aa r X i v : . [ m a t h . N T ] J u l SIMPLICIAL GALOIS DEFORMATION FUNCTORS
Y. CAI AND J. TILOUINE
Abstract.
In [GV18], the authors showed the importance of studying simplicial generalizations of Galoisdeformation functors. They established a precise link between the simplicial universal deformation ring R prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here wefocus on the algebraic part of their study which we complete in two directions. First, we introduce the notionof simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicialpseudo-characters and that of simplicial Galois representations. Secondly, we define the relative cotangentcomplex of a simplicial deformation functor and, in the ordinary case, we relate it to the relative complex ofordinary Galois cochains. Finally, we recall how the latter can be used to relate the fundamental group of R to the ordinary dual adjoint Selmer group, by a homomorphism already introduced in [GV18] and studied ingreater generality in [TU20]. Contents
1. Introduction 12. Classical and simplicial Galois deformation functors 22.1. Classical deformations 22.2. Simplicial reformulation of classical deformations 42.3. Simplicial reformulation of classical framed deformations 62.4. Derived deformation functors 73. Pseudo-deformation functors 113.1. Classical pseudo-characters and functors on
FFS
Introduction
Let p be an odd prime. Let K be a p -adic field, let O be its valuation ring, ̟ be a uniformizing parameterand k = O / ( ̟ ) be the residue field. Let Γ be a profinite group satisfying (Φ p ) the p -Frattini quotient Γ / Γ p (Γ , Γ) is finite. This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participatingin the program Perfectoid spaces (Code: ICTS/perfectoid2019/09). The authors are partially supported by the ANR grantCoLoSS ANR-19-PRC..
For instance, Γ could be Gal( F S /F ) , the Galois group of the maximal S -ramified extension of a number field F with S finite. Let G be a split connected reductive group scheme over O . Let ρ : Γ → G ( k ) be a continuousGalois representation. Assume it is absolutely G -irreducible, which means its image is not contained in P ( k ) for a proper parabolic subgroup P of G . The goal of this paper is to present and develop some aspects ofthe fundamental work [GV18] and the subsequent papers [TU20] and [Cai20], by putting emphasis on thealgebraic notion of simplicial deformation over simplicial Artin local O -algebras of ρ .In the papers mentioned above, it is assumed that the given residual Galois representation is automorphic: ρ = ρ π for a cohomological cuspidal automorphic representatiton on the dual group of G over a numberfield F ; then the (classical and simplicial) deformation problems considered impose certain local deformationconditions satisfied by ρ at primes above p and at ramification primes for π . The fundamental insight of[GV18] is to relate the corresponding universal simplicial deformation ring to a derived version of the Heckealgebra acting on the graded cohomology of a locally symmetric space. Actually, the main result [GV18,Theorem 14.1] (slightly generalized in [Cai20]) is that after localization at the non Eisenstein maximal ideal m of the Hecke algebra corresponding to ρ , the integral graded cohomology in which π occurs is free over thegraded homotopy ring of the universal simplicial deformation ring (and the degree zero part of this ring isisomorphic to the top degree integral Hecke algebra). This is therefore a result of automorphic nature.Here, on the other hand, we want to focus on the purely algebraic machinery of simplicial deformationsand pseudo-deformations and their (co)tangent complex for a general profinite group Γ satisfying (Φ p ) .In [Laf18, Section 11], V. Lafforgue introduced the notion of a pseudo-character for a split connected re-ductive group G . He proved that this notion coincides with that of G -conjugacy classes of G -valued Galoisrepresentations over an algebraically closed field E . The main ingredient of his proof is a criterion of semisim-plicity for elements in G ( E ) n in terms of closed conjugacy class; it is due to Richardson in characteristiczero. It has been generalized to the case of an algebraically closed field of arbitrary characteristic by [BMR05]replacing semisimplicity by G -complete reducibility (see also [Ser05] and [BHKT19, Theorem 3.4]). Note thatabsolute G -irreducibility implies G -complete reducibility.Using this (and a variant for Artin rings), Boeckle-Khare-Harris-Thorne [BHKT19, Theorem 4.10] proveda generalization of Carayol’s result for any split reductive group G : any pseudo-deformation over G of anabsolutely G -irreducible representation ρ is a G -deformation.In section 3.2.2, we reformulate the theory of [BHKT19, Section 4] in the language of simplicial deformation.Our main results are Theorem 3.16 and Theorem 3.20. In Section 3.3, we propose a generalization of thistheory for derived deformations. Unfortunately, the result in this context is only partial, but still instructive.In Sections 4, after recalling the definition of the tangent and cotangent complexes and its calculation fora Galois deformation functor, we introduce a relative version of the cotangent complex. In order to relate thecotangent complex of the universal simplicial ring R prorepresenting a deformation functor to a Selmer group,we shall take Γ = G F,S for a number field F and for S equal to the set of places above p and ∞ , and we shalldeal with the simplest sort of local conditions, namely unramified outside p and ordinary at each place above p . We show that the cotangent complex L R / O ⊗ R T is related to the ordinary Galois cochain complex. Notethat here the base T is arbitrary, whereas in [GV18] and [Cai20] it was mostly the case T = k .Finally, in Section 5, we recall how this is used to define a homomorphism, first constructed in [GV18,Lemma 15.1] and generalized and studied in [TU20], which relates the fundamental group of the simplicialordinary universal deformation ring and the ordinary dual adjoint Selmer group.This work started during the conference on p -adic automorphic forms and Perfectoids held in Bangalore inSeptember 2019. The authors greatly appreciated the excellent working atmosphere during their stay.2. Classical and simplicial Galois deformation functors
Classical deformations.
Let Γ be a profinite group which satisfies (Φ p ) . When necessary, we view Γ asprojective limit of finite groups Γ i . Let Art O be the category of Artinian local O -algebras with residue field k . Recall that the framed deformation functor D (cid:3) : Art O → Sets of ρ is defined by associating A ∈ Art O to IMPLICIAL GALOIS DEFORMATION FUNCTORS 3 the set of continuous liftings ρ : Γ → G ( A ) which make the following diagram commute: Γ ρ / / ¯ ρ ! ! ❈❈❈❈❈❈❈❈❈ G ( A ) (cid:15) (cid:15) G ( k ) Let Z be the center of G over O . We assume throughout it is a smooth group scheme over O . Let b G ( A ) =Ker( G ( A ) → G ( k )) , resp. b Z ( A ) = Ker( Z ( A ) → Z ( k )) . Let g = Lie( G/ O ) , resp. z = Lie ( Z/ O ) be the O -Lie algebra of G , resp. Z , and let g k = g ⊗ O k , resp. z k = z ⊗ O k . The universal deformation functor D = Def ρ : Art O → Sets is defined by associating A ∈ Art O to the set of b G ( A ) -conjugacy classes of D (cid:3) ( A ) .As an application of Schlessinger’s criterion (see [Sch68, Theorem 2.11]), the functor D (cid:3) is pro-representable,and when ¯ ρ satisfies H (Γ , g k ) = z k , the functor D is pro-representable (see [Til96, Theorem 3.3]).We shall consider (nearly) ordinary deformations. In this case, we always suppose Γ = G F,S , where F is anumber field and S = S p ∪ S ∞ is the set of places above p and ∞ . Note that Γ is profinite and satisfies (Φ p ) .For any v ∈ S p , let Γ v = Gal( F v /F v ) . Let B = T N ⊂ G be a Borel subgroup scheme ( T is a maximal splittorus and N is the unipotent radical of B ); all these groups are defined over O . Let Φ be the root systemassociated to ( G, T ) and Φ + the subset of positive roots associated to ( G, B, T ) . Assume that for any place v ∈ S p , we have ( Ord v ) there exists g v ∈ G ( k ) such that ρ | Γ v takes values in g − v · B ( k ) · g v .Let χ v : Γ v → T ( k ) be the reduction modulo N ( k ) of g v · ρ | Γ v · g − v . Let ω : Γ v → k × be the mod. p cyclotomic character. We shall need the following conditions for v ∈ S p : ( Reg v ) for any α ∈ Φ + , α ◦ χ v = 1 , and ( Reg ∗ v ) for any α ∈ Φ + , α ◦ χ v = ω .We can define the subfunctor D (cid:3) , n . o ⊂ D (cid:3) of nearly ordinary liftings by the condition that ρ ∈ D (cid:3) , n . o if andonly if for any place v ∈ S p there exists g v ∈ G ( A ) which lifts ¯ g v such that ρ | Γ v takes values in g − v · B ( k ) · g v .Note that this implies that the homomorphism χ ρ,v : Γ v → T ( A ) given by g v · ρ | Γ v · g − v lifts χ v .Similarly, we define the subfunctor D n . o ⊂ D of nearly ordinary deformations by D n . o ( A ) = D (cid:3) , n . o ( A ) / b G ( A ) .Recall [Til96, Proposition 6.2]: Proposition 2.1.
Assume that H (Γ , g k ) = z k and that ( Ord v ) and ( Reg v ) hold for all places v ∈ S p . Then D n . o (and D (cid:3) , n . o ) is pro-representable, say by the complete noetherian local O -algebra R n . o . Note that the condition ( Reg ∗ v ) will occur later in the study of the cotangent complex in terms of the (nearly)ordinary Selmer complex. As noted in [Til96, Chapter 8], the morphism of functors D n . o → Q v ∈ S p Def χ v givenby [ ρ ] ( χ ρ,v ) v ∈ S p provides a structure of Λ -algebra on R n . o for an Iwasawa algebra Λ called the Hida-Iwasawaalgebra. Remark . A lifting ρ : Γ → G ( A ) of ¯ ρ is called ordinary of weight µ if for any v ∈ S p , after conjugationby g v , the cocharacter ρ | Iv : I v → T ( A ) = B ( A ) /N ( A ) is given (via the Artin reciprocity map rec v ) by µ ◦ rec − v : I v → O × F v → T ( A ) .If we assume that ¯ ρ admits a lifting ρ : Γ → G ( O ) which is ordinary of weight µ , we can also consider theweight µ ordinary deformation problem, defined as the subfunctor D n . o ,µ ⊂ D n . o where we impose the extracondition to [ ρ ] that for any v ∈ S p , after conjugation by some g v , ρ | Iv : I v → T ( A ) = B ( A ) /N ( A ) is given (viathe Artin reciprocity map rec v ) by µ ◦ rec − v : I v → O × F v → T ( O ) → T ( A ) . This problem is prorerepresentableas well, say by R n . o µ . The difference is that R n . o has a natural structure of algebra over an Iwasawa algebra,while, if ρ is automorphic, R n . o µ is often proven to be a finite O -algebra (see [Wi95] or [Ge19] for instance).These functors have natural simplicial interpretations. Y. CAI AND J. TILOUINE
Simplicial reformulation of classical deformations.
In this section, we’ll try to introduce the basicnotions of simplicial homotopy theory and proceed at the same time to give a simplicial definition of thedeformation functor of ¯ ρ .Recall that a groupoid is a category such that all homomorphisms between two objects are isomorphisms.Let Gpd be the category of small groupoids. We have a functor Gp → Gpd from the category Gp of groupsto Gpd sending a group G to the groupoid with one object • and such that End ( • ) = G .A model category is a category with three classes of morphisms called weak equivalences, cofibrations andfibrations, satisfying five axioms, see [Hir03, Definition 7.1.3]. The category of groups is not a model category.But it is known (see [Str00, Theorem 6.7]) that the category of groupoids Gpd is a model category, where amorphism f : G → H is(1) a weak equivalence if it is an equivalence of categories;(2) a cofibration if it is injective on objects;(3) a fibration if for all a ∈ G , b ∈ H and h : f ( a ) → b there exists g : a → a ′ such that f ( a ′ ) = b and f ( g ) = h .If C is a model category, its homotopy category Ho( C ) is the localization of C at weak equivalences. It comeswith a functor C →
Ho( C ) universal for the property of sending weak equivalences to isomorphisms.In Gpd , the empty groupoid is the initial object and the unit groupoid consisting in a unique object witha unique isomorphism is the final object. In a model category, a fibration, resp. cofibration, over the finalobject, resp. from the initial object, is called a fibrant, resp. cofibrant object. Note that every object of
Gpd is both cofibrant and fibrant, and the homotopy category
Ho(
Gpd ) is the quotient category of Gpd moddingout natural isomorphisms. If we regard a group G as a one point groupoid, the functor Gp → Ho(
Gpd ) soobtained has the effect of moding out conjugations, so, for any finite group Γ i , we have Hom Gp (Γ i , G ( A )) /G ad ( A ) ∼ = Hom
Ho(
Gpd ) (Γ i , G ( A )) . To construct the deformation functor, we first need to recall the construction of the classifying simplicialset BG associated to a groupoid G .Let ∆ be the category whose objects are sets [ n ] = { , . . . , n } and morphisms are non-decreasing maps. Itis called the cosimplicial indexing category (see [Hir03, Definition 15.1.8]). Given a category C , the category s C of simplicial objects of C is the category of contravariant functors F : ∆ → C . In particular, s Sets is thecategory of simplicial sets. For any n ≥ , let ∆ [ n ] be the simplicial set [ k ] Hom ∆ ([ k ] , [ n ]) . Note that the category s Sets admits enriched homomorphisms: if
X, Y are two simplicial sets, there is anatural simplicial set sHom ( X, Y ) whose degree zero term is Hom s Sets ( X, Y ) . Actually, sHom ( X, Y ) n = Hom s Sets ( X × ∆ [ n ] , Y ) . For X ∈ s Sets , the morphism ( d , d ) : X → X × X generates an equivalence relation e X . The zerothhomotopy set π X is defined as the quotient set X / e X . Let X be fibrant and let x ∈ X ; one can definefor i ≥ , the i -th homotopy set π i ( X, x ) as the quotient of the set of pointed morphisms Hom s Sets ∗ ( ∆ [ n ] , X ) (morphisms sending the boundary ∂ ∆ [ n ] to x ) by the homotopy relation (see [Weib94, Section 8.3]). Then π i ( X, x ) is naturally a group which is abelian when i ≥ (see [GJ09, Theorem I.7.2]).For X ∈ s Sets , let ∆ X be the category whose objects are pairs ( n, σ ) where n ≥ and σ : ∆ [ n ] → X is amorphism of simplicial sets, and morphisms ( n, σ ) → ( m, τ ) are given by a non-decreasing map ϕ : [ n ] → [ m ] such that σ = τ ◦ ϕ . The category ∆ X is called the category of simplices of X (see [Hir03, Definition 15.1.16]).The following lemma is well-known: Lemma 2.3.
Suppose C is a category admitting colimits; let F : ∆ → C be a covariant functor. Let F ∗ : C → s Sets be the functor which sends A ∈ C to the simplicial set X = ( X n ) n ≥ given by X n = Hom C ( F ([ n ]) , A ) at IMPLICIAL GALOIS DEFORMATION FUNCTORS 5 n -th simplicial degree, and let F ∗ : s Sets → C be the functor which sends X ∈ s Sets to lim −→ ( n,σ ) ∈ ∆ X F ( σ ) . Then F ∗ is left adjoint to F ∗ .Proof. It’s clear that F ∗ is well-defined, and F ∗ is well-defined since every simplicial set morphism f : X → Y induces a functor ∆ X → ∆ Y . For X ∈ s Sets and A ∈ C , we have Hom C ( F ∗ ( X ) , A ) ∼ = lim ←− (∆[ n ] → X ) ∈ ( ∆ X ) op Hom C ( F ([ n ]) , A ) ∼ = lim ←− (∆[ n ] → X ) ∈ ( ∆ X ) op Hom s Sets (∆[ n ] , F ∗ ( A )) ∼ = Hom s Sets ( lim −→ (∆[ n ] → X ) ∈ ∆ X ∆[ n ] , F ∗ ( A )) ∼ = Hom s Sets ( X, F ∗ ( A )) , where the last equation follows from [Hir03, Proposition 15.1.20]. So F ∗ is left adjoint to F ∗ . (cid:3) Example 2.4. (1) Let ∆ → Cat be the functor defined by regarding [ n ] as a posetal category: its objectsare , , . . . n and Hom [ n ] ( k, ℓ ) has at most one element, and is non-empty if and only if k ≤ ℓ . We write P : s Sets → Cat and B : Cat → s Sets for the associate left adjoint functor and right adjoint functorrespectively. The functor B is called the nerve functor. The simplicial set B C = ( X n ) is defined bysets X n ⊂ Ob( C ) [ n ] of ( n + 1) -tuples ( C , . . . , C n ) of objects of C with morphisms C k → C ℓ when k ≤ ℓ ,which are compatible when n varies; it is a fibrant simplicial set if and only if C ∈
Gpd (see [GJ09,Lemma I.3.5]). In a word, for B C to be fibrant, it must have the extension property with respectto inclusions of horns in ∆[ n ] ( ∀ n ≥ ). For n = 2 , it amounts to saying that all homomorphismsin C are invertible; for n > , the extension condition is automatic (details in the reference above).For C ∈
Cat , we have
P B
C ∼ = C , so Hom
Cat ( C , D ) ∼ = Hom s Sets ( B C , B D ) ( ∀C , D ∈
Cat ). Note that B ( C × [1]) ∼ = B C × ∆[1] (product is the degreewise product); in consequence, when
C ∈
Cat and
D ∈
Gpd , two functors f, g : C → D are naturally isomorphic if and only if Bf and Bg are homotopic.(2) As a corollary of (1), we have Hom
Gpd ( GP X, H ) ∼ = Hom s Sets ( X, BH ) for X ∈ s Sets and H ∈ Gpd ,where
GP X is the free groupoid associated to
P X . We remark that
GP X and π | X | (the fundamentalgroupoid of the geometric realization) are isomorphic in Ho(
Gpd ) (see [GJ09, Theorem III.1.1]).Recall that a functor between two model categories is called right Quillen if it preserves fibrations andtrivial fibrations (fibrations which are weak equivalences). Lemma 2.5.
The nerve functor B : Gpd → s Sets is fully faithful and takes fibrant values (Kan-valued).Moreover, it is right Quillen.Proof.
For the first statement, we know by Example 2.2 that:
Hom
Cat ( C , D ) ∼ = Hom s Sets ( B C , B D ) ( ∀C , D ∈
Cat , hence the fully faithfulness. Moreover B C is fibrant for C a groupoid.For the second statement, note that B obviously preserves weak equivalences; moreover, by definition, Bf : BG → BH is a fibration if and only if it has the right lifting property with respect to inclusions of hornsin ∆[ n ] , ∀ n ≥ (see [GJ09, page 10]). For n = 1 this means exactly that f is a fibration, while for n ≥ it’sautomatic (see the proof of [GJ09, Lemma I.3.5]). (cid:3) Let A ∈ Art O . Consider the group G ( A ) of A -points of our reductive group scheme G . Passing to homotopycategories, we get the isomorphism Hom
Ho(
Gpd ) (Γ i , G ( A )) ∼ = Hom
Ho( s Sets ) ( B Γ i , BG ( A )) ∼ = π sHom s Sets ( B Γ i , BG ( A )) . Y. CAI AND J. TILOUINE
Let X = ( B Γ i ) i be the pro-simplicial set associated to the profinite group Γ . We define Hom s Sets ( X, − ) = lim −→ i Hom s Sets ( B Γ i , − ) . Then the Galois representation ¯ ρ : Γ → G ( k ) gives rise to an element of Hom s Sets ( X, BG ( k )) , which wealso denote by ¯ ρ . In order to take into account the deformations of ¯ ρ , we introduce the overcategory M = s Sets / BG ( k ) of pairs ( Y, π ) where Y is a simplicial set and π : Y → BG ( k ) is a morphism of simplicialsets. The category M has a natural simplicial model category structure: the cofibrations, fibrations, weakequivalences and tensor products are those of s Sets (see [GJ09, Lemma II.2.4] for the only nontrivial partof the statement). When we consider X ∈ M , we specify the morphism ¯ ρ : X → BG ( k ) ; similarly, when weconsider BG ( A ) ∈ M for A ∈ Art O , we specify the natural projection BG ( A ) → BG ( k ) . For X, Y ∈ M , wecan define an object of M of enriched homomorphisms sHom M ( X, Y ) for which sHom M ( X, Y ) n consists inthe morphisms X × ∆ [ n ] → Y compatible to the projections to BG ( k ) . Since BG ( A ) → BG ( k ) is a fibration, BG ( A ) ∈ M is fibrant. Similar to the discussion of the preceding paragraph, we have(1) D ( A ) ∼ = Hom
Ho( M ) ( X, BG ( A )) ∼ = π sHom M ( X, BG ( A )) for A ∈ Art O . Note that sHom M ( X, BG ( A )) is the fiber over ¯ ρ of the fibration map sHom s Sets ( X, BG ( A )) → sHom s Sets ( X, BG ( k )) , so it actually calculates the homotopy fiber of π ¯ ρ (see [Hir03, Theorem 13.1.13 and Proposition 13.4.6]).When Γ = G F,S , S = S p ∪ S ∞ and ¯ ρ satisfies ( Ord v ) for v ∈ S p , we reformulate the definition of the nearlyordinary deformation subfunctor D n . o ⊂ D as follows. For each v ∈ S p , we form Γ v = lim ←− i Γ i,v where Γ v → Γ induces morphisms Γ i,v → Γ i of finite groups. Let X v = ( B Γ i,v ) i be the pro-simplicial set associated. For thefixed Borel subgroup B of G , we have a natural cofibration BB ( A ) ⊂ BG ( A ) . Recall that ¯ g v · ¯ ρ | Γ v · ¯ g − v takesvalues in B ( k ) . Let D v ( A ) be π of the fiber over ¯ ρ | Γ v of the fibration map sHom s Sets ( X v , BG ( A )) → sHom s Sets ( X v , BG ( k )) , and let D n . o v ( A ) be π of the fiber over ¯ g v · ¯ ρ | Γ v · ¯ g − v of the fibration map sHom s Sets ( X v , BB ( A )) → sHom s Sets ( X v , BB ( k )) . Then there is a natural functorial inclusion i v of D n . o v ( A ) into D v ( A ) . Let D loc ( A ) = Q v ∈ S p D v ( A ) and D n . oloc ( A ) = Q v ∈ S p D n . o v ( A ) . There is a natural functorial map D ( A ) → D loc ( A ) , resp. D n . oloc ( A ) → D loc ( A ) ,induced by ρ ( ρ | Γ v ) v ∈ S p , resp. by Q v ∈ S p i v .We define D n . o ( A ) as the fiber product D n . o ( A ) = D ( A ) × D loc ( A ) D n . oloc ( A ) . Lemma 2.6.
Suppose ( Reg v ) holds for each place v ∈ S p . Then the functor D n . o is isomorphic to the classicalnearly ordinary deformation functor.Proof. It follows easily from what precedes. See [Cai20] or [TU20]. (cid:3)
Simplicial reformulation of classical framed deformations.
Let
Gpd ∗ and s Sets ∗ be the modelcategories of based groupoids and based simplicial sets (in other words, under categories ∗ \ Gpd and ∗ \ s Sets )respectively. Then we have
Hom Gp (Γ i , G ( A )) ∼ = Hom
Ho(
Gpd ∗ ) (Γ i , G ( A )) . Let M ∗ be the over and under category ∗ \ s Sets / BG ( k ) . Note that X and BG ( A ) for A ∈ Alg O are naturallyobjects of M ∗ . Proceeding as the unframed case, we see that(2) D (cid:3) ( A ) ∼ = Hom
Ho( M ∗ ) ( X, BG ( A )) ∼ = π sHom M ∗ ( X, BG ( A )) . We remark that sHom M ∗ ( X, BG ( A )) is weakly equivalent to hofib ∗ ( sHom M ( X, BG ( A )) → sHom M ( ∗ , BG ( A ))) ,since sHom M ( X, BG ( A )) → sHom M ( ∗ , BG ( A )) is a fibration. IMPLICIAL GALOIS DEFORMATION FUNCTORS 7
Derived deformation functors.
We have defined the functor sHom M ( X, BG ( − )) from Art O to s Sets .Our next goal is to extend this functor to simplicial Artinian O -algebras over k , which we define below.Let s CR be the category of simplicial commutative rings (these are simplicial sets which are rings in alldegrees and for which all face and degeneracy maps are ring homomorphisms). A usual commutative ring A can be regarded as an element of s CR , which consists of A on each simplicial degree with identity face anddegeneracy maps. In this way we regard O and k as objects of s CR . With the natural reduction map O → k ,the over and under category O \ s CR / k has a simplicial model category stucture, such that the cofibrations,fibrations and weak equivalences are those of s CR , and the tensor product of A ∈ O \ s CR / k and K ∈ s Sets is the pushout of
O ← O ⊗ K → A ⊗ K . Note that degreewise surjective morphisms A → B are fibrations.Since s CR is cofibrantly generated, any A ∈ O \ s CR admits a functorial cofibrant replacement c ( A ) : O ֒ → c ( A ) ∼ ։ A. Concretely, for any n ≥ the O -algebra c ( A ) n is a suitable polynomial O -algebra mapping surjectively onto A n . The key property of the cofibrant replacement is that- c ( A ) is a cofibrant object and- c ( A ) → A is a trivial fibration (a fibration which is a weak equivalence).Note that the functor B sHom ( c ( A ) , B ) commutes to weak equivalence (this is called homotopy invari-ance), while it is not necessarily the case of the functor B sHom ( A, B ) .For A ∈ O \ s CR , for any i ≥ , π i A is a commutative group and L i π i A is naturally a graded O -algebra,hence a π A -algebra (see [Gil13, Lemma 8.3.2]). Definition 2.7.
The simplicial Artinian O -algebras over k , which we denote by O \ s Art / k , is the full sub-category of O \ s CR / k consisting of objects A ∈ O \ s CR / k such that:(1) π A is Artinian local in the usual sense.(2) π ∗ A = ⊕ i ≥ π i A is finitely generated as a module over π A .Note that O \ s Art / k is not a model category, and cofibrations, fibrations and weak equivalences in O \ s Art / k are used to indicate those in O \ s CR / k . Nevertheless, O \ s Art / k is closed under weak equivalences since thedefinition only involves homotopy groups. We also remark that every A ∈ O \ s Art / k is fibrant since A → k is degreewise surjective.We define O N • G ∈ Alg ∆ O (i.e., a functor ∆ → Alg O , also called a cosimplicial object in Alg O ) as follows: incodegree p we have O N p G = O ⊗ pG , and the coface and codegeneracy maps are induced from the comultiplicationand the coidentity of the Hopf algebra O G respectively. Then for A ∈ Alg O , the nerve BG ( A ) is nothing but Hom
Alg O ( O N • G , A ) , with face and degeneracy maps induced by the coface and codegeneracy maps in O N • G .When A ∈ O \ s CR , the naïve analogy is the diagonal of the bisimplicial set ([ p ] , [ q ]) Hom
Alg O ( O N p G , A q ) (recall that the diagonal of a bisimplicial set is a simplicial set model for its geometric realization). However,we need to make some modifications using cofibrant replacements to ensure the homotopy invariance. Definition 2.8. (1) For A ∈ O \ s CR , we define Bi( A ) to be the bisimplicial set ([ p ] , [ q ]) Hom O \ s CR ( c ( O N p G ) , A ∆[ q ] ) , with face and degeneracy maps induced by the coface and codegeneracy maps in O N • G and the faceand degeneracy maps in A ∆[ • ] .(2) The diagonal of Bi( A ) , which is denoted by diag Bi( A ) , is the simplicial set induced from the diagonalembedding ∆ op → ∆ op × ∆ op Bi( A ) −−−→ Sets .When A is an O -algebra regarded as a constant object in O \ s CR , we have Bi( A ) p,q = Hom O \ s CR ( c ( O N p G ) , A ∆[ q ] ) ∼ = Hom
Alg O ( O N p G , A ) , Y. CAI AND J. TILOUINE where the latter isomorphism is because the constant embedding functor is right adjoint to π : O \ s CR → Alg O . Hence Bi( A ) is just a disjoint union of copies of BG ( A ) in index q . In particular, for A ∈ O \ s Art / k there is a natural map Bi( A ) • ,q → BG ( k ) for each q ≥ , so we may regard Bi( A ) ∈ M ∆ op via the association [ q ] Bi( A ) • ,q (recall that M is the overcategory s Sets / BG ( k ) ), and diag Bi( A ) is an object of M ). Recallthat any morphism X → Y in s Sets admits a functorial factorisation X ∼ ֒ → e X ։ Y into a trivial cofibration and a fibration. Definition 2.9.
For A ∈ O \ s Art / k , the simplicial set B G ( A ) is defined by the functorial trivial cofibration-fibration factorization diag Bi( A ) ∼ ֒ → B G ( A ) ։ BG ( k ) .It’s clear that B G : O \ s Art / k → M defines a functor. If A ∈ Art O is regarded as a constant simplicialring, then diag Bi( A ) = BG ( A ) ։ BG ( k ) is a fibration, so BG ( A ) is a strong deformation retract of B G ( A ) in M (see [Hir03, Definition 7.6.10]). In particular, these two are indistinguishable in our applications. Remark . Our B G ( A ) is weakly equivalent to the simplicial set Ex ∞ diag Bi( A ) which is the definitionchosen in [GV18, Definition 5.1]. There is a slight difference: we want to emphasize the fibration B G ( A ) ։ BG ( k ) , so that it’s more convenient to handle the homotopy pullbacks.As mentioned above, the reason for taking cofibrant replacements is: Lemma 2.11. If A → B is a weak equivalence, then so is B G ( A ) → B G ( B ) .Proof. If A → B is a weak equivalence, then sHom O \ s CR ( c ( O N p G ) , A ) → sHom O \ s CR ( c ( O N p G ) , B ) is a weakequivalence for each p ≥ , so is diag Bi( A ) → diag Bi( B ) (see [Hir03, Theorem 15.11.11]), and so is B G ( A ) →B G ( B ) . (cid:3) Definition 2.12. (1) The derived universal deformation functor s D : O \ s Art / k → s Sets is defined by s D ( A ) = sHom M ( X, B G ( A )) . (2) The derived universal framed deformation functor s D (cid:3) : O \ s Art / k → s Sets is defined by s D (cid:3) ( A ) = hofib ∗ ( s D ( A ) → sHom M ( ∗ , B G ( A ))) . Remark . In [GV18, Definition 5.4], the derived universal deformation functor is defined by s D ( A ) = hofib ¯ ρ ( sHom s Sets ( X, Ex ∞ diag Bi( A )) → sHom s Sets ( X, BG ( k ))) . Since Ex ∞ diag Bi( A )) and B G ( A ) are weakly equivalent fibrant simplicial sets, sHom s Sets ( X, Ex ∞ diag Bi( A )) is weakly equivalent to sHom s Sets ( X, B G ( A )) . But sHom s Sets ( X, B G ( A )) → sHom s Sets ( X, BG ( k )) is a fibra-tion, so sHom M ( X, B G ( A )) is weakly equivalent to the homotopy fiber.When Γ = G F,S , S = S p ∪ S ∞ and ¯ ρ satisfies ( Ord v ) for v ∈ S p , we can define for each v ∈ S p a functor s D v : O \ s Art / k → s Sets as A sHom s Sets / BG ( k ) ( X v , B G ( A )) , and a functor s D n . o v : O \ s Art / k → s Sets as A sHom s Sets / BB ( k ) ( X v , B B ( A )) . Let s D loc = Q v ∈ S p s D v and let s D n . oloc = Q v ∈ S p s D n . o v . Define s D n . o as thehomotopy fiber product s D n . o = s D × hs D loc s D n . oloc . Definition 2.14.
Let F : O \ s Art / k → s Sets be a functor. We say F is formally cohesive if it satisfies thefollowing conditions:(1) F is homotopy invariant ( i.e. preserves weak equivalences). IMPLICIAL GALOIS DEFORMATION FUNCTORS 9 (2) Suppose that A / / (cid:15) (cid:15) B (cid:15) (cid:15) C / / D is a homotopy pullback square with at least one of B → D and C → D degreewise surjective, then F ( A ) / / (cid:15) (cid:15) F ( B ) (cid:15) (cid:15) F ( C ) / / F ( D ) is a homotopy pullback square.(3) F ( k ) is contractible.We summarize our preceding discussions: Proposition 2.15.
The functors s D , s D (cid:3) , s D ? v (here ? = ∅ or n . o ) and s D n . o are all formally cohesive.Proof. We first verify three conditions in the above definition for s D :(1) If A → B is a weak equivalence, then B G ( A ) → B G ( B ) is a weak equivalence between fibrant objectsin M , so sHom M ( X, B G ( A )) → sHom M ( X, B G ( B )) is also a weak equivalence.(2) First we show that B G ( A ) / / (cid:15) (cid:15) B G ( B ) (cid:15) (cid:15) B G ( C ) / / B G ( D ) is a homotopy pullback square in M . Note that regarding the above diagram as a diagram in s Sets doesn’t affect the homotopy pullback nature. By [GV18, Lemma 4.31], it suffices to check:(a) the functor Ω B G : O \ s Art / k → s Sets preserves homotopy pullbacks, and(b) π B G ( C ) → π B G ( D ) is surjective whenever C → D is degreewise surjective.Part (a) follows from [GV18, Lemma 5.2], and part (b) follows from [GV18, Corollary 5.3].Since small filtered colimits of simplicial sets preserve homotopy pullbacks, we may suppose thepro-object X lies in M . Then sHom M ( X, − ) : M → s Sets is a right Quillen functor, hence its rightderived functor commutes with homotopy pullbacks in the homotopy categories. But we are dealingwith fibrant objects, so in the homotopy category sHom M ( X, − ) is isomorphic to its right derivedfunctor. The conclusion follows.(3) It’s clear that s D ( k ) is contractible.The same argument applies for A → sHom M ( ∗ , B G ( A )) . So s D (cid:3) is formally cohesive because it is thehomotopy pullback of formally cohesive functors.In the nearly ordinary case, we may replace X by X v and replace G by B and the same argument applies.Hence s D ? v ( ? = ∅ or n . o ) is formally cohesive. Since s D n . o is the homotopy limits of formally cohesivefunctors, it is also formally cohesive. (cid:3) Modifying the center.
None of these functors cannot be pro-representable unless G is of adjoint type.If G has a non trivial center Z , we need a variant s D Z , resp. s D n . o Z , of the functor s D , resp. of s D n . o , inorder to allow pro-representability. For this modification, we follow [GV18, Section 5.4]. For a classical ring A ∈ Art , we have a short exact sequence → Z ( A ) → G ( A ) → P G ( A ) → . It yields a fibration sequence BG ( A ) → BP G ( A ) → B Z ( A ) . Indeed, given a simplicial group H and asimplicial sets X with a left H -action, we can form the bar construction N • ( ∗ , H, X ) at each simplicial degree(see [Gil13, Example 3.2.4]), which gives the bisimplicial set ([ p ] , [ q ]) H qp × X p =: N q ( ∗ , H p , X p ) . Considerthe action Z ( A ) × G ( A ) → G ( A ) , and the corresponding simplicial action N p Z ( A ) × N p G ( A ) → N p G ( A ) (note that N • Z ( A ) is a simplicial group because Z ( A ) is abelian). We identify for each p ≥ , BG ( A ) p = N p ( ∗ , ∗ , N p G ( A )) ,BP G ( A ) p = N p ( ∗ , N p Z ( A ) , N p G ( A )) , and we put B Z ( A ) p = N p ( ∗ , N p Z ( A ) , ∗ ) (with diagonal face and degeneracy maps). The desired fibration is given by the canonical morphisms ofsimplicial sets which in degree p are: N p ( ∗ , ∗ , N p G ( A )) → N p ( ∗ , N p Z ( A ) , N p G ( A )) → N p ( ∗ , N p Z ( A ) , ∗ ) . Let us generalize this to A ∈ O \ s Art / k . For this, we note first that BP G ( A ) can also be definedas the functorial fibrant replacement of diag( N ) where N is the trisimplicial set associated to ( p, q, r ) N q ( ∗ , N p Z ( A r ) , N p ( G ( A r )) (replacing O NpG ( A r ) by its functorial cofibrant replacement as above).Then, we define B Z ( A ) as the functorial fibrant replacement of diag( N ′ ) where N ′ is the trisimplicialset associated to ( p, q, r ) N q ( ∗ , N p Z ( A r ) , ∗ ) (replacing O NpG ( A r ) by its functorial cofibrant replacement asabove). The obvious system of maps N q ( ∗ , N p Z ( A r ) , N p G ( A r )) → N q ( ∗ , N p Z ( A r ) , ∗ ) gives the desired map B P G ( A ) → B Z ( A ) . The functor s D Z : O \ s Art / k → s Sets is defined by the homotopy pullback square (here for simplicity weuse M , but the base maps are those induced from B G ( k ) → B P G ( k ) → B Z ( k ) ) s D Z ( A ) / / (cid:15) (cid:15) sHom M ( ∗ , B Z ( A )) (cid:15) (cid:15) sHom M ( X, B P G ( A )) / / Hom M ( X, B Z ( A )) Then s D Z is formally cohesive becasue it is the homotopy pullback of formally cohesive functors. Observethat s D Z and s D coincide when Z is trivial. Remark . (1) We’ll see later that s D Z is pro-representable, under the assumption H (Γ , g k ) = z k .(2) In the nearly ordinary case, one defines similarly s D loc ,Z = Q v ∈ S p s D v,Z and s D n . oloc ,Z = Q v ∈ S p s D n . o v,Z .Note that the construction for s D Z is functorial in X and G , we can form the homotopy pullback s D n . o Z = s D Z × hs D loc ,Z s D n . oloc ,Z . All these functors are formally cohesive. We’ll see later that s D n . o Z is pro-representable, under theassumption H (Γ , g k ) = z k . Proposition 2.17.
When A is homotopy discrete, we have π s D Z ( A ) ∼ = D ( π A ) and π s D ? v,Z ( A ) ∼ = D ? v ( π A ) (here ? = ∅ or n . o ). If in addition ( Reg v ) holds for each v ∈ S p , then π s D n . o Z ( A ) ∼ = D n . o ( π A ) .Proof. We may suppose A ∈ Art O by the formal cohesiveness.From the definition of s D Z it follows that we have a natural fibration sequence s D ( A ) → s D Z ( A ) → sHom M ( ∗ , B Z ( A )) . Since π i sHom M ( ∗ , B Z ( A )) vanishes for i = 2 , we have π s D Z ( A ) = π s D ( A ) . By Equation 1 of section 2.2,we have π s D ( A ) = D ( A ) , hence also π s D Z ( A ) = D ( A ) . IMPLICIAL GALOIS DEFORMATION FUNCTORS 11
By applying the same arguement with X replaced by X v and G replaced by B when necessary, we obtain π s D ? v,Z ( A ) ∼ = D ? v ( A ) ( ? = ∅ or n . o ).We have the exact sequence π s D Z ( A ) ⊕ ( M v ∈ S p π s D n . o v,Z ( A )) → M v ∈ S p π s D v,Z ( A ) → π s D n . o Z ( A ) → π s D Z ( A ) ⊕ ( M v ∈ S p π s D n . o v,Z ( A )) → M v ∈ S p π s D v,Z ( A ) We will see later (Lemma 4.20) that s D v ( A ) is weakly equivalent to holim ∆ X hofib ∗ ( BG ( A ) → BG ( k )) , and(by Lemma 4.22) π s D v ( A ) ∼ = H (Γ v , b G ( A )) . Similarly π s D n . o v ( A ) ∼ = H (Γ v , b B ( A )) .By the assumption ( Reg v ) and Artinian induction, the map π s D n . o v ( A ) → π s D v ( A ) is an isomorphism,and so is π s D n . o v,Z ( A ) → π s D v,Z ( A ) . We deduce that π s D n . o Z ( A ) is the kernel of D ( A ) ⊕ ( L v ∈ S p D n . o v ( A )) → L v ∈ S p D v ( A ) , which is isomorphic to D n . o ( A ) by Lemma 2.6. (cid:3) Pseudo-deformation functors
Classical pseudo-characters and functors on
FFS . Recall the notion of a (classical) G -pseudo-character due to V. Lafforgue (see [Laf18, Définition-Proposition 11.3] and [BHKT19, Definition 4.1]): Definition 3.1.
Let A be an O -algebra. A G -pseudo-character Θ on Γ over A is a collection of O -algebramorphisms Θ n : O ad GN n G → Map(Γ n , A ) for each n ≥ , satisfying the following conditions:(1) For each n, m ≥ and for each map ζ : { , . . . , n } → { , . . . , m } , f ∈ O ad GN m G , and γ , . . . , γ m ∈ Γ , wehave Θ m ( f ζ )( γ , . . . , γ m ) = Θ n ( f )( γ ζ (1) , . . . , γ ζ ( n ) ) , where f ζ ( g , . . . , g m ) = f ( g ζ (1) , . . . , g ζ ( n ) ) .(2) For each n ≥ , for each γ , . . . , γ n +1 ∈ Γ , and for each f ∈ O ad GN n G , we have Θ n +1 ( ˆ f )( γ , . . . , γ n +1 ) = Θ n ( f )( γ , . . . , γ n − , γ n γ n +1 ) , where ˆ f ( g , . . . , g n +1 ) = f ( g , . . . , g n − , g n g n +1 ) .We denote by PsCh( A ) the set of pseudo-characters over A .We want to give a simplicial reformulation of this notion. As a first step, following [Weid18], let usconsider FS the category of finite sets and FFS be the category of finite free semigroups. For any finiteset X , let M X be the finite free semigroup generated by X ; we have Γ X = Hom semGp ( M X , Γ) and G X = Hom semGp ( M X , G ) . For a semigroup M ∈ FFS , note that
Hom semGp ( M X , G ) is a group scheme, so, we candefine a covariant functor FFS → Alg O , M
7→ O
Hom semGp ( M,G ) . We can also define the covariant functor M Map(
Hom semGp ( M, Γ) , A ) . These functors on FFS extend canonically those defined on the category FS by X
7→ O G X and X Map(Γ X , A ) . Moreover, the natural transformation O ad GG X → Map(Γ X , A ) extends uniquely to a natural transformation of functors on FFS . Actually, there are several useful functorson
FFS ; by the canonical extension from FS to FFS mentioned above, it is enough to define them on theobjects [ n ] , as in [Weid18, Example 2.4 and Example 2.5]:(1) The association [ n ] Γ n defines an object Γ • ∈ Sets
FFS op .(2) For A ∈ Alg O , the association [ n ] Map(Γ n , A ) defines an object Map(Γ • , A ) ∈ Alg
FFS O .(3) The association [ n ]
7→ O ad GN n G defines an object O ad GN • G ∈ Alg
FFS O .(4) Let G n //G = Spec( O ad GN n G ) . Then for A ∈ Alg O , the association [ n ] ( G n //G )( A ) defines an object ( G • //G )( A ) ∈ Sets
FFS op . As noted in [Weid18, Theorem 2.12], one sees that a G -pseudo-character Θ of Γ over A is exactly a naturaltransformation from O ad GN • G to Map(Γ • , A ) (we call these natural transformations Alg
FFS O -morphisms). Lemma 3.2.
For A ∈ Alg O , there is a bijection between PsCh( A ) and Hom
Sets
FFS op (Γ • , ( G • //G )( A )) .Proof. It suffices to note that there is a bijection between
Sets
FFS op -morphisms Γ • → ( G • //G )( A ) and Alg
FFS O -morphisms O ad GN • G → Map(Γ • , A ) . (cid:3) For an algebraically closed field A and a (continuous) homomorphism ρ : Γ → G ( A ) , we say that ρ is G -completely reducible if any parabolic subgroup containing ρ (Γ) has a Levi subgroup containing ρ (Γ) . Recallthe following results in [BHKT19, Section 4]: Theorem 3.3. (1) [BHKT19, Theorem 4.5]
Suppose that A ∈ Alg O is an algebraically closed field. Thenwe have a bijection between the following two sets: (a) The set of G ( A ) -conjugacy classes of G -completely reducible group homomorphisms ρ : Γ → G ( A ) , (b) The set of pseudo-characters over A . (2) [BHKT19, Theorem 4.10] Fix an absolutely G -completely reducible representation ¯ ρ : Γ → G ( k ) ,and suppose further that the centralizer of ¯ ρ in G ad k is scheme-theoretically trivial. Let ¯Θ be thepseudo-character, which regarded as an element of Hom
Sets
FFS op (Γ • , ( G • //G )( k )) , is induced from ( γ , . . . , γ n ) (¯ ρ ( γ ) , . . . , ¯ ρ ( γ n )) . Let A ∈ Art O . Then we have a bijection between the followingtwo sets: (a) The set of b G ( A ) -conjugacy classes of group homomorphisms ρ : Γ → G ( A ) which lift ¯ ρ , (b) The set of pseudo-characters over A which reduce to ¯Θ modulo m A . Note that there are similarities between
Sets
FFS op and Sets ∆ op = s Sets . In the following, we shall provesimilar results with
Sets
FFS op replaced by s Sets .3.2.
Classical pseudo-characters and simplicial objects.
Recall that on O N • G there are natural cofaceand codegeneracy maps, and we can regard O N • G as an object in Alg ∆ O ( i.e. a cosimplicial O -algebra). Theadjoint action of G on G • induces an action of G on O N • G , which obviously commutes with the coface andcodegeneracy maps. In consequence, O ad GN • G is well-defined in Alg ∆ O . Definition 3.4.
We define the functor ¯ BG : Alg O → s Sets by associating A ∈ Alg O to Hom
Alg O ( O ad GN • G , A ) with face and degeneracy maps induced from the coface and codegeneracy maps in O ad GN • G .Note that the inclusion O ad GN • G → O N • G gives a natural transformation BG → ¯ BG .3.2.1. Algebraically closed field.
Let A ∈ Alg O be an algebraically closed field. We would like to characterizethe elements of Hom s Sets ( B Γ , ¯ BG ( A )) . They correspond to the quasi-homomorphisms, which we define below. Definition 3.5.
Let Γ and G be two groups. We say a map ρ : Γ → G is a quasi-homomorphism if thereexists a a map φ : Γ → G such that ρ ( x ) − ρ ( xy ) = φ ( x ) ρ ( y ) φ ( x ) − for any x, y ∈ Γ .Obviously a group homomorphism is a quasi-homomorphism. Note that every quasi-homomorphism pre-serves the identity, and the set of quasi-homomorphisms is closed under G -conjugations. Remark . A quasi-homomorphism can fail to be a group homomorphism. We can construct a quasi-homomorphism as follows: let σ : Γ → G be a group homomorphism, let φ : Γ → Z ( σ (Γ)) be a grouphomomorphism and let g ∈ G , then ρ ( x ) = g − σ ( x ) φ ( x ) gφ ( x ) − is a quasi-homomorphism. Such ρ is notnecessarily a group homomorphism, an example could be the following: take G = H × H , σ : Γ → H × { e } and φ : Γ → { e } × H , and choose g such that g / ∈ Z ( φ (Γ)) . Lemma 3.7.
Let ρ be a quasi-homomorphism and let φ as above. Then the map φ induces a group homo-morphism Γ → G/Z ( ρ (Γ)) which doesn’t depend on the choice of φ . IMPLICIAL GALOIS DEFORMATION FUNCTORS 13
Proof.
For x, y, z ∈ Γ , we have φ ( xy ) ρ ( z ) φ ( xy ) − = ρ ( xy ) − ρ ( xyz )= ( φ ( x ) ρ ( y ) φ ( x ) − ) − ( φ ( x ) ρ ( yz ) φ ( x ) − )= φ ( x ) ρ ( y ) − ρ ( yz ) φ ( x ) − = φ ( x ) φ ( y ) ρ ( z ) φ ( y ) − φ ( x ) − . Hence φ ( xy ) − φ ( x ) φ ( y ) ∈ Z ( ρ (Γ)) for any x, y ∈ Γ , and φ induces a group homomorphism Γ → G/Z ( ρ (Γ)) .For any other choice φ such that ρ ( x ) − ρ ( xy ) = φ ( x ) ρ ( y ) φ ( x ) − , we see φ − ( x ) φ ( x ) ∈ Z ( ρ (Γ)) , and theconclusion follows. (cid:3) Lemma 3.8.
Suppose that A ∈ Alg O is an algebraically closed field. Let f ∈ Hom s Sets ( B Γ , ¯ BG ( A )) . Thenwe can associate a quasi-homomorphism ρ : Γ → G ( A ) to f such that f sends ( γ , . . . , γ n ) ∈ B Γ n to the classin ¯ BG ( A ) n represented by ( ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j )) i =1 ,...,n .Proof. For each n ≥ and γ = ( γ , . . . , γ n ) ∈ Γ n , we choose a representative T ( γ ) = ( g , . . . , g n ) ∈ G ( A ) n of f ( γ ) with closed orbit, note that any other representative with closed orbit is conjugated to ( g , . . . , g n ) .Let H ( γ ) be the Zariski closure of the subgroup of G ( A ) generated by the entries of T ( γ ) . Let n ( γ ) be thedimension of a parabolic P ⊆ G A minimal among those containing H ( γ ) , we see n ( γ ) is independent of thechoice of P . Let N = sup n ≥ ,γ ∈ Γ n n ( γ ) . We fix a choice of δ = ( δ , . . . , δ n ) satisfying the following conditions:(1) n ( δ ) = N .(2) For any δ ′ ∈ Γ n ′ satisfying (1), we have dim Z G A ( H ( δ )) ≤ dim Z G A ( H ( δ ′ )) .(3) For any δ ′ ∈ Γ n ′ satisfying (1) and (2), we have π ( Z G A ( H ( δ ))) ≤ π ( Z G A ( H ( δ ′ ))) .Write T ( δ ) = ( h , . . . , h n ) . As in the proof of [BHKT19, Theorem 4.5], we have the following facts:(1) For any ( γ , . . . , γ m ) ∈ Γ m , there exists a unique tuple ( g , . . . , g m ) ∈ G ( A ) m such that ( h , . . . , h n , g , . . . , g m ) is conjugated to T ( δ , . . . , δ n , γ , . . . , γ m ) .(2) Let ( h , . . . , h n , g , . . . , g m ) be as above. Any finite subset of the group generated by ( h , . . . , h n , g , . . . , g m ) which contains ( h , . . . , h n ) has a closed orbit.We define ρ ( γ ) to be the unique element such that ( h , . . . , h n , ρ ( γ )) is conjugated to T ( δ , . . . , δ n , γ ) .Suppose for γ , . . . , γ m ∈ Γ , the unique tuple conjugated to T ( δ , . . . , δ n , γ , . . . , γ m ) is ( h , . . . , h n , g , . . . , g m ) .Consider the following diagram, where the horizontal arrows are compositions of face maps: ( δ , . . . , δ n , γ , . . . , γ m ) / / (cid:15) (cid:15) ( h , . . . , h n , g , . . . , g m ) (cid:15) (cid:15) ( δ , . . . , δ n , Q ij =1 γ j ) / / ( h , . . . , h n , Q ij =1 g j ) Since ( h , . . . , h n , Q ij =1 g j ) has a closed orbit and is a pre-image of f ( δ , . . . , δ n , Q ij =1 γ j ) , we have Q ij =1 g j = ρ ( Q ij =1 γ j ) , and g i = ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j ) ( ∀ i = 1 , . . . , m ) .Let x, y ∈ Γ . Then the element in G ( A ) n +2 associated to ( δ , . . . , δ n , x, δ , . . . , δ n , y ) is ( h , . . . , h n , ρ ( x ) , ρ ( x ) − ρ ( xδ ) , . . . , ρ ( x n − Y j =1 δ j ) − ρ ( x n Y j =1 δ j ) , ρ ( x n Y j =1 δ j ) − ρ ( x n Y j =1 δ j · y )) , and the element in G ( A ) n +1 associated to ( δ , . . . , δ n , δ , . . . , δ n , y ) is ( h , . . . , h n , ρ ( δ ) , . . . , ρ ( n − Y j =1 δ j ) − ρ ( n Y j =1 δ j ) , ρ ( n Y j =1 δ j ) − ρ ( n Y j =1 δ j · y )) . We see both ( ρ ( x Q i − j =1 δ j ) − ρ ( x Q ij =1 δ j )) i =1 ,...,n and ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n have a closed orbit andare pre-images of f ( δ , . . . , δ n ) , so they are conjugated by some φ ( x ) ∈ G ( A ) . Since Z G A ( H ( δ )) is minimal bythe defining property, φ ( x ) must conjugate ρ ( Q nj =1 δ j ) − ρ ( Q nj =1 δ j · y ) to ρ ( x Q nj =1 δ j ) − ρ ( x Q nj =1 δ j · y ) . Wededuce that ∀ x, y ∈ Γ , ρ ( x ) − ρ ( xy ) = φ ( x ) ρ ( y ) φ ( x ) − , and ρ is a quasi-homomorphism. It’s obvious that forany ( γ , . . . , γ n ) ∈ Γ n , ( ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j )) i =1 ,...,n is a pre-image of f ( γ , . . . , γ n ) . (cid:3) Artinian coefficients.
Let ¯ ρ : Γ → G ( k ) be an absolutely G -completely reducible representation, andsuppose that H (Γ , g ) = z . We write ¯ f ∈ Hom s Sets ( B Γ , ¯ BG ( k )) for the map induced from ( γ , . . . , γ n ) (¯ ρ ( γ ) , . . . , ¯ ρ ( γ n )) . Definition 3.9.
For A ∈ Art O , the set aDef ¯ f ( A ) is the fiber over ¯ f of the map Hom s Sets ( B Γ , ¯ BG ( A )) → Hom s Sets ( B Γ , ¯ BG ( k )) . Definition 3.10.
Let A ∈ Art O . We say a map ρ : Γ → G ( A ) is a quasi-lift of ¯ ρ if ρ mod m A = ¯ ρ and ρ isa quasi-homomorphism. Remark . In general, a quasi-lift may not be a group homomorphism. Let → I → A ։ A be aninfinitesimal extension in Art O . Let ρ : Γ → G ( A ) be a group homomorphism, let σ : G ( A ) → G ( A ) be aset-theoretic section of G ( A ) → G ( A ) and let ˜ ρ = σ ◦ ρ . Let’s construct a quasi-lift ρ = exp( X α )˜ ρ where X : Γ → g ⊗ k I is a cochain to be determined.For α, β ∈ Γ , there exists c α,β ∈ g ⊗ k I such that ˜ ρ ( α )˜ ρ ( β ) = exp( c α,β )˜ ρ ( αβ ) since ρ : Γ → G ( A ) is agroup homomorphism. It’s easy to check that c ∈ Z (Γ , g ⊗ k I ) . Let φ ( α ) = exp( Y α ) where Y : Γ → g ⊗ k I isa group homomorphism also to be determined. We require ρ ( αβ ) = ρ ( α ) φ ( α ) ρ ( β ) φ ( α ) − for all α, β ∈ Γ .Note that ρ ( αβ ) = exp( X αβ )˜ ρ ( αβ ) and ρ ( α ) φ ( α ) ρ ( β ) φ ( α ) − = exp( X α )˜ ρ ( α ) exp( Y α ) exp( X β )˜ ρ ( β ) exp( Y α ) − = exp( X α )˜ ρ ( α ) exp( X β + Y α − Ad ˜ ρ ( β ) Y α )˜ ρ ( β )= exp( X α + Ad ˜ ρ ( α ) X β ) exp(Ad ˜ ρ ( α )(1 − Ad ˜ ρ ( β )) Y α )˜ ρ ( α )˜ ρ ( β )= exp( X α + Ad ˜ ρ ( α ) X β ) exp(Ad ˜ ρ ( α )(1 − Ad ˜ ρ ( β )) Y α ) exp( c α,β )˜ ρ ( αβ ) so we need to find a group homomorphism Y : Γ → g ⊗ k I such that Ad ˜ ρ ( α )(1 − Ad ˜ ρ ( β )) Y α ) + c α,β is acoboundary. In particular, in the case H (Γ , g ) = 0 , we can take an arbitrary group homomorphism Y : Γ → g .Note that ρ is a group homomorphism if and only if φ ( α ) = exp( Y α ) ∈ Z ( A ) for any α ∈ Γ . Lemma 3.12.
Let A ∈ Art O and let ρ : Γ → G ( A ) be a quasi-lift of ¯ ρ . Then Z ( ρ (Γ)) = Z ( A ) .Proof. See [Til96, Lemma 3.1] (note that the condition that ρ is a group homomorphism is not used in theproof). (cid:3) Corollary 3.13.
Let A ∈ Art O and let ρ : Γ → G ( A ) be a quasi-lift of ¯ ρ . Then ρ induces a uniquelydetermined group homomorphism φ : Γ → Ker( G ad ( A ) → G ad ( k )) such that ρ ( x ) − ρ ( xy ) = φ ( x ) ρ ( y ) φ ( x ) − for any x, y ∈ Γ .Proof. By combining the above lemma with Lemma 3.7, we see φ : Γ → G ad ( A ) is uniquely determined. Since ¯ ρ is a group homomoprhism, φ mod m A commutes with ¯ ρ (Γ) , and hence φ mod m A is trivial. (cid:3) Now we can characterize aDef ¯ f ( A ) in terms of quasi-lifts. The following propostion owing to [BHKT19]plays a crucial role (see also its use in the proof of [BHKT19, Theorem 4.10]): Proposition 3.14.
Suppose that X is an integral affine smooth O -scheme on which G acts. Let x =( x , . . . , x n ) ∈ X ( k ) be a point with G k · x closed, and Z G k ( x ) scheme-theoretically trivial. We write X ∧ ,x forthe functor Art O → Sets which sends A to the set of pre-images of x under X ( A ) → X ( k ) , and write G ∧ for the functor Art O → Sets which sends A to Ker( G ( A ) → G ( k )) . Then IMPLICIAL GALOIS DEFORMATION FUNCTORS 15 (1)
The G ∧ -action on X ∧ ,x is free on A -points for any A ∈ Art O . (2) Let
X//G = Spec O [ X ] G , let π : X → X//G be the natural map, and let ( X//G ) ∧ ,π ( x ) be the functor Art O → Sets which sends A to the set of pre-images of π ( x ) under ( X//G )( A ) → ( X//G )( k ) . Then π : X → X//G induces an isomorphism X ∧ ,x /G ∼ = ( X//G ) ∧ ,π ( x ) . Proof.
See [BHKT19, Proposition 3.13]. (cid:3)
Corollary 3.15. If ( γ , . . . , γ m ) is a tuple in Γ m such that (¯ ρ ( γ ) , . . . , ¯ ρ ( γ m )) has a closed orbit and a scheme-theoretically trivial centralizer in G ad k , then (¯ ρ ( γ ) , . . . , ¯ ρ ( γ m )) has a lift ( g , . . . , g m ) ∈ G ( A ) m which is apre-image of f ( γ , . . . , γ m ) ∈ ¯ BG ( A ) m , and any other choice is conjugated to this one by a unique element of G ad ( A ) . Theorem 3.16.
Let A ∈ Art O . Then aDef ¯ f ( A ) is isomorphic to the set of b G ( A ) -conjugacy classes ofquasi-lifts of ¯ ρ .Proof. Given a quasi-lift ρ : Γ → G ( A ) , then the association ( γ , . . . , γ m ) ( ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j )) i =1 ,...,m defines an element of aDef ¯ f ( A ) .In the following, we will construct a quasi-lift from a given f ∈ aDef ¯ f ( A ) .Let n ≥ be sufficiently large and choose δ , . . . , δ n ∈ Γ such that (¯ h = ¯ ρ ( δ ) , . . . , ¯ h n = ¯ ρ ( δ n )) is a systemof generators of ¯ ρ (Γ) , then the tuple (¯ h , . . . , ¯ h n ) has a scheme-theoretically trivial centralizer in G ad k . By[BMR05, Corollary 3.7], the absolutely G -completely reducibility implies that the tuple (¯ h , . . . , ¯ h n ) has aclosed orbit. By the above corollary, we can choose a lift ( h , . . . , h n ) ∈ G ( A ) n of (¯ h , . . . , ¯ h n ) which is at thesame time a pre-image of f ( δ , . . . , δ n ) .For any γ ∈ Γ , the tuple (¯ h , . . . , ¯ h n , ¯ ρ ( γ )) obviously has a closed orbit and a trivial centralizer in G ad k ,so we can choose a tuple in G ( A ) n +1 which lifts (¯ h , . . . , ¯ h n , ¯ ρ ( γ )) and is a pre-image of f ( δ , . . . , δ n , γ ) . Forthis tuple, the first n elements are conjugated to ( h , . . . , h n ) by a unique element of G ad ( A ) , so there is aunique g ∈ G ( A ) such that the tuple is conjugated to ( h , . . . , h n , g ) . We define ρ ( γ ) to be this g . It followsimmediately that ρ mod m A = ¯ ρ .Now suppose γ , . . . , γ m ∈ Γ . As above, let ( g , . . . , g m ) be the unique tuple such that ( h , . . . , h n , g , . . . , g m ) lifts (¯ h , . . . , ¯ h n , ¯ ρ ( γ ) , . . . , ¯ ρ ( γ m )) and is a pre-image of f ( δ , . . . , δ n , γ , . . . , γ m ) , consider the following dia-gram, where the horizontal arrows are compositions of face maps: ( δ , . . . , δ n , γ , . . . , γ m ) / / (cid:15) (cid:15) ( h , . . . , h n , g , . . . , g m ) (cid:15) (cid:15) ( δ , . . . , δ n , Q ij =1 γ j ) / / ( h , . . . , h n , Q ij =1 g j ) Then ( h , . . . , h n , Q ij =1 g j ) lifts (¯ h , . . . , ¯ h n , ¯ ρ ( Q ij =1 γ j )) and is a pre-image of f ( δ , . . . , δ n , Q ij =1 γ j ) . Hence Q ij =1 g j = ρ ( Q ij =1 γ j ) , and g i = ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j ) ( ∀ i = 1 , . . . , m ) .Let x, y ∈ Γ . Then the element in G ( A ) n +2 associated to ( δ , . . . , δ n , x, δ , . . . , δ n , y ) is ( h , . . . , h n , ρ ( x ) , ρ ( x ) − ρ ( xδ ) , . . . , ρ ( x n − Y j =1 δ j ) − ρ ( x n Y j =1 δ j ) , ρ ( x n Y j =1 δ j ) − ρ ( x n Y j =1 δ j · y )) , and the element in G ( A ) n +1 associated to ( δ , . . . , δ n , δ , . . . , δ n , y ) is ( h , . . . , h n , ρ ( δ ) , . . . , ρ ( n − Y j =1 δ j ) − ρ ( n Y j =1 δ j ) , ρ ( n Y j =1 δ j ) − ρ ( n Y j =1 δ j · y )) . We see both ( ρ ( x Q i − j =1 δ j ) − ρ ( x Q ij =1 δ j )) i =1 ,...,n and ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n are lifts of (¯ h , . . . , ¯ h n ) and pre-images of f ( δ , . . . , δ n ) , so they are conjugated by some φ ( x ) ∈ G ( A ) . We can even suppose φ ( x ) ∈ Ker( G ( A ) → G ( k )) because the centralizer of (¯ h , . . . , ¯ h n ) is Z . Since φ ( x ) is uniquely determined modulo Z ( A ) , it must conjugate ρ ( Q nj =1 δ j ) − ρ ( Q nj =1 δ j · y ) to ρ ( x Q nj =1 δ j ) − ρ ( x Q nj =1 δ j · y ) . We deduce that ∀ x, y ∈ Γ , ρ ( x ) − ρ ( xy ) = φ ( x ) ρ ( y ) φ ( x ) − , and ρ is a quasi-lift.For the ρ constructed as above, we can recover f from the formula ( γ , . . . , γ m ) ( ρ ( Q i − j =1 γ j ) − ρ ( Q ij =1 γ j )) i =1 ,...,m .So it remains to prove that if ρ and ρ have the same image in aDef ¯ f ( A ) , then they are equal modulo Ker( G ( A ) → G ( k )) -conjugation. Since ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n and ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n are both lifts of (¯ h , . . . , ¯ h n ) and pre-images of f ( δ , . . . , δ n ) , they are conjugated by some g ∈ G ( A ) ,and we may choose g ∈ Ker( G ( A ) → G ( k )) because the centralizer of (¯ h , . . . , ¯ h n ) is Z . After con-jugation by g , we may suppose ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n = ( ρ ( Q i − j =1 δ j ) − ρ ( Q ij =1 δ j )) i =1 ,...,n =( h ′ , . . . , h ′ n ) . Then for γ ∈ Γ , ρ k ( Q nj =1 δ j ) − ρ k ( Q nj =1 δ j · γ ) ( k = 1 , ) is uniquely determined by the condi-tion: ( h ′ , . . . , h ′ n , ρ k ( Q nj =1 δ j ) − ρ k ( Q nj =1 δ j · γ )) lifts (¯ h , . . . , ¯ h n , ¯ ρ ( γ )) and is a pre-image of f ( δ , . . . , δ n , γ ) .In consequence, we have ρ = ρ . (cid:3) For A ∈ Art O , let aDef ¯ f,c ( A ) be the subset of aDef ¯ f ( A ) consisting of f : B Γ → ¯ BG ( A ) which factorizesthrough some finite quotient of Γ . In fact we have aDef ¯ f,c ( A ) = Hom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) (recall that X isthe pro-simplicial set ( B Γ i ) i ). The following corollary is obvious: Corollary 3.17.
Let A ∈ Art O . Then aDef ¯ f,c ( A ) is isomorphic to the set of b G -conjugacy classes of contin-uous quasi-lifts of ¯ ρ . As a by-product of the proof of Theorem 3.16, we also have:
Corollary 3.18.
For A ∈ Art O , the set aDef ¯ f ( A ) (resp. aDef ¯ f,c ( A ) ) is isomorphic to Hom M ( B Γ , BG ( A ) /G ∧ ( A )) (resp. Hom M ( X, BG ( A ) /G ∧ ( A )) ). But unfortunately, the simplicial set BG ( A ) /G ∧ ( A ) isn’t generally fibrant.We attempt to compare the difference between aDef ¯ f,c ( A ) and D ( A ) . Motivated by the front-to-backduality in [Weib94, 8.2.10], we make the following definition. Let the reflection action r act on B Γ and ¯ BG ( A ) as follows:(1) r acts on B Γ n ∼ = Γ × · · · × Γ by r ( γ , . . . , γ n ) = ( γ n , . . . , γ ) .(2) r acts on O N n G by r ( f )( g , . . . , g n ) = f ( g n , . . . , g ) . We see that r preserves O ad GN n G , hence r acts on ¯ BG ( A ) n . Definition 3.19.
For A ∈ Art O , we define bDef ¯ f ( A ) (resp. bDef ¯ f,c ( A ) ) to be the subset of aDef ¯ f ( A ) (resp. aDef ¯ f,c ( A ) ) consisting of f : B Γ → ¯ BG ( A ) which commutes with r . Theorem 3.20.
Let A ∈ Art O . Suppose the characteristic of k is not . Then bDef ¯ f ( A ) is in bijection withthe set of group homomorphisms Γ → G ( A ) which lift ¯ ρ , and bDef ¯ f,c ( A ) is in bijection with D ( A ) .Proof. Let f ∈ bDef ¯ f ( A ) . It suffices to prove that the quasi-lift ρ obtained in Theorem 3.16 is a grouphomomorphism. We choose the tuple ( δ , . . . , δ n ) such that δ i = δ n +1 − i and Q nj =1 δ j = e . Write ρ for the quasi-lift constructed from this tuple as in Theorem 3.16, note that the choice of ( δ , . . . , δ n ) only affects ρ by someconjugation. Let φ : Γ → G ( A ) /Z ( A ) be the group homomorphism such that ρ ( xy ) = ρ ( x ) φ ( x ) ρ ( y ) φ ( x ) − forany x, y ∈ Γ . Note that φ ( x ) mod m A = 1 because ¯ ρ is a group homomorphism.Since f commutes with r , we have(1) ρ ( x ) = ρ ( x − ) − , ∀ x ∈ Γ .(2) ρ ( x ) − ρ ( xy ) = ρ ( yx ) ρ ( x ) − , ∀ x, y ∈ Γ .By substituting (1) into ρ ( xy ) = ρ ( x ) φ ( x ) ρ ( y ) φ ( x ) − , we get ρ ( y − x − ) − = ρ ( x − ) − φ ( x ) ρ ( y − ) − φ ( x ) − ,then consider ( x, y ) ( x − , y − ) and take the inverse, we get ρ ( yx ) = φ ( x ) − ρ ( y ) φ ( x ) ρ ( x ) . Now (2) implies IMPLICIAL GALOIS DEFORMATION FUNCTORS 17 ρ ( xy ) ρ ( x ) = ρ ( x ) ρ ( yx ) , which in turn gives ρ ( x ) φ ( x ) ρ ( y ) φ ( x ) − ρ ( x ) = ρ ( x ) φ ( x ) − ρ ( y ) φ ( x ) ρ ( x ) . So φ ( x ) commutes with ρ (Γ) for any x ∈ Γ , and φ = 1 . Since the characteristic of k is not and φ ( x )mod m A = 1 ∈ G ( k ) /Z ( k ) , we deduce φ = 1 and ρ is a group homomorphism. (cid:3) Derived deformations of pseudo-characters.
The functor aDef ¯ f,c = Hom s Sets / ¯ BG ( k ) ( X, ¯ BG ( − )) isanalogous to the functor D (cid:3) = Hom s Sets /BG ( k ) ( X, BG ( − )) , so it’s natural to consider the function complex sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( − )) and then to extend the domain of definition to O \ s Art / k , as constructing thefunctor s D : O \ s Art / k → s Sets . Definition 3.21.
For A ∈ O \ s Art / k , we define ¯ B G ( A ) to be the Ex ∞ of the diagonal of the bisimplicial set ([ p ] , [ q ]) Hom O \ s CR ( c ( O ad GN p G ) , A ∆[ q ] ) , and define sa D ( A ) = hofib ¯ f ( Hom s Sets ( X, ¯ B G ( A )) → Hom s Sets ( X, ¯ B G ( k ))) . If A ∈ Art O , then the bisimplicial set ([ p ] , [ q ]) Hom O \ s CR ( c ( O ad GN p G ) , A ∆[ q ] ) doesn’t depend on theindex q , and each of its lines is isomorphic to Ex ∞ ¯ BG ( A ) . Hence ¯ f can be regarded as an element of Hom s Sets ( X, ¯ B G ( k )) . As the derived deformation functors s D , we see that sa D : O \ s Art / k → s Sets ishomotopy invariant.Note that the inclusion O ad GN • G ֒ → O N • G induces a natural transformation s D → sa D .We would like to understand π sa D ( A ) . Let’s first analyse the case A ∈ Art O . For simplicity, we don’t takethe Ex ∞ here. Since BG ( A ) → BG ( k ) is a fibration, sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) is a good model for s D ( A ) .However, if ¯ BG ( A ) → ¯ BG ( k ) is a not fibration, then sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) is not weakly equivalent to sa D ( A ) .We have the commutative diagram sHom s Sets / ¯ BG ( k ) ( X, BG ( A )) / / (cid:15) (cid:15) sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) (cid:15) (cid:15) π sHom s Sets /BG ( k ) ( X, BG ( A )) / / π sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) Note that π sa D ( A ) is the coequalizer of sa D ( A ) ⇒ sa D ( A ) = aDef ¯ f,c ( A ) by definition. Proposition 3.22.
The above diagram is naturally isomorphic to D (cid:3) ( A ) / / (cid:15) (cid:15) aDef ¯ f,c ( A ) (cid:15) (cid:15) D ( A ) / / ❦❦❦❦❦❦❦❦ π sHom s Sets / ¯ BG ( k ) ( X, ¯ BG ( A )) And there is a dotted arrow which make the diagram commutative, whose image is bDef ¯ f,c ( A ) ⊆ aDef ¯ f,c ( A ) .Proof. We have sHom s Sets /BG ( k ) ( X, BG ( A )) = Hom M ( X, BG ( A )) , which is exactly D (cid:3) ( A ) , since B : Gpd → s Sets is fully faithful. The other isomorphisms follow by definition.The dotted arrow signifies the inclusion of usual deformations into pseudo-deformations, whose image is bDef ¯ f,c ( A ) by Theorem 3.20. (cid:3) Remark . Note however that the functor sa D : O \ s Art / k → s Sets remains quite mysterious. It may beasked whether there is a more adequate derived deformation functor for pseudo-characters. (Co)tangent complexes and pro-representability Dold-Kan correspondence.
Let’s briefly review the Dold-Kan correspondence. Let R be a commuta-tive ring. Our goal here is to recall an equivalence (of model categories) between the category of simplicial R -modules s Mod R and the category of chain complexes of R -modules concentrated on non-negative degrees Ch ≥ ( R ) . Recall the model category structures on s Mod R and Ch ≥ ( R ) :(1) For s Mod R , the fibrations and weak equivalences are linear morphisms which are in s Sets , and thecofibrations are linear morphisms satisfying a lifting property (see [Hir03, Proposition 7.2.3]).(2) For Ch ≥ ( R ) , the cofibrations, fibrations and weak equivalences are linear morphisms satisfying thefollowing:(a) f : C • → D • is a cofibration if C n → D n is injective with projective cokernel for n ≥ .(b) f : C • → D • is a fibration if C n → D n is surjective for n ≥ .(c) f : C • → D • is a weak equivalence if the morphism H ∗ f induced on homology is an isomorphism.We write M ∈ s Mod R for the simplicial R -module with M n on n -th simplicial degree. Let N ( M ) be thechain complexes of R -modules such that N ( M ) n = n − T i =0 Ker( d i ) ⊆ M n with differential maps ( − n d n : n − \ i =0 Ker( d i ) ⊆ M n → n − \ i =0 Ker( d i ) ⊆ M n − . Obviously M N ( M ) is functorial. We call N ( M ) ∈ Ch ≥ ( R ) the normalized complex of M .The Dold-Kan functor DK : Ch ≥ ( R ) → s Mod R is the quasi-inverse of N . Explicitely, for a chain of R -modules C • = ( C ← C ← C ← . . . ) , we define DK( C • ) ∈ s Mod R as follows:(1) DK( C • ) n = L [ n ] ։ [ k ] C k .(2) For θ : [ m ] → [ n ] , we define the corresponding DK( C • ) n → DK( C • ) m on each component of DK( C • ) n indexed by [ n ] σ ։ [ k ] as follows: suppose [ m ] t ։ [ s ] d ֒ → [ k ] is the epi-monic factorization of thecomposition [ m ] θ → [ n ] σ ։ [ k ] , then the map on component [ n ] σ ։ [ k ] is C k d ∗ → C s ֒ → M [ m ] ։ [ r ] C r . Theorem 4.1. (1) (Dold-Kan) The functors DK and N are quasi-inverse hence form an equivalence ofcategories. Moreover, two morphisms f, g ∈ Hom s Mod R ( M, N ) are simplicially homotopic if and onlyif N ( f ) and N ( g ) are chain homotopic. (2) The functors DK and N preserve the model category stuctures of Ch ≥ ( R ) and s Mod R defined above.Proof. See [Weib94, Theorem 8.4.1] and [GJ09, Lemma 2.11]. Note that (1) is valid for any abelian categoryinstead of s Mod R . (cid:3) Remark . Let Ch ( R ) be the category of complexes ( C i ) i ∈ Z of R -modules and Ch ≥ ( R ) the subcategory ofcomplexes for which C i = 0 for i < . The category Ch ≥ ( R ) is naturally enriched over simplicial R -modules,and we have sHom Ch ≥ ( R ) ( C • , D • ) ∼ = sHom s Mod R (DK( C • ) , DK( D • )) . Given C • , D • ∈ Ch ≥ ( R ) . Let [ C • , D • ] ∈ Ch ( R ) be the mapping complex, more precisely, [ C • , D • ] n = Q m Hom R ( C m , D m + n ) and the differential maps are natural ones. Let τ ≥ be the functor which sends a chaincomplex X • to the truncated complex ← Ker( X → X − ) ← X ← . . . Then there is a weak equivalence
IMPLICIAL GALOIS DEFORMATION FUNCTORS 19 sHom Ch ≥ ( R ) ( C • , D • ) ≃ DK( τ ≥ [ C • , D • ]) (see [Lur09, Remark 11.1]).It’s clear that π i sHom Ch ≥ ( R ) ( C • , D • ) is isomorphic to the chain homotopy classes of maps from C • to D • + n .4.2. (Co)tangent complexes of simplicial commutative rings. We recall Quillen’s cotangent and tan-gent complexes of simplicial commutative rings.Let A be a commutative ring. For R an A -algebra, let Ω R/A be the module of differentials with the canonical R -derivation d : R → Ω R/A . Let
Der A ( R, − ) be the covariant functor which sends an R -module M to the R -module Der A ( R, M ) = { D : R → M | D is A -linear and D ( xy ) = xD ( y ) + yD ( x ) , ∀ x, y ∈ R } . It’s well-known that
Hom R (Ω R/A , − ) is naturally isomorphic to Der A ( R, − ) via φ φ ◦ d .Let T be an A -algebra, and let A \ CR / T be the category of commutative rings R over T and under A .Then for any T -module M and any R ∈ A \ CR / T , we have natural isomorphisms Hom T (Ω R/A ⊗ R T, M ) ∼ = Der A ( R, M ) ∼ = Hom A \ CR / T ( R, T ⊕ M ) . where T ⊕ M is the T -algebra with square-zero ideal M . So the functor R Ω R/A ⊗ R T is left ajoint to thefunctor M T ⊕ M .The above isomorphisms have level-wise extensions to simplicial categories (see [GJ09] Lemma II.2.9 andExample II.2.10). For R ∈ A \ s CR , we can form Ω R/A ⊗ R T ∈ s Mod T .We have sHom s Mod T (Ω R/A ⊗ R T, M ) ∼ = sHom A \ s CR / T ( R, T ⊕ M ) . The functor M T ⊕ M from s Mod T to A \ s CR / T preserves fibrations and weak equivalences (we maysee this via the Dold-Kan correspondence), so the left adjoint functor R Ω R/A ⊗ R T is left Quillen and itadmits a total left derived functor. We introduce the cotangent complex L R/A in the following definition, sothat the total left derived functor has the form R L R/A ⊗ R T . Note that given two simplicial modules M, N over a simplicial ring S , one can form (degreee-wise) a tensor product, denoted M ⊗ S N , which is a simplicial S -module. Definition 4.3.
For R ∈ A \ s CR , we define L R/A = Ω c ( R ) /A ⊗ c ( R ) R ∈ s Mod R , where c ( R ) is the middleobject of some cofibration-trivial fibration factorization A ֒ → c ( R ) ∼ ։ R , and we call L R/A the cotangentcomplex of R .Note that it is an abuse of language, as it should be called cotangent simplicial R -module, because for R simplicial, L R/A ∈ s Mod R but there is no notion of complexes of R -modules.By construction, L R/A ⊗ R T is cofibrant as it’s the image of the cofibrant object c ( R ) under a total leftderived functor, and it is fibrant in s Mod R (all objects are fibrant there). Note also that the weak equivalenceclass of L R/A ⊗ R T is independent of the choice of c ( R ) . It follows from these two observations that L R/A isdetermined up to homotopy equivalence (by the Whitehead theorem [Hir03, Theorem 7.5.10]). Using theDold-Kan equivalence, we can form the normalized complex (determined up to homotopy equivalence) N ( L R/A ⊗ R T ) ∈ Ch ≥ ( T ) . From now on, we keep the functor N understood and simply write L R/A ⊗ R T ∈ Ch ≥ ( T ) Recall that for
M, N ∈ Ch ( T ) , the internal Hom [ M, N ] ∈ Ch ( T ) is defined as [ M, N ] n = Y m Hom T ( M m , N m + n ) . Note that if M ∈ Ch ≥ ( T ) , then [ M, T ] ∈ Ch ≤ ( T ) . For C ∈ Ch ≤ ( T ) , we write C i = C − i for i ≥ ; wethus identify Ch ≤ ( T ) = Ch ≥ ( T ) .Since we’ll consider internal Homs [ L R/A ⊗ R T, M ] , for (classical or simplicial) T -modules M , we define Definition 4.4.
The T -tangent complex t R T of R → T is the internal hom complex [ L R/A ⊗ R T, T ] ∈ Ch ≥ ( T ) . Then t R T is well-defined up to chain homotopy equivalence since it is the case for L R/A ⊗ R T .For R ∈ A \ s CR / T and C • ∈ Ch ≥ ( T ) , we have (by Remark 4.2): sHom A \ s CR / T ( c ( R ) , T ⊕ DK( C • )) ∼ = sHom s Mod T ( L R/A ⊗ R T, DK( C • )) ≃ DK( τ ≥ [ L R/A ⊗ R T, C • ]) ∼ = DK( τ ≥ [ L R/A , C • ]) . Tangent complexes of formally cohesive functors and Lurie’s criterion.
In [GV18, Section 4],the authors define the tangent complexes of formally cohesive functors. To summarize, we have the followingproposition:
Proposition 4.5.
Let F : O \ s Art / k → s Sets be a formally cohesive functor. Then there exists L F ∈ Ch ( k ) such that F ( k ⊕ DK( C • )) is weakly equivalent to DK( τ ≥ [ L F , C • ]) for every C • ∈ Ch ≥ ( k ) with H ∗ ( C • ) finiteover k .Proof. See [GV18, Lemma 4.25]. (cid:3)
Definition 4.6.
Let F : O \ s Art / k → s Sets be a formally cohesive functor.(1) We call L F the cotangent complex of F .(2) The tangent complex t F of F is the chain complex defined by the internal hom complex [ L F , k ] .Note that L F and t F are uniquely determined up to quasi-isomorphism. We shall use t i F to abbreviatethe homology groups H − i t F . Remark . If R ∈ O \ s CR / k is cofibrant, then the functor F R = sHom O \ s CR / k ( R, − ) : O \ s Art / k → s Sets is formally cohesive. Since
DK( τ ≥ [ L F R , k [ n ]]) ≃ sHom O \ s CR / k ( R, k ⊕ k [ n ]) ≃ DK( τ ≥ [ L R/ O , k [ n ]]) , thecotangent complexes L F R and L R/ O ⊗ R k are quasi-isomorphic. Definition 4.8.
We say a functor F : O \ s Art / k → s Sets is pro-representable, if there exists a projective sys-tem R = ( R n ) n ∈ N with each R n ∈ O \ s Art / k cofibrant, such that F is weakly equivalent to lim −→ n sHom O \ s Art / k ( R n , − ) .In this case we say R = ( R n ) n ∈ N is a representing ring for F . We shall write sHom O \ s Art / k ( R, − ) for lim −→ n sHom O \ s Art / k ( R n , − ) . Remark . The pro-representability defined above is called sequential pro-representability in [GV18], butwe will only deal with this case.
Theorem 4.10 (Lurie’s criterion) . Let F be a formally cohesive functor. If dim k t i F is finite for every i ∈ Z ,and t i F = 0 for every i < , then F is (sequentially) pro-representable.Proof. See [Lur04, Corollary 6.2.14] and [GV18, Theorem 4.33]. (cid:3)
The following lemma illustrates the conservativity of the tangent complex functor:
Lemma 4.11.
Suppose F , F : O \ s Art / k → s Sets are formally cohesive functors. Then a natural transfor-mation F → F is a weak equivalence if and only if it induces isomorphisms t i F → t i F for all i .Proof. If the natural transformation induces isomorphisms t i F → t i F , then F ( k ⊕ k [ n ]) → F ( k ⊕ k [ n ]) is a weak equivalence. So by simplicial artinian induction [GV18, Section 4], it induces a weak equivalence F ( A ) → F ( A ) for A ∈ O \ s Art / k . (cid:3) IMPLICIAL GALOIS DEFORMATION FUNCTORS 21
Pro-representability of derived deformation functors.
In the following, we suppose p > , and Γ = G F,S for S = S p ∪ S ∞ . Suppose further that ¯ ρ satisfies ( Ord v ) and ( Reg v ) for v ∈ S p , and H (Γ , g k ) = z k .Recall that we’ve introduced derived deformation functors s D and s D n . o , as well as the modifying-centervariants s D Z and s D n . o Z . These functors are all formally cohesive. Their tangent complexes are related to theGalois cohomology groups H i ∗ (Γ , g k ) of adjoint representations, where ∗ = ∅ or n . o .4.4.1. Galois cohomology.
We briefly review the Galois cohomology theory. To define the nearly ordinarycohomology, we fix the standard Levi decomposition B = T N of the standard Borel of G ; it induces adecomposition of Lie algebras over k : b k = t k ⊕ n k . Recall the definition of the Greenberg-Wiles nearlyordinary Selmer group e H . o (Γ , g k ) = Ker H (Γ , g k ) → Y v ∈ S p H (Γ v , g k ) L v where L v = im( H (Γ v , b k ) → H (Γ v , g k )) .For v ∈ S p , let ˜ L v ⊆ Z (Γ v , g k ) be the preimage of L v . Let C • n . o (Γ , g k ) be the mapping cone of the naturalcochain morphism / / C (Γ , g k ) / / (cid:15) (cid:15) C (Γ , g k ) / / (cid:15) (cid:15) C (Γ , g k ) (cid:15) (cid:15) / / . . . / / / / L v ∈ S p C (Γ v , g k ) / ˜ L v / / L v ∈ S p C (Γ v , g k ) / / . . . Then we define the nearly ordinary cohomology groups H ∗ n . o (Γ , g k ) as the cohomology of the complex C • n . o (Γ , g k ) .They fit into the exact sequence ( ⋆ ): → H . o (Γ , g k ) → H (Γ , g k ) → → H . o (Γ , g k ) → H (Γ , g k ) → M v ∈ S p H (Γ v , g k ) /L v → H . o (Γ , g k ) → H (Γ , g k ) → M v ∈ S p H (Γ v , g k ) → H . o (Γ , g k ) → In particular, e H . o (Γ , g k ) = H . o (Γ , g k ) . Definition 4.12.
For a finite O [Γ] -module M , we write M ∨ = Hom O ( M, K/ O ) and M ∗ = Hom O ( M, K/ O (1)) .In particular if M is a k -vector space, M ∨ = Hom k ( M, k ) and M ∗ = Hom k ( M, k (1)) .Recall the local Tate duality H (Γ v , g k ) × H (Γ v , g ∗ k ) → k . Let L ⊥ v ⊆ H (Γ v , g ∗ k ) be the dual of L v . Wedefine similarly the cohomology groups H ∗ n . o , ⊥ (Γ , g ∗ k ) . In particular M v ∈ S p L v → H (Γ , g ∗ k ) ∨ → H . o , ⊥ (Γ , g ∗ k ) ∨ → is exact. By fitting this into the Poitou-Tate exact sequence (see [Mil06, Theorem I.4.10]), we obtain the exactsequence ( ⋆⋆ ): H (Γ , g k ) → M v ∈ S p H (Γ v , g k ) /L v → H . o , ⊥ (Γ , g ∗ k ) ∨ → H (Γ , g k ) → M v ∈ S p H (Γ v , g k ) → H (Γ , g ∗ k ) ∨ → We deduce the Poitou-Tate duality:
Theorem 4.13.
For each i ∈ { , , , } , there is a perfect pairing H i n . o , ⊥ (Γ , g ∗ k ) × H − i n . o (Γ , g k ) → k. Proof.
For i ∈ { , } , it suffices to compare the exact sequences ( ⋆ ) and ( ⋆⋆ ). The cases i ∈ { , } followby duality. (cid:3) Tangent complex.
Lemma 4.14. (1)
We have t i s D ∼ = H i +1 (Γ , g k ) for all i ∈ Z . On the other hand, t i s D Z ∼ = t i s D when i = − , and t − s D Z = 0 . (2) Let v ∈ S p . Then we have t i s D v ∼ = H i +1 (Γ v , g k ) for all i ∈ Z . On the other hand, t i s D v,Z ∼ = t i s D v when i = − , and t − s D v,Z ∼ = H (Γ v , g k ) / z k . (3) Let v ∈ S p . Then we have t i s D n . o v ∼ = H i +1 (Γ v , b k ) for all i ∈ Z . On the other hand, t i s D n . o v,Z ∼ = t i s D n . o v when i = − , and t − s D n . o v,Z ∼ = H (Γ v , b k ) / z k . Moreover, t s D n . o v = 0 if ( Reg ∗ v ) holds.Proof. Note that t j − i F ∼ = π i F ( k ⊕ k [ j ]) for any formally cohesive functor F and any i, j ≥ . Later in section4.5 we shall give a slightly generalized version of the lemma. See also [GV18, Section 7.3]. (cid:3) In particular, by Lurie’s criterion (Theorem 4.10), this lemma together with the finiteness of the cohomologygroups, implies
Corollary 4.15.
The center-modified functor s D Z is pro-representable. Now we treat the nearly ordinary case s D n . o Z . Let’s recall that s D loc ,Z = Q v ∈ S p s D v,Z , s D n . oloc ,Z = Q v ∈ S p s D n . o v,Z ,and s D n . o Z = s D Z × hs D loc ,Z s D n . oloc ,Z . Recall that ¯ ρ satisfies ( Ord v ) and ( Reg v ) for v ∈ S p , so s D n . o Z is indeedthe derived generalization of D n . o , i.e., π s D n . o Z ( A ) ∼ = D n . o ( π A ) for homotopy discrete A ∈ O \ s Art / k (seeProposition 2.17). Lemma 4.16.
Suppose furthermore ( Reg ∗ v ) for v ∈ S p . Then t i s D n . o Z ∼ = H i +1n . o (Γ , g ) when i ≥ , and t i s D n . o Z =0 when i < .Proof. We have the Mayer-Vietoris exact sequence (see [GV18, Lemma 4.30 (iv)] and [Weib94, Section 1.5]) t i s D n . o Z → t i s D Z ⊕ t i s D n . oloc ,Z → t i s D loc ,Z [1] → . . . . IMPLICIAL GALOIS DEFORMATION FUNCTORS 23
By Lemma 4.14, we obtain an exact sequence → t − s D n . o Z → M v ∈ S p H (Γ v , b k ) / z k → M v ∈ S p H (Γ v , g k ) / z k → t s D n . o Z → H (Γ , g k ) ⊕ ( M v ∈ S p H (Γ v , b k )) → M v ∈ S p H (Γ v , g k ) → t s D n . o Z → H (Γ , g k ) → M v ∈ S p H (Γ v , b k ) → t s D n . o Z → By assumtion ( Reg v ) , the map H (Γ v , b k ) / z k → H (Γ v , g k ) / z k is an isomorphism. The conclusion followsfrom comparing the above exact sequence with ( ⋆ ). (cid:3) In particular t − s D n . o Z = 0 (note that for this we don’t need ( Reg ∗ v ) ). By Lurie’s criterion (Theorem 4.10)and the finiteness of the cohomology groups, we have the following corollary: Corollary 4.17.
The functor s D n . o Z is pro-representable. Let R s, n . o be a representing (pro-)simplicial ring. Since π s D n . o Z ( A ) ∼ = D n . o ( A ) for A ∈ Art O , the ring π R s, n . o represents the classical nearly ordinary deformation functor D n . o .4.5. Relative derived deformations and relative tangent complexes.
Let T ∈ Art O and let ρ T : Γ → G ( T ) be a nearly ordinary representation. For v ∈ S p , we write ρ T,v for the restriction of ρ T on Γ v and wesuppose the image of ρ T,v lies in B ( T ) (more precisely, we should say the image of some conjugation of ρ T,v lies in B ( T ) , but there is no crucial difference). Let X and X v be the pro-simplicial sets associated to theprofinite groups Γ and Γ v . We identify ρ T as a map of (pro-)simplicial sets X → BG ( T ) → B G ( T ) (here BG ( T ) is the classical classifying space of the finite group G ( T ) and B G ( T ) is a fibrant replacement, seeDefinition 2.9) and identify ρ T,v as X v → BB ( T ) → B B ( T ) → B G ( T ) .Let’s consider the derived deformations functors over ρ T . Definition 4.18. (1) Let s D ρ T : O \ s Art / T → s Sets be the functor A hofib ρ T ( sHom s Sets ( X, B G ( A )) → sHom s Sets ( X, B G ( T ))) . (2) For v ∈ S p , let s D ρ T,v : O \ s Art / T → s Sets be the functor A hofib ρ T,v ( sHom s Sets ( X v , B G ( A )) → sHom s Sets ( X v , B G ( T ))) . (3) For v ∈ S p , let s D n . o ρ T,v : O \ s Art / T → s Sets be the functor A hofib ρ T,v ( sHom s Sets ( X v , B B ( A )) → sHom s Sets ( X v , B B ( T ))) . Our goal is to prove the following proposition (see also [GV18, Example 4.38 and Lemma 5.10]):
Proposition 4.19.
Let M be a finite module over an arbitrary Artin ring T . Then for i, j ≥ we have π i s D ρ T ( T ⊕ M [ j ]) ∼ = H j − i (Γ , g T ⊗ T M ) . Note that sHom s Sets ( X, − ) is defined by the filtered colimit lim −→ i sHom s Sets ( B Γ i , − ) , which commutes withhomotopy pullbacks. So it suffices to prove the proposition with Γ replaced by Γ i and X replaced by B Γ i . Tosimplify the notations, we suppose Γ is a finite group during the proof. Lemma 4.20.
Let A ∈ O \ s Art / T . Then s D ρ T ( A ) is weakly equivalent to holim ∆ X hofib ∗ ( B G ( A ) → B G ( T )) . Proof.
By [Hir03, Proposition 18.9.2], X is weakly equivalent to hocolim ( ∆ X ) op ∗ (i.e., the homotopy colimitof the single-point simplicial set indexed by ( ∆ X ) op ). Hence (see [Hir03, Theorem 18.1.10]) sHom s Sets ( X, B G ( A )) ≃ holim ∆ X sHom s Sets ( ∗ , B G ( A )) ≃ holim ∆ X B G ( A ) , and sHom s Sets ( X, B G ( T )) ≃ holim ∆ X sHom s Sets ( ∗ , B G ( T )) ≃ holim ∆ X B G ( T ) . Note that ρ T , as the single-point simplicial subset of sHom s Sets ( X, B G ( T )) , is identified with holim ∆ X ∗ → holim ∆ X B G ( T ) . Since homotopy limits commute with homotopy pullbacks, we conclude that s D ρ T ( A ) ≃ holim ∆ X hofib ∗ ( B G ( A ) → B G ( T )) . (cid:3) Let’s first analyse hofib ∗ ( B G ( A ) → B G ( T )) . Lemma 4.21.
The homotopy groups of hofib ∗ ( B G ( A ) → B G ( T )) are trivial except at degree j + 1 , where itis g T ⊗ T M .Proof. Note that A hofib ∗ ( B G ( A ) → B G ( T )) preserves weak equivalences and homotopy pullbacks.Since T ⊕ M [ j ] → T is j -connected, the map B G ( T ⊕ M [ j ]) → B G ( T ) is ( j + 1) -connected (see [GV18,Corollary 5.3]), and the homotopy groups of the homotopy fiber vanish up to degree j . Since the functor A hofib ∗ ( B G ( A ) → B G ( T )) maps the homotopy pullback square T ⊕ M [ j − / / (cid:15) (cid:15) T (cid:15) (cid:15) T / / T ⊕ M [ j ] to a homotopy pullback square, we get π j + k hofib ∗ ( B G ( T ⊕ M [ j ]) → B G ( T )) ∼ = π j + k − hofib ∗ ( B G ( T ⊕ M [ j − → B G ( T )) for any k ≥ . Consequently π j + k hofib ∗ ( B G ( T ⊕ M [ j ]) → B G ( T )) ∼ = π k hofib ∗ ( B G ( T ⊕ M [0]) → B G ( T )) , and hofib ∗ ( B G ( T ⊕ M [ j ]) → B G ( T )) has homotopy groups concentrated on degree j + 1 , where it is g T ⊗ T M . (cid:3) Let Y be the ∆ X -diagram in s Sets (i.e, functor ∆ X → s Sets ) which takes the value hofib ∗ ( B G ( A ) →B G ( T )) . Then Y is a local system (see [GV18, Definition 4.34], it’s called the cohomological coefficient systemin [GM13, Page 28]) on X . There is hence a π ( X, ∗ ) -action on the homotopy group g T ⊗ T M . By unwindingthe constructions, we see this is the conjugacy action of ρ T on g T ⊗ T M .It suffices to calculate holim Y . Under the Dold-Kan correspondence, we may identify hofib ∗ ( B G ( A ) →B G ( T )) with the chain complex with homology g T ⊗ T M concentrated on degree j + 1 . But in fact it’s moreconvenient to regard hofib ∗ ( B G ( A ) → B G ( T )) as a cochain complex with cohomology g T ⊗ T M concentratedon degree − ( j + 1) , because the homotopy limit of cochain complexes is drastically simple (see [Dug08, Section19.8]). By shifting degrees, it suffices to suppose that the cohomology is concentrated on degree . Lemma 4.22.
Let N be a T [Γ] -module, and we regard N as a cochain complex concentrated on degree .Let Y be the ∆ X -diagram in Ch ≥ ( T ) (i.e, functor ∆ X → Ch ≥ ( T ) ) which takes the value N . Then holim Y ≃ C • (Γ , N ) . Here C • (Γ , N ) is the cochain which computes the usual group cohomology. IMPLICIAL GALOIS DEFORMATION FUNCTORS 25
Proof.
By [Hir03, Lemma 18.9.1], holim Y is naturally isomorphic to the homotopy limit of the cosimplicialobject Z in Ch ≥ ( T ) whose codegree [ n ] term is Q σ ∈ X n Y σ = Q σ ∈ X n N . We have to explain the coface mapsof Z . For this purpose we describe Z = ( Z n ) n as follows:The T [Γ] -module N defines a functor D from the one-object groupoid • with End ( • ) = Γ to Ch ≥ ( T ) ,such that D ( • ) = N , and D (Γ) acts on N by the Γ -action. Then Z n is Q i →···→ i n D ( i n ) (all i k ’s are equal tothe object • here, but keeping the difference helps to clarify the process). Let d k be the k -th face map from Γ n +1 to Γ n , in other words, d k maps ( i → · · · → i n +1 ) to ( j → · · · → j n ) by "covering up" i k . Then thecorresponding D ( j n ) → D ( i n +1 ) is the identity map if k = n + 1 , and is D ( i n → i n +1 ) if k = n + 1 .By [Dug08, Proposition 19.10], holim Z is quasi-isomorphic to the total complex of the alternating doublecomplex defined by Z . Since each Z n is concentrated on degree , the total complex is simply · · · → Y Γ n N → Y Γ n +1 N → . . . and the alternating sum Q Γ n N → Q Γ n +1 N is exactly the one which occurs in computing group cohomology. Weconclude that holim Y ≃ holim Z ≃ C • (Γ , N ) . (cid:3) Now we can prove Proposition 4.19:
Proof.
From the above discussions, s D ρ T ( T ⊕ M [ j ]) corresponds to τ ≤ C • + j +1 (Γ , g T ⊗ T M ) under the Dold-Kan correspondence (with Ch ≥ ( T ) replaced by Ch ≤ ( T ) ). Hence π i s D ρ T ( T ⊕ M [ j ]) ∼ = H j − i (Γ , g T ⊗ T M ) . (cid:3) We can define the modifying-center version s D ρ T ,Z as in Section 2.4.1. Note the fibration sequence (see[GV18, (5.7)]) hofib( sHom s Sets ( X, B Z ( A )) → sHom s Sets ( X, BZ ( T ))) → s D ρ T ( A ) → s D ρ T ,Z ( A ) . From this,we deduce that π i s D ρ T ,Z ( T ⊕ M [ j ]) ∼ = π i s D ρ T ( T ⊕ M [ j ]) when i = j + 1 , and π i +1 s D ρ T ,Z ( T ⊕ M [ i ]) = 0 .For each v ∈ S p , there is also a modifying-center version s D ρ T,v ,Z , resp. s D n . o ρ T,v ,Z of s D ρ T,v , resp. s D n . o ρ T,v .Similarly to the global situation, we have: π i s D ρ T,v ,Z ( T ⊕ M [ j ]) ∼ = (cid:26) H j − i (Γ v , g T ⊗ T M ) when i = j + 1; H (Γ v , g T ⊗ T M ) / ( z T ⊗ T M ) when i = j + 1 . And π i s D n . o ρ T,v ,Z ( T ⊕ M [ j ]) ∼ = (cid:26) H j − i (Γ v , b T ⊗ T M ) when i = j + 1; H (Γ v , b T ⊗ T M ) / ( z T ⊗ T M ) when i = j + 1 . The global nearly ordinary derived deformation functor over ρ T is defined as follows: s D n . o ρ T ,Z = s D ρ T ,Z × h Q v ∈ Sp s D ρT,v,Z Y v ∈ S p s D n . o ρ T,v ,Z . Then π i s D ? ρ T ,Z ( T ⊕ M [ j ]) ( ? = n . o or ∅ ) depends only on j − i . We denote t j − iT,M s D ? ρ T ,Z = π i s D ? ρ T ,Z ( T ⊕ M [ j ]) . Proposition 4.23.
Suppose ( Reg v ) and ( Reg ∗ v ) . Let j ≥ i ≥ and let M be a finitely generated (classical) T -module. Then π i s D n . o ρ T ,Z ( T ⊕ M [ j ]) ∼ = H j − i n . o (Γ , g T ⊗ T M ) . Proof.
By preceding discussions, we have the exact sequence → t − T,M s D n . o ρ T ,Z → M v ∈ S p H (Γ v , b T ⊗ T M ) / ( z T ⊗ T M ) → M v ∈ S p H (Γ v , g T ⊗ T M ) / ( z T ⊗ T M ) → t T,M s D n . o ρ T ,Z → H (Γ , g T ⊗ T M ) ⊕ ( M v ∈ S p H (Γ v , b T ⊗ T M )) → M v ∈ S p H (Γ v , g T ⊗ T M ) → t T,M s D n . o ρ T ,Z → H (Γ , g T ⊗ T M ) → M v ∈ S p H (Γ v , b T ⊗ T M ) → t T,M s D n . o ρ T ,Z → Note that we have used H (Γ v , b T ⊗ T M ) = 0 for v ∈ S p . To see this, it suffices to show H (Γ v , b k ) = 0 byArtinian induction. By local Tate duality, it suffices to prove H (Γ v , b ∗ k ) = 0 . But we have a Galois-equivariantismomorphism b ∗ k ∼ = g k / n k (1) , so the result follows from the assumption ( Reg ∗ v ) .Under the condition ( Reg v ) , the map H (Γ v , b T ⊗ T M ) → H (Γ v , g T ⊗ T M ) is an isomorphism. Let L v,T,M = im( H (Γ v , b T ⊗ T M ) → H (Γ v , g T ⊗ T M )) , then we have the following exact sequence similar to( ⋆ ): → H . o (Γ , g T ⊗ T M ) → H (Γ , g T ⊗ T M ) → → H . o (Γ , g T ⊗ T M ) → H (Γ , g T ⊗ T M ) → M v ∈ S p H (Γ v , g T ⊗ T M ) /L v,T,M → H . o (Γ , g T ⊗ T M ) → H (Γ , g T ⊗ T M ) → M v ∈ S p H (Γ v , g T ⊗ T M ) → H . o (Γ , g T ⊗ T M ) → By comparing the two exact sequences above, we get t iT,M s D n . o ρ T ,Z ∼ = H i +1n . o (Γ , g T ⊗ T M ) . (cid:3) Recall that we have a pro-simplicial ring R s, n . o which represents s D n . o Z . Then ρ T defines a map R s, n . o → π R s, n . o → T. With this specified map, we regard R s, n . o ∈ pro − O \ s Art / T , and it’s easy to see that R s, n . o represents s D n . o ρ T ,Z .Write R s, n . o = ( R k ) for a projective system ( R k ) in O \ s Art / T . Then π i s D n . o Z ( T ⊕ M [ j ]) ∼ = π i lim −→ k sHom O \ s Art / T ( R k , T ⊕ M [ j ]) ∼ = π i lim −→ k DK( τ ≥ [ L R k , M [ j ]]) ∼ = H i lim −→ k [ L R k , M [ j ]] . Let’s define [ L R/ O , M ] = lim −→ k [ L R k / O , M ] for R = ( R k ) ∈ pro − O \ s Art / T . Then [ L R s, n . o / O , M ] , when regardedas a cochain complex, has the same cohomology groups as the complex τ ≥ C • +1n . o (Γ , g T ⊗ T M ) . We thus obtainthe following corollary: Corollary 4.24.
For every finite T -module M , there is a quasi-isomorphism [ L R s, n . o / O , M ] ≃ τ ≥ C • +1n . o (Γ , g T ⊗ T M ) . Comments:
Let ρ T : Γ → G ( T ) be an ordinary representation of weight µ , which satisfies ( Reg v ) for all v ∈ S p . This means that the cocharacter given by ρ T | I v : I v → B ( T ) /N ( T ) is given (via Artin reciprocity)by µ ◦ rec − v : I v → O × v → Θ( T ) (here Θ =
B/N is the standard maximal split torus of B ). In this whole IMPLICIAL GALOIS DEFORMATION FUNCTORS 27 section, if ρ T is ordinary of weight µ , we could consider instead of the functor s D n . o ρ T the subfunctor s D n . o ,µρ T ofordinary deformations of fixed weight µ . This means we impose as local condition at v ∈ S p that s D n . o ,µρ T,v ( A ) = hofib µ ◦ rec − v (cid:16) s D n . o ρ T,v ( A ) → sHom ( BI v , B Θ( A )) (cid:17) . Then, s D n . o ,µρ T is prorepresentable by a simplical pro-artinian ring R s, n . o µ and we have an analogue of Proposi-tion 4.23: Proposition 4.25.
Suppose ( Reg v ) and ( Reg ∗ v ) . Let j ≥ i ≥ and let M be a finitely generated (classical) T -module. Then π i s D n . o ,µρ T ,Z ( T ⊕ M [ j ]) ∼ = H j − i n . o ,str (Γ , g T ⊗ T M ) . Here H j − i n . o ,str (Γ , g T ⊗ T M ) is the cohomology of the subcomplex C • n . o ,µ (Γ , g T ⊗ T M ) defined as in Section4.4.1, replacing ( L v , e L v ) by ( L ′ v , e L ′ v ) where L ′ v is the image in H (Γ v , g T ⊗ M ) of the kernel of H (Γ v , b T ⊗ M ) → H ( I v , ( b T / n T ) ⊗ M ) , and e L ′ v is the inverse image of L ′ v in Z (Γ v , g T ⊗ M ) .The proof is identical to Proposition 4.23. As a corollary we get Corollary 4.26.
For every finite T -module M , there is a quasi-isomorphism [ L R s, n . o µ / O , M ] ≃ τ ≥ C • +1n . o ,µ (Γ , g T ⊗ T M ) . In the next section, we shall use these objects with a fixed weight µ .5. Application to the Galatius-Venkatesh homomorphism
Let
Γ = Gal( F S /F ) for S = S p ∪ S ∞ . Let ¯ ρ : Γ → G ( k ) be an ordinary representation of weight µ ,which satisfies ( Reg v ) for all v ∈ S p . Let T be a finite local O -algebra and ρ T : Γ → G ( T ) be an ordinarylifting of weight µ of ¯ ρ . Let M be a T -module which is of O -cofinite type, that is, whose Pontryagin dual Hom O ( M, K/ O ) is finitely generated over O . We use the notations of Definition 4.12. Recall that if ¯ ρ : Γ → G ( k ) is ordinary automorphic, it is proven under certain assumptions (see [CaGe18, Th.5.11] and [TU20,Lemma 11]) that H . o (Γ , g T ⊗ T M ) is finite and H . o ⊥ (Γ , g ∗ T ⊗ T M ) ∨ is of O -cofinite type. Let T n = T / ( ̟ n ) ;it is a finite algebra over O n = O / ( ̟ n ) . Let R = R s, n . o µ , which prorepresents simplicial ordinary deformationsof weight µ . We consider the simplicial ring homomorphism φ n : R → T n given by the universal property for the deformation ρ n = ρ m (mod ( ̟ n )) . Let T n = T / ( ̟ n ) ; it is a finitealgebra over O n = O / ( ̟ n ) . We consider the simplicial ring homomorphism φ n : R → T n given by the universal property for the deformation ρ n = ρ m (mod ( ̟ n )) . Let M n be a finite T n -module.Consider the simplicial ring Θ n = T n ⊕ M n [1] concentrated in degrees and up to homotopy. It is endowedwith a simplicial ring homomorphism pr n : Θ n → T n given by the first projection. Let L n ( R ) be the setof homotopy equivalence classes of simplicial ring homomorphisms Φ :
R → Θ n such that pr n ◦ Φ = φ n . ByProposition 4.25, there is a canonical bijection L n ( R ) ∼ = H . o ,str (Γ , g T n ⊗ T n M n ) . Moreover, as noticed in [GV18, Lemma 15.1], there is a natural map π ( n, R ) : L n ( R ) → Hom T ( π ( R ) , M n ) defined as follows. Let [Φ] be the homotopy class of Φ ∈ Hom T ( R , Θ n ) ; then π ( R )(Φ) is the homomorphismwhich sends the homotopy class [ γ ] of a loop γ to Φ ◦ γ ∈ Hom s Sets (∆[1] , M n [1]) = M n . Recall a loop γ is amorphism of s Sets γ : ∆[1] → Θ n from the simplicial interval ∆[1] to the simplicial set Θ n which sends the boundary ∂ ∆[1] to . For G = GL N and F a CM field (assuming Calegari-Geraghty assumptions), it is proven in [TU20] that Proposition 5.1.
For any n ≥ , the map π ( n, R ) is surjective. Then, we choose M n = Hom ( T, ̟ − n O / O ) ; we take the Pontryagin dual π ( n, R ) ∨ and apply Poitou-Tateduality H . o ,str (Γ , g T n ⊗ T n M n ) ∼ = H . o ,str (Γ , ( g T n ⊗ T n M n ) ∗ ) . We obtain a T -linear homomorphism called the mod. ̟ n Galatius-Venkatesh homomorphism: GV n : Hom T ( π ( R ) , M n ) ∨ ֒ → H . o ,str (Γ , ( g T n ⊗ T n M n ) ∗ ) The left hand side is π ( R ) ⊗ ̟ − n / O and the right hand side is Sel n . o ,str (Ad( ρ n )(1)) . Taking inductivelimit on both sides we obtain Proposition 5.2.
There is a canonical T -linear injection GV T : π ( R s, n . o ) ⊗ O K/ O ֒ → Sel(Ad( ρ T ) ∨ (1)) For G = GL N , F CM and under Calegary-Geraghty assumptions, and for T the non Eisenstein localizationof the Hecke algebra acting faithfully on the Betti cohomology, it follows from [CaGe18, Theorem 5.11] thatthe left-hand side is ̟ -divisible of corank rk ( T ) and it is proven in [TU20, Lemma 11] that the right-handside has corank rk ( T ) . For any O -finitely generated ordinary Γ -module M such that the Selmer group H . o ,str (Γ , M ⊗ Q / Z ) is O -cofinitely generated, we define its Tate-Shafarevitch module as X ( M ) = H . o ,str (Γ , M ⊗ Q / Z ) /H . o ,str (Γ , M ⊗ Q / Z ) ̟ − div . It is the torsion quotient of H . o ,str (Γ , M ⊗ Q / Z ) . For any O -algebra homomorphism λ : T → O , let ρ λ = ρ T ⊗ λ O . For M = Ad( ρ λ ) ∨ (1)) , one shows in [TU20, Lemma 11], using Poitou-Tate duality, that X (Ad( ρ λ ) ∨ (1))) is Pontryagin dual to Sel n . o ,str (Ad( ρ λ )) .It follows from [TU20, Lemma 11] that the cokernel of GV λ can be identified to the Tate-Shafarevitch group X (Ad( ρ λ ) ∨ (1)) in the sense of Bloch-Kato. So that Coker GV λ ∼ = Sel n . o ,str (Ad( ρ λ )) ∨ . References [A18] P. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne,Potential automorphy over CM fields, arXiv:1812.09999v1 [math.NT].[Cai20] Y. Cai, Derived minimal deformation rings for small groups, preprint[CHT08] L. Clozel, M. Harris, R. Taylor, Automorphy for some ℓ -adic lifts of automorphic modulo ℓ Galois representations,Publ. Math. Inst. Hautes Études Sci. (2008),[BHKT19] G. Boeckle, M. Harris, C. Khare, J. Thorne, b G -local systems on smooth projective curves are potentially automorphic,arXiv preprint arXiv:1609.03491 (2016) to appear in Acta Mathematica[BMR05] M. Bate, B. Martin, G. Rohrle, A geometric approach to complete reducibility . Inventiones mathematicae, 2005,161(1): 177-218[CaGe18] F. Calegari, D. Geraghty, Modularity lifting beyond the Taylor-Wiles method, Invent. Math. 211, pp 297-433 (2018)[CR05] A. M. Cegarra, J. Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set,Topology and its Applications 153.1 (2005): 21-51.[Dug08] D. Dugger,
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