Canonical Witt formal scheme extensions and p-torsion groups

We study the nth arithmetic jet space of the p-torsion subgroup attached to a smooth commutative formal group scheme. We show that the nth jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length n and the p-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.


Introduction
Buium in [8] introduced the theory of arithmetic jet spaces on (formal) abelian schemes over p-adic rings and showed that the jet spaces of an abelian scheme A are naturally affine fibrations over A. Since then the theory of arithmetic jet spaces has been developed in several articles such as [1], [9], [12], [6], [7], [16], and has found remarkable applications in diophantine geometry as in [8] and [13].
In this paper, we study the structure of the jet space functors associated to the ptorsion subgroup G[p ∞ ] of a smooth commutative formal group scheme G over a fixed p-adic basis.Here we show that for any n, the n-th jet space J n (G[p ∞ ]) is canonically an extension of G[p ∞ ] by a power of the unipotent formal group scheme W n−1 , where W n−1 is A n , the n-dimensional formal affine space endowed with the group scheme structure of the additive Witt vectors of length n.This generalizes results obtained by Buium in [11] for G the multiplicative group scheme.
Before stating our main result in detail, let us introduce some notation.Let K be a finite extension of Q p with ramification index e, uniformizer π and ring of integers O.We denote by k the residue field of O and let q be its order.Then the identity map of O is a lift of q-Frobenius.Fix a π-adically complete π-torsion free O-algebra R with a lifting of Frobenius φ, i.e., an endomorphism of R such that φ(r) − r q ∈ πR for all r ∈ R. As an example, consider the ring of restricted power series O x with φ the O-algebra endomorphism given by φ(x) = x q .Let W n be the functor of ramified Witt vectors of length n + 1 (following Borger's convention, details in § 2.2).
Let Nilp R be the category of R-algebras on which π (or equivalently, p) is nilpotent.
Its opposite category is a site with respect to the Zariski topology.Any adic R-algebra A with ideal of definition I containing π gives rise to a sheaf (of sets) Spf(A) such that Given a sheaf X on Nilp op R we define its n-th π-jet by (1.1) for any C ∈ Nilp R .If X is a formal scheme over R the same is J n X and it holds (1.2) Hom R (Spf(C), J n X) = Hom R (Spf(W n (C)), X) [8], [4], [7], [2].
For a smooth commutative formal group scheme G over R there is a short exact sequence of formal group schemes as sheaves on Nilp op R ; see Lemma 4.2.
Consider the natural projection map u : denote the kernel of u.Buium in [11,Corollary 1.2] shows that if e = 1, p > 2 and G is the formal multiplicative group scheme over R, then the sheaf representable by a formal R-scheme and is isomorphic to A n , the n-dimensional affine space over Spf(R).
In this paper, we will enrich Buium's result and extend it to any smooth commutative formal R-group scheme G of relative dimension d ≥ 1.In fact, in Theorem 3.18 we A n endowed with the group structure of Witt vectors of length n.Further, we deduce the following result (see Theorem 4.8).
Main Theorem.Assume p ≥ e+2.Given a smooth commutative formal group scheme G of relative dimension d over R, for any positive integer n the natural morphism We remark that by Lemma 4.
] as sheaves on Nilp op R ; hence there is no possible ambiguity in the above statement.
1.1.Plan of the paper.In Section 2 we recall the definition and properties of π-jets in the setting of formal schemes, with particular attention to the adjunction between jet algebras and Witt vectors (2.14).
In Section 3 we focus on the notion of shifted Witt vectors W + n , introduced by [7], and show that W + n induces an adjoint functor to N n (Theorem 3.8).This is an important result, analogous to the adjunction formula that involves W n and J n (1.2).Then we show that given a smooth formal R-group scheme G of dimension d, we have ii) Assume p ≥ e+2.Then there is a natural isomorphism of formal group schemes The proof of the first fact reduces to a local computation in coordinates which is detailed in the Appendix section.Another important ingredient is the notion of lateral Frobenius introduced in [7].Both results i) and ii) are generalized to the case of m-shifted Witt vectors in [17] by the third author.
In Section 4 we apply the previous results to the study of the sheaves and N n (G[p ∞ ]) and deduce a statement similar to Theorem 3.18 where G is replaced by G[p ∞ ], see Theorem 4.8.
In this paper all rings are assumed to be commutative with unit and Alg R denotes the category of R-algebras, i.e., of ring homomorphisms R → B.

Arithmetic jets
2.1.Conventions.Let R be the base ring fixed in the introduction.Given a formal scheme X and a fixed point x : Spf(R) → X, one can consider the fibre of x under the natural map J n X → X, which is the closed formal subscheme Note that if G is a formal group scheme then J n G is naturally a formal group scheme too and we set N n G = (J n G) ε = ker(J n G → G) to be the fibre along the unit section ε.
If X is a functor on Alg R , we will usually denote by X the restriction of X to Nilp R .
Let R x 1 , ..., x n be the π-adic completion of the R-polynomial algebra in n variables.
Let G a := Spf(R x ) be the additive formal group scheme over R. Note that this formal group scheme should not be confused with the (x)-adic formal group G for a = Spf(R[[x]]), the formal completion of G a along the zero section.
] is a commutative formal group law of dimension g, let F{n} be the formal group law given by π −n F(π n x, π n y), for any n ≥ 1.Note that F{n} endows Spf(R x ) with a structure of formal group scheme over R.
If B is an R-algebra, ρ = ρ B : R → B always denote the corresponding ring homomorphism.If the context is clear, we will write r in place of ρ(r) ∈ B.

2.2.
Witt vectors over R. In the following pages W n denotes the functor of π-typical Witt vectors of length n+1 on R-algebras.Hence, for any R-algebra B, the ring W n (B) is always considered with its natural R-algebra structure, which depends on φ.We explain this briefly.
As functor on O-algebras W n coincides with the so-called functor of ramified Witt vectors of length n + 1 (see [15], [3]).Let w : W n (R) → n i=0 R be the ghost map.Then for any Witt vector a = (a 0 , . . ., a n ), w(a) = (w 0 (a), . . ., w n (a)) where w i are the ghost polynomials (2.1) Since R has a lifting of Frobenius φ, by the universal property of Witt vectors there exists a ring homomorphism of O-algebras exp δ making the following diagram commute (see [2, (2.9)]) In [7, §3.2] the authors give an equivalent construction of the functor W n .The ghost map w in (2.2) is O-linear, but not R-linear in general, if the ring n i=0 R is endowed with the direct product R-module structure.It is then preferable to change the Rmodule structure on the product ring so that w becomes R-linear.Let be the direct product algebra.Its underlying ring is i B and there is a commutative diagram of R-algebras Then Frobenius and Verschiebung maps can be described in terms of ghost components as in the case of ramified Witt vectors, with caution when considering the R-algebra structure.As for example, the Frobenius ring homomorphism described in terms of ghost components as the left shift is φ-semilinear.We prefer then to write it as the homomorphism of R-algebras corresponding to the homomorphism of R-algebras Similarly, the Verschiebung map V : W n (B) → W n+1 (B) is described on ghost components as the right shift multiplied by π.Clearly it is O-linear but not R-linear in general.We prefer then to write it as the homomorphism of R-modules corresponding to the homomorphism of R-modules Then F V is multiplication by π on W n ( φ B).
Since φ might not be invertible, one can not write B in place of φ B in (2.5) and (2.7).However, since φ is the identity on O, the O-module structure on B and φ B are the same.
Remark 2.9.If B is a π-adic R-algebra (by this we mean π-adically complete and separated) then the same is W n (B) for any n.The proof works as in [19, Proposition 3].

Shifted Witt vectors.
We recall the construction of 0-shifted Witt vectors as introduced in [7] and [17].Here we simply refer to them as shifted Witt vectors.The general theory of m-shifted Witt vectors is developed in [17].
Let B be an R-algebra and set-theoretically define Also define the product ring where n φ (B) was introduced in (2.3).Note that there is an isomorphism of R-algebras Define the apriori set-theoretic map w + : for all i = 0, . . ., n.Then note that W + n (B) naturally is endowed with the Witt ring structure of addition and multiplication making w + a ring homomorphism.Hence we have the following commutative diagram where pr 0 is the projection onto the 0-th component.The R-algebra W + n (B) was denoted by Wn (B) in [7, §4] and by W 0n (B) in [17].
Since the lower horizontal arrows in (2.11) are homomorphisms of R-algebras, the same are the upper horizontal arrows.Hence the left hand square in (2.11) is a diagram of R-algebras and, up to the above identifications, it can be illustrated as (2.12) where we have written r in place of ρ(r) in B and w i are the ghost polynomials in (2.1).

Prolongation sequences.
For any formal schemes Y and Z over Spf(R) we say A morphism of prolongation sequences T * → P * is a system of morphisms of formal schemes f n : T n → P n that satisfy the expected commutations: Then the fixed π-derivation δ on R makes S * into a prolongation sequence.Let C S * denote the category of prolongation sequences defined over S * .
2.5.Jet spaces.Given a formal scheme X over S 0 = Spf(R), Buium constructs the canonical prolongation sequence J * X = {J n X} ∞ n=0 where J 0 X = X and by [9, Proposition 1.1], J * X satisfies the following universal property: for any Moreover, by [5] and [2] we have the following functorial description: for any C ∈ Nilp R .In particular in the affine case with X = Spf(A) and J n X = Spf(J n A), we have a natural adjunction such that w 0 • Θ(g) = g • ι with ι : A → J n A the natural morphism.
Here we make the above adjunction explicit when X = Spf(R x ) is the formal affine line over Spf(R).Let A = R x .Then J n X = Spf(J n A) and (2.15) where x, x ′ , . . ., x (n) are the Buium-Joyal coordinates and p 0 , . . .p n are the Witt coordinates and they satisfy p 0 = x, p 1 = x ′ while the general relation between the above two coordinate systems can be found in [2, Proposition 2.10].
Note that when G = G a we have the following isomorphism of formal group schemes (2.16) where W n is A n+1 endowed with the additive group structure of Witt vectors of length n + 1.

The Kernel as a π-jet space
For any sheaf of groups G on Nilp op R one defines where u is the natural morphism.Scope of this section is to take a closer look at the kernel N n G in the case G is representable by a smooth formal scheme.Since the kernel is the fibre at the unit section, we will first consider more general fibres.
Let X be a smooth formal scheme over R with a marked point a and let u : in particular, N n X is formal affine and isomorphic to N n U .Up to shrinking U , we may assume U = Spf(A) with A a π-adically complete separated R-algebra.Then the point 3.1.Adjunction.Let Alg + R denote the category of augmented (commutative) Ralgebras.Its objects are commutative R-algebras A together with an augmentation ε, i.e. an R-algebra morphism ε : A → R; morphisms in Alg + R are morphisms of Ralgebras h : A 1 → A 2 respecting augmentations, i.e. ε 2 • h = ε 1 .For any (A, ε) in Alg + R we define the R-algebra Note that shifted Witt vectors yield objects in Alg + R .Indeed, let B be an R-algebra and let w + 0 : W + n (B) → R denote the projection onto the first component, i.e., the composition of the upper horizontal arrows in (2.11).Then W + n (B) together with w + 0 is an augmented R-algebra.Hence the above construction defines a functor on the category of R-algebras.
We now prove a key result: N n and W + n is a pair of adjoint functors.
Theorem 3.4.For any augmented R-algebra (A, ε) and any R-algebra B there is a natural bijection , where the first equality follows by (3.3), the second by (2.14) taking f = Θ(g), the third by definition of W + n (B) in (2.10).
By Remark 2.9 an analogous adjunction holds when working with the category of augmented formal R-algebras fAlg + R .We make this fact explicit.
The higher dimensional case is analogous.Let A = R x with x a collection of r indeterminates {x 1 , . . ., x r } and let ε be the zero section.Then J n A ≃ R p 0 , p 1 , . . ., p n where p i denotes a collection of polynomials {p i,1 , . . ., p i,r } and p i,j ∈ R[x j , x ′ j , ..., x j ] plays the role of p i in (2.15).Then (3.7) N n A ≃ R p 0 , . . .p n ⊗R x ,ε R ≃ R p + 1 , . . ., p + n , where p + i denotes the collection of polynomials {p + i,1 , . . ., p + i,r } with p + i,j obtained by evaluating p i,j at x j = 0. Finally for a homomorphism g : N n A → B, Θ + (g) maps x i to (0, g(p + 1,i ), . . ., g(p + n,i )).
We can now describe the functor N n X on R-algebras as done in (1.2) for J n X.
Theorem 3.8.Let X be a smooth formal scheme over R with a marked point x and let B be in Nilp R .Then , where on the right we are considering morphisms of R-pointed formal schemes.
Proof.The result is clearly true if X is affine by Theorem 3.4.For the general case, assume first that X is an R-scheme and consider the following diagram Spec(B) where w 0 is induced by the projection on the first component on algebras, ρ is the structure morphism and u is the natural map.Note that Spec(W + n (B)) is the pushout of w, ρ in the category of all schemes [18, 07RS].Then N n X(B) = X(W + n (B)).Indeed, If X is a formal scheme, then the above holds for all schemes X × Spec R/(π m ) and thus one concludes.

A special case.
Let G be a formal group scheme over R and denote by G for the formal completion of G along the unit section.Let F ∈ R[[x 1 , . . ., x r , y 1 , . . ., y r ]] be the formal group law on G for , F φ the one obtained by acting on the coefficients of F by φ and F φ {1} := π −1 F φ (πx, πy), where πx := (πx 1 , . . ., πx r ).By [8, Lemma 2.2] it is We give below a direct computation of this fact.
Lemma 3.9.Let the notation be as above.Then the formal group law on the formal completion of N 1 G at the origin is isomorphic to F φ {1}.
Proof.As seen in Remark 3.6, we may write ] be the π-derivation compatible with that of R and such that δ(x) = x ′ , δ(y) = y ′ .If F(x, y) is the formal group law of G, the formal group law of N 1 G is δ(F(x, y)) evaluated at x = 0, y = 0. Write F(x, y) = α,β a α,β x α y β with α, β varying in N r {0}, x α := x α1 1 x α2 1 . . .x αr r , and the coefficients of the monomials of degree 1 equal to 1.By induction, applying the usual rules of π-derivations, one checks that 3.3.Lateral Frobenius.Let X be a formal R-scheme.As n varies, the π-jet spaces J n X form an inverse system of formal schemes and, more precisely, a prolongation sequence, whence a lifting of Frobenius φ J exists on the limit.Clearly, the transition maps u = u n+1 n : J n+1 X → J n X induce homomorphisms N n+1 X → N n X, but the image of φ restricted to N n+1 X is not necessarily contained in N n X and hence φ does not induce a lifting of Frobenius on the sequence of the kernels.For this reason, the notion of lateral Frobenius was introduced and studied in [7,17].
On shifted Witt vectors, the lateral Frobenius F + is defined as the homomorphism of R-algebras making the following diagram (3.10) W + n (B) commute, where vertical identifications are meant as sets, and F is the Frobenius on Witt vectors recalled in (2.5).The homomorphism F + then corresponds to the homomorphism of R-algebras via ghost map, i.e., it makes the following diagram (3.12) commute.Then, the homomorphism F + induces via (2.14) a natural morphism called again lateral Frobenius.It is showed in [7,Theorem 4.3] that f is a lift of Frobenius and satisfies where u denotes the immersion N m X → J m X and φ denotes the Frobenius morphism J m X → J m−1 X for any m.
Remark 3.13.For later use, note that the element (0, b .) = (0, b 1 , . . ., b n ) ∈ W + n (B) traces in (3.12) the following images Hence πc 1 = πb q 1 + π 2 b 2 implies c 1 = b q 1 + πb 2 (for B without π-torsion and hence for any R-algebra B by standard arguments).By recursion one sees If G is smooth over R, the same are J n G and N n G for all n.As seen in the previous section, N n G = Spf(N n A) is an affine space over R. In particular the Ralgebras N n A are π-torsion free and therefore the lateral Frobenius homomorphisms (1) Assume G = G a = Spf(R x ) with the comultiplication map- where the group law is described by with the group law described by ) and the group law on the latter maps m {1} as formal group schemes.Now, G for m {1}, as formal group law, has invariant differential (1 + πT ) −1 dT and the corresponding logarithm is Assume p ≥ e + 1.We prove that π −1 log(1 + πT ) ∈ O[[T ]] and indeed in O T .It suffices to check that v π (a j ) ≥ 0 tends to infinity as j tends to infinity.Let r > 0 and note that v π (a p r ) = p r − 1 − er ≥ 0 since Further v π (a p r ) < v π (a p r+1 ) and for p r ≤ j < p r+1 we have It is an isomorphism under the stronger hypothesis that p ≥ e + 2. Indeed the inverse of π −1 log(1 + πT ) is and the π-adic valuation of the j-th coefficient is where s p (j) denotes the sum of the digits in the base-p expansion of j.Clearly if p ≥ e + 2 this valuation tends to infinity as j tends to infinity and hence and, with arguments as in (1), that The next result is an extension of [8, Lemma 2.3].
Lemma 3.17.Let F be a commutative formal group law over R of dimension d.If n(p − 1) ≥ e + 1 then F{n} ≃ ( G a ) d as formal group schemes over R.
Proof.In [8, Lemma 2.3] R is a complete discrete valuation ring with algebraically closed residue field.The proof in our hypothesis works the same.Indeed Buium applies results in [15] that are valid for any Z (p) -algebra and the key-point is showing that the coefficients of the logarithm and exponential series of F{n} over R[1/p] are indeed in πR.This is done by explicit estimates for the π-valuation of those coefficients.
Theorem 3.18.Let G be a smooth commutative formal group scheme of relative dimension d over Spf(R).Assume p ≥ e + 2. Then there is a natural isomorphism of formal group schemes Proof.Let G for be the formal completion of G along the unit section Spf(R) → G. Let be the formal group law on G for , F φ the one obtained by acting the coefficients of F by φ and F φ {1} : = π −1 F φ (πx ., πy .).By Lemma 3.9 (see also [8,Lemma 2.2]) it is N 1 G ≃ F φ {1} as formal group schemes.Note that since φ(π) = π it is F φ {1} = (F{1}) φ .Now by hypothesis and Lemma 3.17 Remark 3.19.Assume R = O, p > 2 and let G be as in the previous theorem.Then passing to limit on n we have a short exact sequence .We can naturally pass to limit on ν.
Lemma 4.2.Let the notation be as above.Then as sheaves on Nilp op R .
Proof.Recall (1.1) and (4.1).For any C in Nilp R it is Whence the first isomorphism.The second one follows by the fact that J n is left exact (being a right adjoint) and hence Recall that N n F is defined as ker(J n F → F ) for any sheaf of groups on Nilp op R .We then deduce from Lemma 4.2 the following result.Lemma 4.3.Let notation be as above.Then as sheaves on Nilp op R .
Proof.By Theorem 3.8 the functor N n is left exact; hence We say that a commutative formal R-group scheme is triangular if it admits a finite filtration by formal subgroup schemes whose successive quotients are isomorphic to G a .
It is called triangular of level 0 in [8, p. 322 for any n ≥ 0.
Proof.Only the right exactness needs to be proved.Let s be a section of G such that p m s = 0.It lifts to a section s ′ of J n G. Now p m s ′ comes from a section of N n G and thus is p-power torsion by the previous theorem.Hence s ′ is p-power torsion.
We are now ready to proof the Main Theorem stated in the introduction.
Proof.This follows directly by applying Theorems 3.18, 4.5 and Corollary 4.6.
We can now conclude the study of Examples 3.16.
(1) Assume G = G a .We have seen in Example 3.16(1) that ) is isomorphic to G a = Spf(R x as formal group scheme mapping x to x ′ on algebras.In particular, for any ν ≥ 1, it is This result can be checked directly.Indeed G a [p ν ] = Spf(R x /(p ν x)) and hence Proof.The second assertion is [2, Proposition 2.10].The proof of the first one is similar and we write below the main steps: Hence cancelling the common terms on both sides of the above equality we get π i+j q n−1−i j x q(q n−1−i −j) i and one can divide by π n .
We now prove that prolongation sequences B * as above satisfying condition (A.6) for all n are unique up to unique isomorphism.
Theorem A.8. Let B * be the prolongation sequence in (A.3).Assume that the indeterminates x i satisfy (A.6) for all n and let z i ∈ J ∞ R[x 0 ] be the elements defined just below (A.5).Then we have (i) The inclusion R[z 0 , . . ., z n ] → J n R[x 0 ] is an isomorphism for any n.
(ii) For any n ≥ 0 the R-algebra homomorphism h n : 0 → ∂ i x 0 is an isomorphism and the following square R x 0 , . . ., x Proof.The first assertion was proved in [2, Lemma 2.20] with z i = P i (x).Commutativity of the squares is immediate by definition of h n .We are then left to prove that h n is an isomorphism.Note hat J n R[x 0 ] = R[z 0 , . . ., z n ] by point (i) and B n = R[x 0 , . . ., x n ].If we prove that h n (z i ) = x i , for all 0 ≤ i ≤ n, the result is clear.

Spfn
Hom R (A/I n , B) , for any B in Nilp R .By a formal scheme over R we mean a sheaf on Nilp op R admitting an open cover by open subfunctors of the type Spf(A) for A as above.
which we call the canonical Witt formal scheme extension of G because of Theorem 3.18 below.Let G[p ν ] be the p ν -torsion formal subgroup scheme of G and let G[p ∞ ] denote the sheaf on Nilp op R such that G[p ∞ ](C) = lim − →ν G[p ν ](C), for any C ∈ Nilp R .For each ν the closed immersions G[p ν ] ֒→ G[p ν+1 ] induce closed immersions of π-jets J n (G[p ν ]) ֒→ J n (G[p ν+1 ]) and φ n B denote the ring B with the R-algebra structure induced by ρ B • φ n : R → B, and let (2.3) be an étale chart around a, where x denotes here a finite family of indeterminates.Hence U is an open affine formal subscheme of X, u is étale, a factors through u and u • a is the zero section 0 of the affine space A. By [10, Proposition 3.13 & Corollary 3.16] (see also [2, Proposition 3.12]), we have J n U ≃ J n A × A U for all n and hence (3.1) Here we discuss examples for G = G a and G m in the context of Theorem 3.15 Examples 3.16.
we have F{1} ≃ ( G a ) d .Hence F{1} φ ≃ (( G a ) d ) φ = ( G a ) d and hence N 1 G ≃ ( G a ) d .By Theorem 3.15 and definition of J n−1 it is is the reduction map, and we recover the fact that the kernel of the reduction map is isomorphic to O d = W (k) d as groups.4.p-power torsionFor any formal commutative R-group scheme G let G[p ν ] denote the kernel of the multiplication by p ν on G and let G[p ∞ ] be the sheaf on Nilp op R such that(4.1)G[p ∞ ](C) = lim − → ν G[p ν ](C) for any C in Nilp R .Note that G[p ∞ ]is a sheaf since the above colimit commutes with equalizers [18, 04AX].If G is a formal torus or a formal abelian scheme G[p ∞ ] is p-divisible, but not in general.For each ν > 0 the closed immersions G[p ν ] ֒→ G[p ν+1 ] induce closed immersions of π-jets J n (G[p ν ]) ֒→ J n (G[p ν+1 ]) [9, Proposition 1.7], [2, Lemma 3.8, Theorem 3.9]

Theorem 4 . 8 .
Assume p ≥ e + 2. Given a smooth commutative formal group scheme G of relative dimension d over R, for any positive integers n the natural morphism J n G → G gives an exact sequence 0 R [x 0 , . . ., x n ] (u,∂) / / R [x 0 , . . ., x n+1 ]commutes, where u denotes the inclusion map on both levels.
].The formal R-group scheme W n is triangular.Lemma 4.4.If H is a triangular formal R-group scheme then H[p ∞ ] = H as sheaves on Nilp op R .Proof.We proceed by induction on the length m of the filtration.The result is clearly true for H = G a since p is nilpotent in any object C of Nilp R .Assume m > 1.Then H is extension of G a by G where G is a triangular formal group scheme with a filtration of length m − 1.One concludes by induction hypothesis.Moreover, we have the following stronger result.Theorem 4.5.Assume G is a smooth commutative formal R-group scheme and p ≥ e + 2. Then N n (G[p ∞ ]) = N n G as sheaves on Nilp op R .Proof.By Theorem 3.18, the formal R-group scheme N n G is triangular.Hence the result follows by Lemmas 4.3 and 4.4.