ − (cid:1) , (3.2)in the ∞ -category Pro(Coh b ( X )). Moreover, under our assumptions, it follows that ( ?? ) lies in the fullsubcategory Coh + ( X ) ⊆ Pro(Coh b ( X )) , via the equivalence of ∞ -categories provided in [GR17b, Corollary 1.4.4.2]. Since the right hand side of (3.2) is1-connective, it follows from [GR17b, §8, 5.5] that the extension X ( n ) → X ( n +1) , for n ≥ , can be identified with the (usual) square-zero extension of X ( n ) by means of the derivation (3.2). This provesthe second part of claim (3) and the proof of the proposition is thus concluded. (cid:3) Remark . Assume that f : X → Y is a closed immersion of derived schemes locally almost of finitepresentation. Then Construction 3.1 can be simplified by taking directly Y instead of the formal completion Y ∧ X in the defining pullback square for (3.1). Indeed, we claim that the simplicial object obtained as the fiberproduct ( D • X/Y ) Y × A k Map / A k (B • scaled , X × A k ) Map / A k (B • scaled , Y × A k ) , in Fun( ∆ op , dSt laft k ), agrees with D • X/Y . Observe that we have a natural identification( D • X/Y ) × A k G m ’ X × Y • , and the latter agrees with the formal groupoid associated with ( X → Y ∧ X ) ∈ FMP X × A k . Moreover, we have anatural identification D X/Y × A k { } ’ T • X/Y , where T • X/Y denotes the groupoid object associated to the usual shifted relative tangent bundle T
X/Y [1].Under our assumption that f : X → Y is a closed immersion it follows from [Lur12b, Corollary 8.4.3.2] thatthe relative cotangent complex L X/Y ∈ Coh + ( X ) , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 43 is 1-connective. As a consequence, we deduce that the natural morphism X → T • X/Y is a termwise nil-isomorphism. As a consequence, we conclude immediately that T • X/Y agrees with the formal groupoid, over X ,associated to the formal moduli problem( X → T X/Y [1] ∧ ) ∈ FMP X/ . Remark . Consider the sequence of deformations in Proposition 3.4, { X ( n ) } n ≥ . The latter defines a filtrationon the global sections of the formal completion Y ∧ X , which we shall refer to as the Hodge filtration associated tothe morphism f . Therefore, Proposition 3.4 can be interpreted as a spreading out result of the Hodge filtrationfrom de Rham cohomology to the global sections of the formal completion. Remark . The left-lax action of the multiplicative monoid A k can be made explicit at the fiber of themorphism p : D X/Y → A k at { } ⊆ A k . Indeed, under the natural identification( D X/Y ) ’ T ∧ X/Y [1] , we obtain that the filtration on D X/Y induces the natural filtration on T ∧ X/Y [1] which is obtained by considering T X/Y ∈ IndCoh( X ) gr , =1 ⊆ IndCoh( X ) gr , ≥ , using notations as in [GR17b, §9.2.5.2]. Moreover, the construction of the Hodge filtration in Proposition 3.4(3) is naturally A k left-lax equivariant and the equivalence in Proposition 3.4 (4) is A k left-lax equivariant, aswell.Our goal now is to identify the Hodge filtration on Y ∧ X , in the case where f : X → Y is a closed immersion: Proposition 3.8.
Let f : X → Y be a morphism of affine derived schemes. Denote by A = Γ( X, O X ) and B = Γ( Y, O Y ) , the corresponding derived global sections and let I := fib( B → A ) denote the fiber of the inducedmorphism of derived k -algebras B → A . The following assertions hold:(1) If f : X → Y be a locally of complete intersection closed morphism in the ∞ -category (dAff k ) laft , thenthe Hodge filtration on the formal completion Y ∧ X constructed in Proposition 3.4 identifies with the usual I -adic filtration on B ∧ I , by taking derived global sections.(2) Assume that f : X → Y is a closed immersion. Then the Hodge filtration, { Fil nH } , on the derived globalsections Γ( Y ∧ X , O Y ∧ X ) ’ B ∧ I , produces, for each n ≥ , natural equivalences B/ Fil nH ’ dR A/B / Fil nH , of derived k -algebras. In particular, we obtain a natural equivalence of derived k -algebras B ∧ I ’ lim n ≥ dR A/B / Fil nH . Proof.
We first prove claim (2) of the Proposition. Since f : X → Y is a closed immersion between affines wededuce that the relative cotangent complex L X/Y ∈ Coh + ( X ) is 1-connective. For this reason, the sequencemorphisms X × A k = X (0) , → X (1) , → · · · , → X ( n ) , → . . . , of Proposition 3.4 (4) correspond to successive square-zero extensions. Moreover, it follows from [GR17b,Corollary 9.5.2.5] that, after applying the Serre duality functor on IndCoh, one has an equivalence O Y ∧ X ’ lim n ≥ O ( X ( n ) ) λ , in CAlg k , for any for λ = 0. Henceforth, at the level of derived global sections, one obtains a naturalidentification B ∧ I ’ lim n ≥ Γ(( X ( n ) ) λ , ( O X ( n ) ) λ ) , in the ∞ -category CAlg k . Similarly, thanks to the proof of [GR17b, §9, Theorem 5.1.3] in the case of vectorgroups (c.f. [GR17b, §9.5.5]) we deduce a natural equivalencedR A/B ’ lim n ≥ Γ(( X ( n ) ) , O ( X ( n ) ) ) , in CAlg k . By the naturality of the construction of [GR17b, §9.5.1] combined with the argument provided in[GR17b, §9.5.5] to the reduction to the case of vector groups, we deduce that p : X ( n ) → A k , has constant fibers. Indeed, for n = 0, the natural morphism X (0) → X (1) , corresponds, by construction, to the natural square-zero extension L A → L A/B , classifying the extension L A/B [ − → Sym ≤ ( L A/B [ − → A, of A -modules. Assuming the result for n ≥
0, we have that L A ( n ) /B ∈ Coh + ( A ( n ) ) , where A ( n ) := Γ( X ( n ) , O X ( n ) ), identifies with the relative cotangent complex of the natural morphism B → Sym ≤ n ( L A/B [ − , produced by induction and the naturality of the construction. Moreover, thanks to [GR17b, §9.5.5] the relativecotangent complex L Sym ≤ n ( L A/B [ − /B ∈ Coh + (Sym ≤ n ( L A/B [ − , is equipped with a decreasing filtration, starting in degree n , whose n -th piece identifies withSym n +1 ( L A/B [ − ∈ Coh + (Sym ≤ n ( L A/B [ − . For this reason, the derivation L A ( n ) → L A ( n ) /B → Sym n +1 ( L A/B [ − , identifies with the natural derivation L Sym ≤ n ( L A/B [ − /B → Sym n +1 ( L A/B [1]) , classifying the extensionSym n +1 ( L A/B [ − → Sym ≤ n +1 ( L A/B [ − → Sym ≤ n ( L A/B [ − . The claim now follows by induction on n ≥
0. As a consequence of the previous considerations we deduce thatΓ(( X ( n ) ) λ , ( O X ( n ) ) λ ) ’ Γ(( X ( n ) ) , ( O X ( n ) ) ) , for every λ ∈ A k , and therefore we have natural equivalences B ∧ I ’ dR A/B and B/ Fil nH ’ dR A/B / Fil nH , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 45 in CAlg k , as desired. We now prove statement (1) of the Proposition. Assume first that A and B are ordinary k -algebras and the (classical) ideal I ⊆ B is generated by a regular sequence. In this case, we have anidentification L A/B ’ I/I [1] , see for instance [Sta13, Tag 08SJ]. In particular, the n -th graded piece of the Hodge filtration in Proposition 3.4(3) can be identified with Sym n ( L A/B [ − ’ I n /I n +1 , where the latter equivalence follows again by the fact that I is generated by a regular sequence in π ( B ). Inparticular, we deduce, by induction on n ≥
1, a natural identification between the fiber sequenceSym n +1 ( L A/B [ − → Sym ≤ n +1 ( L A/B [ − → Sym ≤ n ( L A/B [ − , and the the I -adic one I n /I n +1 → B/I n +1 → B/I n , as B -modules. By naturality of [GR17b, Theorem 9.5.1.3] and induction on n ≥ X ( n ) ’ Spec(
A/I n +1 ) , of affine derived schemes, as desired. The assertion in general follows along the same reasoning noticing that L A/B ∈ Mod A , is concentrated in homological degrees [1 , (cid:3) Remark . It follows from [GR17b, §9, Theorem 5.5.4], that in the situation of Proposition 3.8, the naturalcomposite morphism Sym n ( L A/B [ − → Sym ≤ n ( L A/B [ − → Sym n +1 ( L A/B [ − , corresponds to the usual de Rham differential, where Sym ≤ n ( L A/B [ − → Sym n +1 ( L A/B [ − n +1 ( L A/B [ − → Sym ≤ n +1 ( L A/B [ − → Sym ≤ n ( L A/B [ − , associated to the n -th piece of the Hodge filtration, Remark . Let f : X → Y be a closed immersion of affine derived schemes. Let A := Γ( X, O X ) and B := Γ( Y, O Y ). In [Bha12, Proposition 4.16], the author proves a similar statement to Proposition 3.8. It turnsout that both filtrations on B ∧ I agree in the case of [Bha12] or in the situation of [GR17b, §9]. A comparisoncan be stated as follows: it is immediate that both the filtrations of Bhatt and Gaitsgory-Rozemblyum on B ∧ I ,denoted respectively Fil nB and Fil nH , agree in the case where f is a local complete intersection morphism. Moreover, it follows from the fact that thecotangent complex commutes with sifted colimits that, for each n ≥
0, the natural morphismsSym ≤ n ( L A/B [ − → Sym n +1 ( L A/B [ − A, I ) where A is a simplicial k -algebra and I ⊆ A a simplicial ideal given by the description: a pair( A, I ) is a fibration (resp., trivial fibration) if and only if both A and I are simultaneously fibrations (resp.,trivial fibrations) of simplicial sets. Moreover, the cofibrant objects correspond to pairs ( P, J ) where, for each[ n ] ∈ ∆ , P n is a polynomial k -algebra on a set X n and J n an ideal defined by a subset Y n ⊆ X n , both preserved by degeneracies and the relative situation. The ∞ -categorical localization at the class of weak equivalencesagrees furthermore with the ∞ -category Fun closed (∆ , CAlg k ). In particular, considering a simplicial resolution( P • , I • ) for the morphism B → A consisting of polynomial algebras and regular ideals we deduce thatFil nH ’ Fil nB , for n ≥ , in the general case, as well.3.2. The construction of the deformation in the affinoid case.
Consider the object B an , • scaled : ∆ op → (dAnSt k ) / A k , obtained as the analytification of the simplicial object B • scaled ∈ Fun( ∆ op , dSt laft k ) described in the previoussection.Our goal in this section, is to construct the deformation to the normal cone in the non-archimedean setting,in the case where Y is derived k -affinoid. We shall further assume that f : X → Y, exhibits X as an analytic formal moduli problem over Y . In particular, X admits a deformation theory and wecan consider the relative cotangent complex L an X/Y ∈ Pro ps (Coh + ( X )) . Definition 3.11.
We define the parametrized non-archimedean deformation to the normal bundle via thepullback diagram D an , • X/Y Y × A k Map / A k ( B an , • scaled , X × A k ) Map / A k ( B an , • scaled , Y × A k ) , computed in the ∞ -category Fun( ∆ op , dAnSt k ). Remark . As in the algebraic case, the object B an , • scaled admits a left-lax action of the multplicative monoid A k ∈ dAnSt k . This action induces a natural left-lax action of A k on the deformation D an , • X/Y , as well.
Remark . It follows immediately from the definition of D an , • X/Y that the latter commutes with filtered colimitsin X ∈ dAnSt k . Moreover, Proposition 2.45 implies that the natural morphismcolim S ∈ (AnNil cl /X ) /Y D an , • S/Y → D an , • X/Y , is an equivalence in dAnSt k . For this reason, we will often reduce the construction to the case where f : X → Y is a nil-isomorphism of derived k -affinoid spaces. Remark . Let f : X → Y be a nil-isomorphism in the ∞ -category dAfd k . Then we can consider the diagramid Y : Y → X G X red Y → Y, which exhibits ( X f −→ Y ) as a retrat of the induced morphism ( X g −→ X t X red Y ), both regarded as objects inAnFMP X/ . In this case, it follows by the naturality of the parametrized Hodge filtration that we have a retractdiagram D • X/Y → D • X/X F X red Y → D • X/Y , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 47 of simplicial objects in AnFMP X × A k / . This observation will allow us to perform reduction arguments byreplacing the nil-isomorphism f : X → Y by the nil-embedding g : X → X t X red Y .Let f : X → Y be a nil-isomorphism in the ∞ -category dAfd k . Thanks to [PY20, Lemma 6.9.], we deducethat the induced morphism f alg : X alg → Y alg , is again a nil-isomorphism of affine derived schemes. We shall need a few preparations: Lemma 3.15.
Let Y ∈ dAfd k and ( f : X → Y ) ∈ AnFMP /Y . Then one has a natural equivalence ( f alg ) an Y ’ f, in the ∞ -category Fun( ∆ op , dAfd k ) .Proof. The statement of the Lemma is a direct consequence of [PY20, Theorem 6.12]. (cid:3)
Lemma 3.16.
Let f alg : X alg → Y alg be a closed embedding of derived schemes. There exists a filtered ∞ -category I and a diagram F : I op → Fun(∆ , dAff laft k ) , such that, for each α ∈ I , we have that F ( α ) := ( f α : X α → Y α ) , is a closed immersion in the ∞ -category dAff laft k and such that f ’ lim I op F, in the ∞ -category Fun(∆ , dAff k ) .Proof. Consider the full subcategory Fun closed (∆ , CAlg k ) ⊆ Fun(∆ , CAlg k ) spanned by morphisms A → B such that the induced homomorphism of ordinary rings π ( A ) → π ( B ) , is surjective. As in the proof of Proposition 3.4 the ∞ -category Fun closed (∆ , CAlg k ) is obtained as an ∞ -categorical localization at weak equivalences of the simplicial model category whose objects correspond topairs ( C, I ) where C is a simplicial k -algebra and I ⊆ C a simplicial ring. Moreover, thanks to [Qui67, §2 IV,Theorem 4] it follows that finite projective generators in the latter correspond to pairs of the form ( P, I ) where P is some polynomial algebra in a finite number of generators (concentrated in non-negatively homologicallydegrees) and I an ideal spanned by a regular sequence.In particular, finite projective generators in Fun closed (∆ , CAlg k ) correspond to morphisms g : P → Q where P is a finite free derived k -algebra and the morphism g is a complete intersection morphism. As in the proof of[Lur12b, Proposition 8.2.5.27] we then deduce that the ∞ -category Fun closed (∆ , CAlg k ) is compactly generatedby morphisms g : P → Q which can be obtained from finite projective generators via a finite sequence of retractsand finite colimits.Arguing as in [Lur12b, Proposition 8.2.5.27], we are able to describe the set of compact generators ofFun closed (∆ , CAlg k ): they correspond to locally of finite presentation morphisms between locally finitepresentation derived k -algebras. In particular, we can find a presentation of f alg : X alg → Y alg , by closed immersions between almost finite presentation affine derived schemes f α : X α → Y α , for α ∈ I op , where I is a filtered ∞ -category, as desired. (cid:3) Remark . Let f : X → Y be a nil-embedding in the ∞ -category dAfd k . Find a presentation of f alg byclosed immersions between almost of finite presentation affine derived schemes of the formlim α ∈ I op ( f α : X α → Y α ) . Consider the object D • X alg /Y alg ∈ Fun( ∆ op , dAff k ) as in Construction 3.1. It is clear from the descritption of D • X alg /Y alg given in Construction 3.1 combined with Remark 3.5 that one has a natural equivalence of derivedstacks D • X alg /Y alg ’ lim α ∈ I op D • X alg α /Y alg α , in the ∞ -category Fun( ∆ op , dSt k ). Lemma 3.18.
Let Y ∈ dAfd k . Consider a morphism f : X → Y in dAnSt k which exhibits X as an analyticformal moduli problem over Y . Then one has a natural equivalence θ X/Y : D an , • X/Y ’ ( D • X alg /Y alg ) an Y , where ( − ) an Y denotes the relative analytification with respect to Y .Proof. We first observe that both D an , • X/Y and ( D • X alg /Y alg ) an Y are stable under filtered colimits in X . For thisreason, we reduce ourselves to the case where f : X → Y itself is a nil-isomorphism in the ∞ -category dAfd k .Similarly, θ X/Y is stable under retracts. Therefore, we are allowed to replace the morphism X → Y by X → Y G X red X, and therefore assume that f : X → Y is a nil-embedding. As in Lemma 3.16, write the nil-embedding betweenaffine derived schemes f : X alg → Y alg , as a cofiltered limit of closed immersionslim α ∈ I op f α : lim α ∈ I op X alg α → lim α ∈ I op Y alg α , where for each index α ∈ I , both X α and Y α are affine derived schemes almost of finite presentation. Since thenatural projection morphism B • scaled → A k , where A k is considered as a constant simplicial object in dSt k , is a proper morphism, we deduce from [HP18,Theorem 6.13] that for every α , the natural morphismsMap / A k (B • scaled , X alg α × A k ) an → Map / A k ( B • , anscaled , ( X alg α ) an × A k )Map / A k (B • scaled , Y alg α × A k ) an → Map / A k ( B • , anscaled , ( Y alg α ) an × A k )are equivalences in the ∞ -category dAnSt k . Therefore, the natural morphism( D • X alg α /Y alg α ) an → D • , an( X alg α ) an / ( Y alg α ) an , is an equivalence in the ∞ -category dAnSt k . Observe further that for every α ∈ A the mapping stacksMap / A k (B • scaled , X alg α × A k ) and Map / A k (B • scaled , Y alg α × A k )are affine schemes and therefore it follows by the construction of the analytification functor( − ) an : dAff k → dAnSt k , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 49 as a right Kan extension of the usual analytification functor( − ) an : dAff laft k → dAn k , that we have natural equivalences(lim α Map / A k (B • scaled , X alg α × A k )) an ’ Map / A k (B • , anscaled , ( X alg ) an Y × A k )(lim α Map / A k (B • scaled , Y alg α × A k )) an ’ Map / A k (B • , anscaled , ( Y alg ) an Y × A k ) . in the ∞ -category (dAnSt k ) X × A //Y × A k . The result now follows from the existence of a commutative cube( D • X alg /Y alg ) an Y Y × A k ( D • X alg /Y alg ) an ( Y alg × A k ) an Map / A k ( B • , X × A k ) Map / A k ( B • , Y × A k ) Map / A k ( B • , ( X alg ) an × A k ) Map / A k ( B • , ( Y alg ) an × A k )whose top and front squares are fiber products in dAnSt k , and thus so it is the back square, as desired. (cid:3) Corollary 3.19.
With notations as above, one has a canonical equivalence ( D • , an X/Y ) ’ ( T an , • X/Y ) ∧ , where the latter denotes the commutative group object associated to the formal completion of the tangent bundleof f along the zero section s : X → T an , • X/Y .Proof.
The result follows from [GR17b, Proposition 9.2.3.6] combined with the fact that relative analytificationcommutes with tangent bundles and formal completions, see the proof of [HP18, Lemma 5.31] and [HP18,Corollary 5.20]. (cid:3)
The following result is an immediate consequence of Lemma 3.18:
Lemma 3.20.
For each [ n ] ∈ ∆ , the object D an , [ n ] X/Y ∈ (dAnSt k ) X × A k //Y × A k admits a deformation theory.Proof. We shall prove that each component of the simplicial object D an , • X/Y ∈ Fun( ∆ op , AnFMP X × A k //Y × A k ) , admits a deformation theory. By [GR17b, Lemma 2.3.2], it follows that D • X alg //Y alg is an object in FMP X alg //Y alg .The result now follows by applying the relative analytification functor ( − ) an Y together with [PY20, Proposition6.10]. (cid:3) Remark . Let ( X f −→ Y ) ∈ AnFMP /Y , with Y ∈ dAfd k . It follows immediately from Lemma 3.20 that thedatum ( X × A k → D • , an X/Y → Y × A k ) ∈ (dAnSt k ) X × A k //Y × A k , belongs to the full subcategory AnFMP X × A k //Y × A k . Construction . The ∞ -category AnFMP X × A k //Y × A k is presentable and in particular it admits siftedcolimits. Thanks to Lemma 3.20, we can consider the object D an X/Y := colim ∆ op D an , • X/Y ∈ AnFMP X × A k //Y × A k . Similarly, we consider the sifted colimit D X alg /Y alg := colim ∆ op D • X alg /Y alg ∈ FMP X alg × A k //Y alg × A k , computed in the ∞ -category FMP X alg × A k //Y alg × A k . We can consider the latter as an object in D X alg /Y alg ∈ dSt X alg × A k //Y alg × A k , and therefore consider its relative analytification ( D X alg /Y alg ) an Y ∈ dAnSt X alg × A k //Y alg × A k . Thanks toLemma 3.18 it follows that we have a natural morphism θ X/Y : D an X/Y → ( D X alg /Y alg ) an Y , in the ∞ -category (dAnSt k ) X × A k //Y × A k .Our next goal is to prove that the morphism θ X/Y is an equivalence of deformations. In order to prove thelatter statement, we shall need a preliminary lemma:
Lemma 3.23.
Let f : X → Y be a nil-embedding in the ∞ -category dAfd k . Consider approximation of thenil-embedding morphism f alg lim α ∈ I op f α : lim α ∈ I op X α → lim α ∈ I op Y α , as in Lemma 3.16. Then, we have a natural morphism of the form θ : D X alg /Y alg → lim α ∈ I op D X α /Y α , (3.4) which is furthermore an equivalence in the ∞ -category AnFMP X × A k /Y alg × A k Proof.
It is clear that the natural morphisms X α → D X α /Y α assemble into a morphism X alg ’ lim α ∈ I op X α → lim α ∈ I op D X α /Y α , which admits a deformation theory, therefore the morphism X alg → lim α ∈ I op D X α /Y α , exhibits lim α ∈ I op D X α /Y α as an object in the ∞ -category FMP X alg × A k / . Notice that Theorem 2.65 holds truein this setting, proved in an analogous way, to provide us with an equivalence of ∞ -categoriesB X ( • ) : FGrpd( X alg × A k ) → FMP X alg × A k / , where FGrp( X alg × A k ) denotes the ∞ -category of formal groupoids over X alg × A k ∈ dSt k . For this reason,we are reduced to show the existence of a natural equivalence between formal groupoids over X alg × A k , D • X alg /Y alg ’ X × lim α D Xα/Yα • , where the right hand side denotes the Čech nerve of the morphism X → lim α D X α /Y α . Since limits commutewith limits in the ∞ -category dAnSt k , we obtain a chain of natural equivalences of the form( X alg ) × lim α D Xα/Yα • ’ lim α (cid:0) X × D Xα/Yα • α (cid:1) ’ lim α D • X α /Y α ’ D • X alg /Y alg , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 51 in the ∞ -category FGrp( X ). Only the the second equivalence needs a justification: for each α ∈ I , we havenatural identifications D X α /Y α ’ B X α ( D • X α /Y α ) , by construction, where B X α ( − ) is as in [GR17b, §5.2]. Therefore, we deduce by adjunction that X × D Xα/Yα • ’ D • X α /Y α , is an equivalence in the ∞ -category FGrpd( X α ), as well. The proof is thus concluded. (cid:3) The following result implies that the relative analytification commutes with the deformation to the normalbundle:
Proposition 3.24.
The natural morphism of (3.4) , θ X/Y : D an X/Y → ( D X alg /Y alg ) an Y , is an equivalence of derived k -analytic stacks.Proof. Since the morphism θ X/Y commutes with filtered colimits in X , we reduce the statement of theProposition to the case where X ∈ AnNil /Y . In this case, we have that Y × A k ∈ AnFMP X × A k / . By naturalityof the morphism θ X/Y we are allowed to prove the statement of the Proposition up to a retract. For this reason,we can replace Y by Y t X red X and therefore assume that f : X → Y is a nil-embedding. Consider now therelative analytification functor ( − ) an Y : FMP /Y alg × A k → AnFMP /Y × A k , introduced in [PY20, §6.1]. Thanks to [PY20, Theorem 6.12] the natural functor displayed above is anequivalence of ∞ -categories. Observe further that we have a well defined functor( − ) an Y : FMP X alg × A k //Y alg × A k → AnFMP X × A k //Y × A k , (3.5)induced by the usual relative analytification functor. Indeed, we have that X × A k ’ ( X alg ) an Y × A k , as the pair(( − ) alg , ( − ) an Y ) forms an equivalence itself. It is further clear that (3.5) is an equivalence of ∞ -categories as well. In particular, it commutes with all small colimits. Therefore, we have a chain of naturalequivalences ( D X alg /Y alg ) an Y ’ (colim ∆ op D • X alg /Y alg ) an Y ’ colim ∆ op ( D • X alg /Y alg ) an Y ’ colim ∆ op D an , • X/Y , in AnFMP X × A k //Y × A k , thanks to Proposition 3.24. The conclusion now follows from the observation that theforgetful functor AnFMP X × A k //Y × A k → AnFMP X × A k / , commutes with colimits and it is moreover conservative, thus it reflects sifted colimits. (cid:3) Lemma 3.25.
Consider the natural projection morphism q : D an X/Y → A k . Then its fiber at λ = 0 coincides with the formal completion Y ∧ X , and its fiber at { } ⊆ A k with the completion of the shifted tangent bundle T an X/Y [1] along the zero section s : X → T an X/Y [1] .Proof.
The above assertions follow immediately from Proposition 3.24, Corollary 3.19 and Proposition 3.4. (cid:3)
Construction . Let Y ∈ dAfd k and ( X f −→ Y ) ∈ AnFMP /Y . Denote by g : U → Y a morphism in dAfd k .Consider the pullback diagram X U UX Y, f computed in the ∞ -category dAfd k . It follows from the definitions that the morphism X U → U admits adeformation theory and that we have a natural pullback square of simplicial objects D an , • X U /U D an , • X/Y
U Y, in the ∞ -category dAnSt k . For this reason, we obtain a natural commutative diagram D an X U /U D an X/Y
U Y, in the ∞ -category dAnSt k . Similarly, [PY18b, Proposition 3.17] implies that we have a pullback diagram X alg U U alg X alg Y alg f alg , in the ∞ -category dAff k . Reasoning as above, we have a natural commutative square D X alg U /U alg D X alg /Y alg U alg Y alg , in the ∞ -category dSt k . Proposition 3.27.
The commutative square D an X U /U D an X/Y
U Y, of Construction 3.26, is a pullback square in the ∞ -category dAnSt k . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 53
Proof.
Since the pullback diagram above is preserved under filtered colimits in X , we can assume without lossof generality that f : X → Y is a nil-isomorphism. Similarly, we are allowed to replace the retract : X → Y bythe nil-embedding g : → X → Y G X red X. In order to show the assertion of the proposition, we are reduced to show that the natural morphism D an X U /U → D an X/Y × Y U, is an equivalence in the ∞ -category AnFMP X U //U . Moreover, Proposition 3.24 reduce us to show that thenatural morphism D X alg U /U alg → D X alg /Y alg × Y alg U alg , is an equivalence in dSt k (see also Proposition 3.24). Moreover, Lemma 3.23 allows us to perform Noetherianapproximation on the nil-isomorphism f alg : X alg → Y alg . For this reason, we can find an approximation of themorphism f : X alg → Y alg an approximation of the formlim α ∈ I op ( f α : X α → Y α ) , where I is a filtered ∞ -category and for each α ∈ I , the morphism f α is a closed immersion of affine derivedschemes almost of finite presentation over k . Similarly, find an almost of finite presentation approximation U alg ’ lim β ∈ J op U β . By almost finite presentation we deduce that for every α ∈ I there exists an index β ( α ) ∈ J such that thecomposite U alg → Y alg → Y α , (3.6)factors as U alg → U β ( α ) → Y α , for a well defined morphism U β ( α ) → Y α in dAff laft k . Let K denote the filtered subcategory of the product I × J spanned by those pairs ( α, β ) ∈ I × J such that the composite displayed in (3.6) factors as U β → Y α . We thushave that the morphism U alg → Y alg , can be written as lim ( γ ,γ ) ∈ K ( U γ → Y γ ). For each ( γ , γ ) ∈ K consider the deformations to the normalbundle D X γ /Y γ and D X Uγ /U γ , where X U γ is defined as the pullback of the diagram U γ → Y γ ← X γ . We claim that for each ( γ , γ ) ∈ K , we have a natural equivalence D X Uγ /U γ ’ D X γ /Y γ × Y γ U γ , of derived k -stacks almost of finite presentation. Indeed, by conservativity of the relative tangent complex, c.f.[GR17b, §5, Theorem 2.3.5], it suffices to show that T X Uγ / D XUγ /Uγ → T X Uγ / D Xγ /Yγ × Yγ U γ , is an equivalence in the ∞ -category IndCoh( X U γ ). Thanks to [GR17b, §5, Corollary 2.3.6], we are reduced toshow that the natural morphism of simplicial objects { T X Uγ / D • XUγ /Uγ } → { T X Uγ / D • Xγ /Yγ × Yγ U γ } , is an equivalence in Fun( ∆ op , IndCoh( X U γ )). The latter assertion further reduce us to show that thecommutative diagram D • X Uγ /U γ D • X γ /Y γ U γ Y γ is a pullback diagram component-wise in dAff laft k . By unwinding the definitions, it suffices to prove that thecommutative diagram of simplicial objects Map / A k ( B • scaled , X U γ × A k ) Map / A k ( B • scaled , U γ × A k )) Map / A k ( B • scaled , X γ × A k ) Map / A k ( B • scaled , Y γ × A k ))is a pullback square. But the latter assertion is obvious as derived k -analytic mapping stacks commute withfiber products in dAnSt k . (cid:3) Gluing the Deformation.
In this §, we globalize the results proved so far in §3.2. Let f : X → Y be amorphism of locally geometric derived k -analytic stacks. Consider as before the deformation to the normalbundle of the morphism f constructed via the pullback diagram D an , • X/Y Y ∧ X × A k Map / A k ( B an , • scaled , X × A k ) Map / A k ( B an , • scaled , Y ∧ X × A k ) , in the ∞ -category dAnSt k . As in the previous §, one can show that the simplicial object D an , • X/Y ∈ AnFMP X × A k //Y ∧ X × A k . Let now g : U → Y ∧ X be a morphism in dAnSt k , where U ∈ dAfd k . Consider the pullback diagram X U XU Y ∧ X , g fg in the ∞ -category dAnSt k . We have: Lemma 3.28.
The morphism X U → U admits a deformation theory, and thus exhibits X U as an object in AnFMP /U .Proof. It is clear that X U is both nilcomplete and infinitesimally cartesian as a derived k -analytic stack. Itsuffices thus to show that the structural morphism X U → U admits a cotangent complex. We claim that L X U /U identifies with ( g ) ∗ L X/Y ∈ Coh + ( X U ) . PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 55
Indeed, let Z → X U be any morphism with h : Z ∈ dAfd k . Then we obtain that h ∗ ( g ) ∗ L X/Y ’ ( g ◦ h ) ∗ ( L X/Y ) , in Coh + ( Z ), which corepresents the functorDer X/Y ( Z, − ) : Coh + ( Z ) → S , see [PY17, Definition 7.6] for the definition of the latter. Since fiber products commute with fibers, we have anatural identification Der X U ( Z, − ) ’ Der X ( Z, − ) × Der Y ( Z, − ) Der U ( Z, − ) . We thus conclude that Der X U /U ( Z, − ) ’ fib (cid:0) Der X U ( Z, − ) → Der U ( Z, − ) (cid:1) , identifies naturally with Der X/Y ( Z, − ) ’ fib (cid:0) Der X ( Z, − ) → Der Y ( Z, − ) (cid:1) . The claim of the lemma now follows from the fact that Der
X/Y ( Z, − ) : Coh + ( Z ) → S is corepresentable by h ∗ ( g ) ∗ L X/Y , as desired. (cid:3)
Thanks to the above lemma we conclude that the canonical map X U → U exhibits X U as an analytic formalmoduli problem over U . We are thus in the case of the previous section. The following result implies that thedeformation to the normal bundle in the non-archimedean setting glues: Proposition 3.29.
The simplicial object D an , • X/Y : ∆ op → AnFMP
X//Y , admits a colimit D an X/Y ∈ AnFMP X/ .Proof. Let ( Y ∧ X ) afd denote the ∞ -category consisting of morphisms U → Y ∧ X , where U is a derived k -affinoid space. We have a natural equivalence of ∞ -categoriesΨ : (dAnSt k ) /Y → lim U ∈ ( Y ∧ X ) afd (dAnSt k ) /U . We deduce from the construction of D an , • X/Y that the simplicial object D an , • X/Y satisfiesΨ( D an , • X/Y ) ’ { D an , • X U /U }∈ lim U ∈ ( Y ∧ X ) afd (dAnSt k ) /U . For each U ∈ ( Y ∧ X ) afd the ∞ -categories AnFMP X U //U are presentable. For this reason, we can consider thegeometric realization D an X U /U ∈ AnFMP X U /U , for every [ n ] ∈ ∆ . Thanks to Proposition 3.27 we deduce that the above deformations glue to form a uniquelydefined object D an X/Y ∈ dAnSt k , satisfiying the relation Ψ( D an X/Y ) ’ { D an X U /U }∈ lim U ∈ ( Y ∧ X ) afd (dAnSt k ) /U . It is clear from the definitions that D an X/Y ∈ AnFMP
X//Y . We claim that the latter is a colimit of the diagram D an , • X/Y . We need to show that for every Z ∈ AnFMP
X//Y together with a morphism D an , • X/Y → Z, then there exists a uniquely defined (up to a contractible space of choices) morphism D an X/Y → Z, in the ∞ -category AnFMP X//Y . Moreover, we are reduced to check this property after base change along any U → Y ∧ X , in which case the assertion follows immediately from the construction. (cid:3) Definition 3.30.
Let f : X → Y be a morphism of locally geometric derived k -analytic stacks. Then the deformation to the normal bundle associated to f is by definition the object D X/Y ∈ AnFMP X × A k //Y × A k , asin Proposition 3.29.3.4. The Hodge filtration.
Let f : X → Y be a morphism between locally geometric derived k -analyticstacks. In this §, we will describe the construction of the Hodge filtration on the deformation to the normalbundle D an X/Y associated to f . We first deal with the k -affinoid case: Construction . Let f : X → Y be a nil-embedding in the ∞ -category dAfd k . Consider the induced morphism f alg : X alg → Y alg , where X alg = Spec A and Y alg = Spec B , with A := Γ( X, O alg X ) and B := Γ( Y, O alg Y ) . The morphism f alg is a nil-embedding of affine derived schemes (combined [PY20, Lemma 6.9] and [PY18b,Proposition 3.17]). Consider then the nil-embedding g := f alg × id A k : X × A k → Y × A k , in dAff k . By Noetherian approximation, we can write g as an inverse limit of the formlim α ∈ I op g α : lim α ∈ I op X α × A k → lim α ∈ I op Y α × A k , where I is a filtered ∞ -category and for each index α ∈ I , we have that g α : X α × A k → Y α × A k , is a closed immersion in the ∞ -category dAff laft k . Fix some α ∈ I . Thanks to Proposition 3.4, there exists asequence of square-zero extensions of the form X α × A k = X (0) α , → X (1) α , → · · · , → X ( n ) α , → · · · → Y α × A k , such that each term comes equipped with a natural left-lax action of multiplicative monoid A k ∈ dSt k . Lemma 3.32.
Let n ≥ , and let α → β be a morphism in I op . Then the transition morphism X α × A k → X β × A k , lifts to a well defined induced morphism X ( n ) α → X ( n ) β , in dAff laft k .Proof. The result follows immediately from the naturality of the construction in [GR17b, §9.5.1]. (cid:3)
PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 57
We now introduce the (algebraic) Hodge filtration associated with the closed immersion f alg : X alg → Y alg : Definition 3.33.
Let f : X → Y be a nil-isomorphism in the ∞ -category dAfd k . Consider the morphism g := f × id : X alg × A k → Y alg × A k above. Lemma 3.32 implies that for each n ≥
0, we have a well definedobject X alg , ( n ) := lim α ∈ I op X ( n ) α ∈ dAff k , which fits into a sequence of square-zero extensions X × A k = X alg , (0) , → X alg , (1) , → · · · , → X alg , ( n ) , → · · · → Y alg , in dAff k . We shall refer to the sequence of the X alg , ( n ) as the algebraic Hodge filtration associated to themorphism f alg . In the more general case, where f : X → Y exhibits X as an analytic formal moduli problemwe define, for each n ≥
0, the n -th piece of the Hodge filtration as X ( n ) := colim S ∈ (AnNil cl X//Y ) S alg , ( n ) , the filtered colimit being computed in the ∞ -category AnFMP X × A k / . Construction . Let f : X → Y be a nil-embedding in the ∞ -category dAfd k . It follows from Proposition 3.4(4) that we have a natural morphism θ X/Y : colim n ≥ X alg , ( n ) → lim α ∈ I op D X α /Y α (3.7) ’ D X alg /Y alg , (3.8)in the ∞ -category FMP X × A k //Y × A k . Passing to filtered colimits we produce a natural morphism as in (3.7) inthe case where ( X f −→ Y ) ∈ AnFMP /Y and Y ∈ dAfd k .The following result implies that the Hodge filtration associated to f alg does not depend on choices: Proposition 3.35.
Let ( X f −→ Y ) ∈ AnFMP /Y and Y ∈ dAfd k . Then the natural morphism θ X/Y : colim n ≥ X alg , ( n ) → D X alg /Y alg , is an equivalence of derived k -stacks.Proof. Since both sides of θ X/Y are stable under filtered colimits it suffices to treat the case where f : X → Y isa nil-isomorphism of derived k -affinoid spaces. We observe that X → Y exhibits Y as a retract of the morphism X → X G X red Y. Since θ X/Y is stable under retracts, we are further reduced to the case where f is a nil-embedding. In this case,the relative cotangent complex L X alg /Y alg ’ L an X/Y , is 1-connective, see [PY20, Lemma 6.9]. The proof of Lemma 3.16 allow us to produce a presentationlim α ∈ I op ( f α : X α → Y α ) , of the morphism f alg by closed immersions f α : X α → Y α of affine derived schemes almost of finite presentationparametrized by a filtered ∞ -category I . The proof of Proposition 3.8, implies that, for each α ∈ I , the naturalmorphism X ( n ) α → X ( n +1) α , exhibits X ( n +1) α as a square-zero extension of X ( n ) α via a natural morphism L X ( n ) α → i n, ∗ Sym n +1 ( L X α /Y α [ − , in the ∞ -category Coh + ( X ( n ) α ). In particular, we deduce that, for each n ≥
0, the natural morphisms X α × A k → X ( n ) α , are nil-embeddings of affine derived schemes and so are the natural morphisms X alg × A k → X alg , ( n ) . We are now able to show that θ X/Y is an equivalence in the ∞ -category dSt k . Thanks to [Lur18, Corollary4.4.1.3] combined with [Lur18, Corollary 4.5.1.3], the relative cotangent complex satisfies L X alg × A k /X ( n ) ’ colim α ∈ I h ∗ α L X α × A k /X ( n ) α , in the ∞ -category QCoh( X alg × A k ), where h α : X → X α denotes the corresponding transition morphisms.Similarly, we have that L X alg × A k /Y × A k ’ colim α ∈ I h ∗ α L X α × A k /Y α × A k , in the ∞ -category QCoh( X alg × A k ). Moreover, the latter equivalences are compatible with the left-lax actionsof the multiplicative monoid A k ∈ dSt k on X alg , ( n ) and X ( n ) α , for each α ∈ I, over X alg × A k (resp. X α × A k ). Let λ ∈ A k be such that λ = 0. By passing to Serre duals, we deduce fromProposition 3.8 that, for each α ∈ I and n ≥
0, the morphism L ( X ( n ) α ) λ → Sym n +1 ( L X α /Y α [ − , (3.9)classifies the square-zero extension associated to the fiber sequenceSym n +1 ( L X α /Y α [ − → Sym ≤ n +1 ( L X α /Y α [ − → Sym ≤ n ( L X α /Y α [ − . Since (3.9) are stable under filtered colimits, we deduce, for each n ≥
0, that the morphism L X alg , ( n ) → Sym n +1 ( L X alg /Y alg [ − , classifies the square-zero extension given bySym n +1 ( L X alg /Y alg [ − → Sym ≤ n +1 ( L X alg /Y alg [ − → Sym ≤ n ( L X alg /Y alg [ − . Thanks to [GR17b, §9.5.5.2] combined with an analogous Noetherian approximation reasoning as the oneemployed before we obtain a natural equivalence L colim n ≥ X alg , ( n ) λ ’ L X alg /Y alg . In particular, thanks to the algebraic version of Proposition 2.36 we deduce that the natural morphismcolim n ≥ X alg , ( n ) → D X alg /Y alg , is an equivalence in FMP X alg , for any λ A k such that λ = 0. For λ = 0, the precise same reasoning applies using[GR17b, §9, Theorem 5.5.4] for the case of vector groups. The proof is thus concluded. (cid:3) We now introduce the non-archimedean Hodge filtration:
PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 59
Definition 3.36.
Let Y ∈ dAfd k and ( X f −→ Y ) ∈ AnFMP /Y . For each n ≥
0, we define the square-zeroextension X × A k , → X ( n ) , as the relative analytification, ( − ) an Y , of the square-zero extension X alg × A k , → X alg , ( n ) , introduced in Definition 3.33. By construction, for each n ≥
0, we have structural morphisms X ( n ) → Y × A k . We shall refer to the sequence X × A k := X (0) , → X (1) , → · · · , → X ( n ) , → · · · → Y × A k , as the non-archimedean Hodge filtration associated to the morphism f .Putting together the above results we can easily deduce: Corollary 3.37.
There exists a natural morphism colim n ≥ X ( n ) → D an X/Y , which is furthermore an equivalence in the ∞ -category AnFMP X × A k //Y × A k .Proof. Since the natural functorAnFMP X × A k //Y × A k → (dAnSt k ) X × A k //Y × A k , commutes with filtered colimits, we are reduced to prove the statement of the Corollary in the ∞ -category(dAnSt k ) X × A k //Y × A k . The result is now an immediate consequence of Proposition 3.24 and the fact thatrelative analytification commutes with filtered colimits (as it is defined as left Kan extension). (cid:3) Corollary 3.38.
Let f : X → Y be a closed immersion of derived k -affinoid spaces. Denote by A := Γ( X, O alg X ) and B := Γ( Y, O alg Y ) and I := fib( B → A ) . Then the Hodge filtration on D an X/Y induces a natural filtration on B ∧ I , { Fil n, an H } n ≥ , together with natural equivalences B/ Fil nH ’ Sym ≤ n, an ( L an X/Y [ − , in the ∞ -category CAlg k .Proof. The claim of the proof follows from Proposition 3.8 (2) combined with Noetherian approximation andthe fact that the relative analytification functor sends the relative cotangent complex L X/Y to L an X/Y , c.f. [PY17,Theorem 5.21]. (cid:3)
We now globalize the Hodge filtration to geometric derived k -analytic stacks: Construction . Let f : X → Y be a morphism of locally geometric derived k -analytic stacks. Consider theformal completion X → Y ∧ X → Y, of the morphism f . Let g : U → Y ∧ X , be a morphism in the ∞ -category dAnSt k , such that U ∈ dAfd k . Form the pullback diagram X U UX Y ∧ Xf in dAnSt k . Let V → U be a morphism of derived k -affinoid spaces. Consider the pullback diagram X V X U V U, in the ∞ -category dAfd k . By naturality of the Hodge filtration we deduce that, for each n ≥
0, we have naturalcommutative diagrams X ( n ) V X ( n ) U V U, (3.10)of the n -th pieces of the Hodge filtrations on D X U /U and D X V /V , respectively. Lemma 3.40.
The commutative square in (3.10) is a pullback square in the ∞ -category dAfd k .Proof. As in the proof of Proposition 3.35, we can reduce the statement to the case where X U → U is anil-embedding in the ∞ -category dAfd k in which case so it is X V → V . Thanks to Lemma 3.15 it suffices toprove that for each n ≥
0, the induced diagram X alg , ( n ) V X alg , ( n ) U V alg U alg , is a pullback square in dAff k . By a standard argument of Noetherian approximation, we might assume that U := U alg and V := V alg are both affine derived schemes almost of finite presentation. We observe that foreach n ≥
0, the morphisms g n : X ( n ) V → X ( n ) U , are left-lax A k -equivariant. For this reason, they induce morphisms g IndCoh n, ∗ ( T X ( n ) V /V )) → T X ( n ) U /U , in the ∞ -category IndCoh( X ( n ) V ) fil . In particular, we have an induced morphism at the n -th piece of thefiltration g IndCoh n, ∗ (cid:0) Sym n +1 ( T X ( n ) U /U [ − (cid:1) → Sym n +1 (cid:0) T X ( n ) V /V [ − (cid:1) [1] , (c.f. [GR17b, Theorem 9.5.1.3]). The result is now a direct consequence of Proposition 3.4 (3) combined with[PY17, Proposition 5.12] and [Lur18, Proposition 2.5.4.5]. (cid:3) Construction . Let f : X → Y be a locally geometric derived k -analytic stack. Denote by ( Y ∧ X ) afd the ∞ -category of morphisms in dAnSt k U → Y ∧ X , PREADING OUT THE HODGE FILTRATION IN NON-ARCHIMEDEAN GEOMETRY 61 where U is derived k -affinoid. Consider the relative analytification functorlim U ∈ ( Y ∧ X ) afd ( − ) an U : lim U ∈ ( Y ∧ X ) afd (dSt k ) /U alg × A k → lim U ∈ ( Y ∧ X ) afd (dAnSt k ) /U × A k ’ (dAnSt k ) /Y ∧ X × A k . Thanks to Lemma 3.40, for each n ≥
0, the algebraic Hodge filtration defines a well defined object in the limit { X alg , ( n ) U } U ∈ ( Y ∧ X ) afd ∈ lim n ≥ (dSt k ) /U alg . Therefore, after taking its relative analytification we produce well defined objects X ( n ) ∈ (dAnSt k ) Y ∧ X × A k , which we shall refer to as the Hodge filtration associated to the morphism f . Moreover, it follows essentially byconstruction that the latter restricts to the usual Hodge filtration, for each g : U → Y ∧ X , in ( Y ∧ X ) afd . Moreover, by construction we have a natural morphismcolim n ≥ X ( n ) → D an X/Y , which is an equivalence in the ∞ -category dAnSt k .We can thus summarize the previous results in the form of a main theorem: Theorem 3.42.
Let f : X → Y be a morphism of locally geometric derived k -analytic stacks. Then thefollowing assertions hold:(1) There exists a deformation to the normal bundle D an X/Y ∈ (dAnSt k ) A k such that its fiber at { } ⊆ A k coincides with the shifted k -analytic tangent bundle T an X/Y [1] ∧ , completed at the zero section s : X → T an X/Y [1] . Moreover, its fiber at λ = 0 coincides with the formalcompletion Y ∧ X of Y at X along f ;(2) The object D an X/Y admits a left-lax equivariant action of the (multiplicative) monoid-object A k in dAnSt k ;(3) There exists a sequence of left-lax A k -equivariant morphisms admitting a deformation theory X × A k = X (0) , → X (1) , → · · · , → X ( n ) , → · · · → Y × A k . Assume further that f : X → Y is a closed immersion of derived k -analytic spaces then, for each n ≥ ,the relative k -analytic cotangent complex L an X ( n ) /Y ∈ Coh + ( X ( n ) ) , is equipped with a (decreasing) filtration such that its n -th piece identifies canonically with Sym n +1 ( L an X/Y [ − .Moreover, for each n ≥ , the morphisms X ( n ) → X ( n +1) are square-zero extensions obtained via acanonical composite L an X ( n ) → L an X ( n ) /Y → Sym n +1 ( L an X/Y [ − , in Coh + ( X ( n ) ) ;(4) We have a natural morphism colim n ≥ X ( n ) → D an X/Y , which is furthermore an equivalence in the ∞ -category dAnSt k ; (5) Assume that f : X → Y is a locally complete intersection morphism of derived k -affinoid spaces with A := Γ( X, O alg X ) and B := Γ( Y, O alg Y ) . Then the corresponding filtration on (derived) global sections ofthe formal completion Y ∧ X (via points (1) and (3)) identifies canonically with the I -adic filtration on B ,where I := fib( B → A ) .Proof. Claim (1) follows immediately from Lemma 3.25. Assertion (2) follows immediately from Proposition 3.24,as in the algebraic case, the deformation D X alg /Y admits a natural left-lax A k -equivariant action. The statementin point (4) is essentially the content of Corollary 3.37 and point (3) follows from the construction of the Hodgefiltration in the non-archimedean setting combined with Noetherian approximation (for the relative cotangentcomplex) and Proposition 3.4 (3). Finally, claim (4) follows readily from Proposition 3.8 and the fact that boththe Hodge filtration and the I -adic filtrations are compatible via relative analytification. (cid:3) References [Ant17] Jorge António. Moduli of p -adic representations of a profinite group. arXiv preprint arXiv:1709.04275 , 2017.[Ant19a] Jorge António. Modèles intègres dérivés et ses applications à l’étude de certains espaces des modules rigides analytiquesdérivés . PhD thesis, Université de Toulouse, Université Toulouse III-Paul Sabatier, 2019.[Ant19b] Jorge António. Moduli of ‘ -adic pro-étale local systems for smooth non-proper schemes. arXiv preprint arXiv:1904.08001 ,2019.[Ant20] Jorge António. Koszul duality for unbounded derived noetherian rings. 2020.[AP19] Jorge António and Mauro Porta. Derived non-archimedean analytic hilbert space. arXiv preprint arXiv:1906.07044 , 2019.[AT19] Piotr Achinger and Mattia Talpo. Betti realization of varieties defined by formal laurent series. arXiv preprintarXiv:1909.02903 , 2019.[Ber93] Vladimir G. Berkovich. Étale cohomology for non-Archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math. ,(78):5–161 (1994), 1993.[Bha12] Bhargav Bhatt. Completions and derived de rham cohomology. https://arxiv.org/abs/1207.6193 , 2012.[Bra18] Arthur-César Le Bras. Overconvergent relative de rham cohomology over the fargues-fontaine curve. arXiv preprintarXiv:1801.00429 , 2018.[BZN12] David Ben-Zvi and David Nadler. Loop spaces and connections.
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