p-Jets of Finite Algebras, II: p-Typical Witt Rings

We study the structure of the p-jet spaces of the p-typical Witt rings of the p-adics. We also study the p-jets of the comonad map. These data can be viewed as an arithmetic analogue, for the ring Z, of the Lie groupoid of the line and hence as an infinitesimal version of the Galois group of Q over “F1”. 2010 Mathematics Subject Classification: 13 K 05

1. Introduction 1.1.Motivation.This paper is the second in a series of papers where we investigate p-jet spaces (in the sense of [6]) of finite flat schemes/algebras.The understanding of such p-jet spaces seems to hold the key to a number of central questions about arithmetic differential equations [7].This paper is logically independent of its predecessor [8].In [8] we dealt with the case of p-divisible groups; in the present paper we investigate the case of algebras of Witt vectors of finite length.Another example of a class of finite algebras whose p-jet spaces are arithmetically significant is that of Hecke algebras; we hope to undertake the study of this example in a subsequent work.The present paper is partly motivated by the quest for "absolute geometries" (the so-called "geometries over the field with one element, F 1 "); cf.[12] for an overview of various approaches and some history.In particular, according to Borger's approach [3], the geometry over F 1 should correspond to λ-geometry (i.e.algebraic geometry in which all rings appearing come equipped with a structure of λ-ring in the sense of Grothendieck).For the case of one prime p the "p-adic completion" of λ-geometry is the δ-geometry developed by the author [6,7], where δ is a p-derivation (morally a "Fermat quotient operator").Now Borger established in [3] an elegant categorical framework which predicts what actual objects should correspond to the basic hypothetical constructions over F 1 .According to his framework the hypothetical tensor product Z ⊗ F1 Z (which was one of the first objects sought in the quest for F 1 ) should correspond to the big Witt ring W(Z) of the integers.Then the hypothetical groupoid structure on Z⊗ F1 Z should correspond to the commonad structure ∆ : W(Z) → W(W(Z)).The main interest in Z ⊗ F1 Z comes from the fact that this tensor product should be viewed as an arithmetic analog of a surface X × X where X is a curve (algebraic, analytic, C ∞ ).With this analogy in mind one is immediately tempted to ask for an arithmetic analogue of the Lie groupoid of X, in the sense of Lie and Cartan, and more recently Malgrange [11].(Recall that, roughly speaking, a point of the Lie groupoid of X is by definition a pair of points of X together with a formal isomorphism between the germs of X at these two points.)Since the Lie groupoid of X is the infinitesimal version of an automorphism group we should view any arithmetic analogue of the Lie groupoid of X as an infinitesimal version of the "Galois group of Q/F 1 ".Now the Lie groupoid of X is an open set in the inverse limit, as n → ∞, of the manifolds J n (X × X/X) of "n-jets of formal sections, at various points, of the second projection X × X → X".Since the arithmetic analogue of the second projection X × X → X is the structure morphism Spec W(Z) → Spec Z, one candidate for an arithmetic analogue of the Lie groupoid of X could be the jet spaces (in the sense of [4]) of the Witt ring W(Z) over Z.We will not recall the definition of these jet spaces here (because we don't need it) but let us note that they are constructed using derivations and knowing them essentially boils down (in this easy case) to knowing the Kähler differentials Ω W(Z)/Z .By the way, the module of Kähler differentials Ω W(Z)/Z is also the starting point for the construction of the deRham-Witt complex of Z [9].However using Kähler differentials (equivalently usual derivations) arguably looks like "going arithmetic only half way".What we propose in this paper is to "go arithmetic all the way" and consider p-jet spaces (in the sense of [6]) of Witt rings rather than usual jet spaces (in the sense of [4]) of the same Witt rings.The former are an arithmetic analogue of the latter in which usual derivations are replaced by p-derivations.A few adjustments are in order.First since p-jet spaces are "local at p" we replace the big Witt ring functor W( ) by the p-typical Witt functor W ( ).Also we replace Z by Z p or, more generally, by the Witt ring R = W (k) on a perfect field k of characteristic p. Finally since W (R) is not of finite type over R we replace W (R) by its truncations W m (R) (where we use the labeling in [1], so W m (R) = R m+1 as sets.)So after all what we are going to study are the p-jet algebras J n (W m (R)) and the p-jet maps induced by the comonad maps ∆ : W m+m ′ (R) → W m (W m ′ (R)); cf. the review of J n and W m in the next subsection.Since W m (R) and W m (W m ′ (R)) are finite flat R-algebras our investigation here is part of the more general effort to study p-jets J n (C) of finite flat R-algebras C; the case when Spec C is a finite flat p-group scheme was addressed in [8].
1.2.Main concepts and results.For R = W (k) the Witt ring on a perfect field k of characteristic p = 2, 3 we let φ = W (Frob) be the automorphism of R defined by the p-power Frobenius Frob of k.(The main examples we have in mind are the ring of p-adic integers Z p = W (F p ) and the completion of the maximum unramified extension of Z p , Z ur p = W (F a p ); here F a p is the algebraic closure of F p .)Let x, x ′ , x ′′ , ..., x (n) , ... be families of variables x = (x α ) α∈Ω , x ′ = (x ′ α ) α∈Ω , etc., indexed by the same set Ω, and let φ : R[x, x ′ , x ′′ , ...] → R[x, x ′ , x ′′ , ...] be the unique endomorphism extending φ on R and satisfying φ(x for r ≥ 0. Following [6] we define the map of sets (referred to as a p-derivation) δ : R[x, x ′ , x ′′ , ...] → R[x, x ′ , x ′′ , ...] by the formula Then for any R-algebra C = R[x]/(f ), where f is a family of polynomials, we define the p-jet algebras of C: Note that each J n+1 (C) has a natural structure of J n (C)-algebra and we have naturally induced set theoretic maps δ : (For C of finite type over R we also defined in [6] the p-jet spaces of Spec C as the formal schemes J n (Spec C) := Spf (J n (C))ˆwhere ˆmeans p-adic completion; these spaces are very useful when one further looks at non-affine schemes but here we will not need to take this step.) We need one more piece of terminology.First, for any ring B and element b ∈ B we let B = B/pB and we let b ∈ B be the image of b.Assume now the finitely generated R-algebra C comes equipped with an R-algebra homomorphism C → R which we refer to as an augmentation.Then there is a unique lift of the augmentation to an R-algebra homomorphism J ∞ (C) → R that commutes with δ.Composing the latter with the natural homomorphism J n (C) → J ∞ (C) and reducing mod p we get an induced homomorphism J n (C) → k.Let P n be the kernel of the latter.Consider the ring J n (C) ′ defined (up to isomorphism) by asking that Spec J n (C) ′ be the connected component of Spec J n (C) that contains the prime ideal P n ; we refer to J n (C) then we call J n (C) ′′ the complement of the identity component of J n (C). Clearly Let now C be the Witt ring W m (R), m ≥ 1. Recall that W m (R) is the set R m+1 equipped with the unique ring structure which makes the ghost map w : R m+1 → R m+1 , w i (a 0 , ..., a m ) = a p i 0 + pa p i−1 1 + ...+ p i a i , a ring homomorphism.Let v i = (0, ..., 0, 1, 0, ..., 0) ∈ W m (R), (1 preceded by i zeroes, i = 1, ..., m), set π = 1 − δv 1 ∈ J 1 (W m (R)), and let Ω m = {1, ..., m}.As usual we denote by π the ring of fractions of J n (W m (R)) with denominators powers of π.The ring W m (R) comes with a natural augmentation W m (R) → R given by the first projection.So we may consider the identity component of J n (W m (R)).The following gives a compete description of this component and also shows this component is ) is idempotent and we have an isomorphism

Indeed by the theorem
Next let C be one of the iterated Witt rings W m (W m ′ (R)), m, m ′ ≥ 1, (cf. the next section for more details).Set There is a natural augmentation of W m (W m ′ (R)) given by composing the two obvious first projections.Then we have the following complete description for the identity component of ) is idempotent and we have an isomorphism sending each δ r v i,i ′ into the class of the variable x Again the theorem shows that 1−Π are isomorphic to the identity component, respectively to the complement of the identity component, of J n (W m (W m ′ (R))).Finally we have the following complete description of the reduction mod p of the map induced by the comonad map: and into the class of Remark 1.4.The above results give a complete description of the identity components of our objects.On the other hand one can ask for a description of the complements of the identity components.Take for instance J n (W m (R)).This is not a group object so the components different from the identity component cannot be expected to necessarily "look like" the identity component.And this is indeed what happens (in spite of the comonad structure floating around): 1−π of the identity component looks quite differently (more degenerate) than the identity component J n (W m (R)) π .Indeed the identity component is a polynomial ring in m variables over a local Artin ring (cf.Theorem 1.1) and hence has Krull dimension m; by contrast, for the complement of the identity component, we have: Remark 1.6.The simplicity modulo p of all these p-jet rings and maps may be deceptive.The structure of these objects in characteristic zero is actually extremely complicated and, as in [8], the whole point of this paper is to manage the complexity of the situation in such a way that, eventually, mod p, the situation becomes transparent.On a conceptual level the results of this paper are best understood as an attempt to unravel the "differential geometry" of the "automorphisms of Z over F 1 "; cf. the beginning of our Introduction.The objects introduced and studied in the present paper could then be viewed as an "infinitesimal" replacement (at p) for the elusive absolute Galois group of Q over F 1 .
1.3.Plan of the paper.In Section 2 we review (and give some complements to) the basic theory of Witt vectors.Section 3 is devoted to computing J n (C) for certain finite flat R-algebras C whose structure constants satisfy some simple divisibility/vanishing axioms.These axioms are in particular satisfied in the cases C = W m (R) and C = W m (W m ′ (R)).Using this we derive, in Section 4, the main results of this note, stated above.
1.4.Acknowledgment.This material is based upon work supported by the National Science Foundation under Grant No. 0852591, and by IHES, Bures sur Yvette, and MPI, Bonn.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation, IHES, or MPI.Also the author is indebted to James Borger for explaining some of his insights (and also some technical points) about Witt vectors.

Witt rings
In this section we review some basic facts about the rings of Witt vectors which we are going to need in the sequel.For most proofs we refer to [9] and [1].However, for the convenience of the reader, we will provide proofs for the facts for which we could not find an explicit reference.Note that the labeling of Witt rings in [9] and [1] are different (W m in [9] is W m−1 in [1]); we follow here the labeling in [1].Fix a prime p and m a non-negative integer or ∞.For any ring A we may consider the ghost maps w i : Then there is a unique functor W m from the category of rings to itself such that, for any ring A, we have that W m (A) = A m+1 as sets and the ghost map w = (w 0 , ..., w m ) : W m (A) = A m+1 → A m+1 is a ring homomorphism where the target A m+1 is given the product ring structure.We use the convention The rings W m (A) are called the (p-typical) rings of Witt vectors of length m + 1.
These maps are related by the following identities: It is also convenient to introduce the maps m (a) = (0, ..., 0, a, 0, ..., 0) where a is preceded by i zeroes.We have the identities: w(V i m (a)) = (0, ..., 0, p i a, p i a p , p i a p 2 , ...).Also, for any N ∈ Z we have the following formula for the Teichmüller map [9]: where c 0 (N ) = N and If A is p-torsion free so is W m (A).Now if A is p-torsion free and φ : A → A is a ring homomorphism lifting the p-power Frobenius on A/pA then there is a unique ring homomorphism λ φ : A → W m (A) such that w i (λ φ (a)) = φ i (a) for all i; if in addition A/pA is perfect then λ φ induces an isomorphism A/p m+1 A ≃ W m (A/pA).
Lemma 2.1.Let A be a p-torsion free ring equipped with a ring automorphism φ : A → A lifting the p-power Frobenius on A/pA.Let 0 ≤ m < ∞ and view W m (A) as an A-algebra via the homomorphism Proof.The case A = Z p is in [1].The general case is similar but for convenience we recall the argument.If w : W m (A) → A m+1 , by (2.2) and by the injectivity of φ, we have that w(v i ) are A-linearly independent in A m+1 (the latter viewed as an A-algebra via (1, φ, ..., φ m ) : A → A m+1 ).Hence v i are A-linearly independent.To check that v i span W m (A) we proceed by induction on m.For m = 0 this is clear.Assume spanning holds for m − 1.The kernel of the map Proof.The equalities We are left to prove that This follows from the following computation: −→ W m (B).If B is a finitely generated A-algebra (respectively a finite A-module) then W m (B) is also a finitely generated A-algebra (respectively a finite A-module).
Proof.The ghost map w : W m (B) → B m+1 is injective and integral.Now if B is a finitely generated A-algebra (respectively a finite A-module) then so is B m+1 (with the A-algebra structure given by (1, φ, ..., φ m ) : A → A m+1 → B m+1 ), because φ is bijective.In the finite case, by Noetherianity W m (B) is a finite A-algebra.In the finitely generated case it follows that B m+1 is finite over W m (B) and hence, by the Artin-Tate lemma W m (B) is a finitely generated A-algebra.
Next we discuss iterated Witt vectors.One proves (cf.e.g.[9]) that F : W (A) → W (A) lifts the p-power Frobenius on W (A)/pW (A).So for A ptorsion free, since W (A) is also p-torsion free, we have at our disposal a ring homomorphism ∆ = λ F : W (A) → W (W (A)) which composed with any ghost map w i : W (W (A)) → W (A) equals the i-th iterate F i .Then one trivially checks that the composition where, if we write the elements of (A ∞ ) ∞ as ((a00, a01, a02, ...), (a10, a11, a12, ...), (a20, a21, a22, ...), ... Using this plus the injectivity of the map w one immediately checks that if ).So we have induced ring homomorphisms These homomorphisms (and ∆) were constructed for A p-torsion free but, as usual, one extends this construction uniquely to all rings in a functorial manner.Also one immediately checks (composing with ghost maps) that the following diagram is commutative: (2.4) Lemma 2.4.Let R = W (k) be the Witt ring on a perfect field of characteristic p and φ : R → R the Frobenius.Let 0 ≤ m, m ′ < ∞.Then: ) is a finite R-algebra, where the structure morphism is given by R Proof.The first assertion follows from Lemma 2.3.The second assertion follows from the "coassociativity" property in [9], p 15. Lemma 2.5.For any a ∈ A, s ∈ Z + , and 0 ≤ i ≤ m < ∞ we have the following formula in W m (A): Proof.
Lemma 2.6.Let R = W (k), k a perfect field of characteristic p ≥ 5, φ : R → R the lift of Frobenius on R, u : R → A a p-torsion free finite R-algebra (e.g. , in which case ν = i ′ ).Then for any s ≥ 0 and 0 ≤ i ≤ m we have (For i = m the sum in the right hand side is, by definition, zero.) Also set M i m = N i m = 0 for i > m.Since W m (A) is a finite R-algebra the modules M i m and N i m are finitely generated.By Lemma 2.5, for s ≥ 1 we have because for p ≥ 5, s ≥ 1, t ≥ 1, ν ≥ 1, r ≥ 0 we have p s−1 |c t (p s ) and p 2 |c r (p (p t −1)ν ).In particular because M i m is a finitely generated R-module and hence N i m is p-adically separated.So M i m = N i m .So for all s ≥ 0 we have where the latter is viewed as an R-algebra via the map Proof.First it is trivial to check that the composition Next note that the images of w ∞ (w(v i,i ′ )) in (R m ′ +1 ) m+1 are R-linearly independent (where (R m ′ +1 ) m+1 is an R-algebra via the map (2.6)); indeed the matrix with the first i rows and i ′ columns zero.The assertion of the Lemma now follows.
Lemma 2.8.Let R = W (k), k a perfect field of characteristic p ≥ 5.For 0 ≤ i ≤ m < ∞ and 0 ≤ i ′ ≤ m ′ < ∞ let v i,i ′ be as in ( 2.5).Then for 0 ≤ i ≤ j ≤ m, i ′ , j ′ ∈ {0, ..., m ′ }, and 1 ≤ t ≤ m − j there exist unique elements c ii ′ jj ′ t ∈ R such that the following equalities hold in W m (W m ′ (R)): Proof.Uniqueness of the c's follows from Lemma 2.7.Let us prove the existence of the c's.We have Proof.Linear independence was proved in Lemma 2.7.To prove generation we fix m ′ , set A = W m ′ (R), and proceed by induction on m.The case m = 0 is Lemma 2.1.For the induction step we need to show that the kernel of the map W m (A) → W m−1 (A) (which equals V m (A) = {(0, ..., 0, a); a ∈ A}) is generated as an R-module by v m,j ′ .By Lemma 2.1 the R-module A is generated by the v j ′ 's and note that v m,j ′ = V m (v j ′ ).So to conclude it is enough to show that the map V m : A φ m → V m (A) is an isomorphism of R-modules where which by the way is given by a → (0, ..., 0, p m a)), is an R-module homomorphism where A m+1 is an R-algebra via the map (λ φ ) m+1 • (1, φ, ..., φ m ) : R → R m+1 → A m+1 .This is however trivial to check.
Lemma 2.10.With notation as in Lemmas 2.1 and 2.9 the comultiplication where a i,i ′ ,i ′′ ∈ Z. Moreover we have the following relations: Note that the relations above allow one to recurrently determine all the coefficients a i,i ′ ,i ′′ .
Proof of Lemma 2.10.Let K = F rac(R) and let M (i ′′ ) be the linear subspace of the space of all (m+1)×(m ′ +1)-matrices (K m ′ +1 ) m+1 consisting of all matrices (r i,i ′ ) with r i,i ′ = 0 for i + i ′ < i ′′ .Since the elements w ∞ (w(v i,i ′ )) ∈ M (i ′′ ) for i + i ′ ≥ i ′′ and since these elements are K-linearly independent it follows that these elements form a basis of M (i ′′ ).By (2.2) and ( 2.3) we have that with i ′′ zeros on the first line.So w ∞ (w(∆(v i ′′ ))) belongs to M (i ′′ ), hence we get an equality as in (2.8) with a i,i ′ ,i ′′ ∈ K and relation 1) holding.Since v i,i ′ form a basis of W m (W m ′ (R)) we get that a i,i ′ ,i ′′ ∈ R. Picking out the (i, i ′ )-entry in (2.8) and using (2.7) we get the relation (2.10) Relations 2) follows immediately.Relation 3) follows by induction.To prove relation 6) subtract the equality (2.10) with i ′ replaced by i ′ − 1 from the equality (2.10) and divide by p i+i ′ .Relation 4) for i = 0 follows by induction from (2.10).Relation 4) for arbitrary i follows by induction from 6).Relations 1), 2), and 6) imply relation 5) by induction.By 1), 2), and 5) we have Remark 2.11.It is easy to see directly from the definitions that for any Z palgebra C we have an isomorphism of Q p -algebras which for any c ∈ C sends c φ s := φ s (c) into 1 ⊗ ... ⊗ 1 ⊗ c ⊗ 1 ⊗ ... ⊗ 1 (c on position s with positions labled from 0 to n).Hence we have Q p -algebra isomorphisms (2.11) where N means N -fold product in the category of rings.If we set v 0 = 1 and Hence the isomorphism (2.11) is defined by the family of orthogonal idempotents (2.12) In particular the whole complexity of J n (W m (Z p )) disappears after tensorization with Q p and hence it is an "integral" phenomenon.On the other hand, by the above, the Z p -algebra is a free Z p -module of rank (m + 1) n+1 that retains most (but not all) of the complexity of J n (W m (Z p )).For instance if one considers the surjection (2.13) then the target of this surjection is an F p -vector space of dimension (m + 1) n+1 whereas the source of this surjection is, by Theorem 1.1, an infinite dimensional F p -vector space; in fact this source, J n (W m (Z p )), is a product of two algebras: the identity component of J n (W m (Z p )) and the complement of the identity component.By Theorem 1.1, the identity component is a polynomial algebra in over an Artin local subring of J n (W m (Z p )) whose dimension as an F p -vector space is 2p n−1 m n .Indeed one can take as an F p -vector space basis for this Artin ring the elements It is interesting to compare the two families (2.12) and (2.14).

Documenta Mathematica 18 (2013) 971-996
Remark 2.12.We end by discussing the link between p-jets and Witt vectors.The discussion that follows will be helpful to set up notation for later and to put things into the right perspective.However, the adjunction properties that will be explained below have, by themselves, very little impact on the unraveling of the structure of p-jet spaces.
The following concept was introduced independently by Joyal [10] and the author [6].A p-derivation from a ring A into an A-algebra B is a map of sets δ : A → B such that the map A → W 1 (B), a → (a, δa) is a ring homomorphism.
(Here we identify a with a•1 B .)A δ-ring is a ring A equipped with a p-derivation A → A. The category of δ-rings is the category whose objects are the δ-rings and whose morphisms are the ring homomorphisms that commute with δ.By definition a p-derivation δ : where C p is the polynomial: If δ is as above then φ : A → B, φ(x) = x p + pδx, is a ring homomorphism.Note that δ(xy) = x p δy + φ(y)δx = y p δx + φ(x)δy.Also δ and φ commute.If A is p-torsion free then δ is, of course, uniquely determined by φ; also where Note that for any ring A the ring W (A) has a structure of δ-ring which functorial in A; it is given by the composition W (A) According to a result of Joyal [10] (which will not be needed in the sequel) for any ring A and any δ-ring B we have where ! is the forgetful functor from δ-rings to rings.More generally if R is a δ-ring by a δ-ring over R we shall mean a δ-ring equipped with a δ-ring homomorphism from R into it.Similarly a ring over R will mean an R-algebra.Then the above adjunction property implies that for any δ-ring B over R and any ring A over R, where W (A) is an R-algebra via R → W (R) → W (A). Let now R = W (k) with k a perfect field of characteristic p. Recall that for any R-algebra we defined in the Introduction R-algebras J n (C) and J ∞ (C).The set theoretic maps δ : J n (C) → J n+1 (C) and δ : J ∞ (C) → J ∞ (C) constructed in the Introduction are then p-derivations and we have the following adjunction property: for any δ-ring D over R and any ring C over R we have Putting together the two adjunction properties above we get for any rings A and C over R. It is sometimes useful to use a universality property that is more refined than that of J ∞ .To that purpose let us define a prolongation sequence to be a sequence Note that for any ring A over R the morphisms ∆ : This is a prolongation sequence over R because of the φ-linearity of F : So by the universality property for prolongation sequences we have a natural (compatible) family of R-homomorphisms: Documenta Mathematica 18 (2013) 971-996 for i = 1, ..., m, where v 0 = 1.Indeed it is enough to show this for A = R in which case this follows from Lemmas 2.1 and 2.2.Finally note that by the commutativity of (2.4) if m ′ is fixed and m varies the morphisms ∆ : ) fit into a morphism of prolongation sequences.This induces commutative diagrams (2.18) Remark 2.13.If the upper row of the diagram 2.18, for m, m ′ , n variable, is viewed as the "Lie groupoid of the integers" (i.e. an arithmetic analogue, for the integers, of the Lie groupoid of the line) then one is tempted to view the bottom row of the above diagram as an analogue of a "subgroupoid" of that "Lie groupoid".However this candidate for a "subgroupoid" is contained in the "complement of the identity component" of the "Lie groupoid of integers"; cf.Remark 4.7.

p-jets and p-triangular bases
Let R be any ring, and let C be a commutative unital R-algebra, equipped with an R-algebra homomorphism C → R. Let C + be the kernel of this homomorphism and assume C + is a free R-module of finite rank.Let {v α ; α ∈ Ω} be an R-basis of C + where Ω is a finite set equipped with a total order ≤.
for α ≤ β, where c αβγ ∈ R. Let x be a collection of variables x α indexed by α ∈ Ω and Proof.Indeed the source is generated as an R-module by 1 and the classes of x α so the algebra map above is injective (because 1 and the v α 's are linearly independent) and surjective (because 1 and the v α 's generate C).
For the rest of this section, we assume that C + possesses a p-triangular basis v α , α ∈ Ω.We also assume R = W (k) for k a perfect field of characteristic p ≥ 3. n) ].We start by recalling some filtrations from [8].Let Also let Consider the ideal I [p] ⊂ A generated by all elements of the form pf and f p where f ∈ I; equivalently I [p] ⊂ A is the ideal generated by all elements of the form px (j) α and (x (j) α ) p where α ∈ Ω, j ≥ 0. It is trivial to check (cf.[8]) that For any set S let us denote by [S] an arbitrary element of S. In particular for our algebra C and the V -basis v α of C + , and Q αβ depends only on the variables x γ with γ ≥ α.
Finally let Q (n) ⊂ A n be the ideal generated by Note that if F, G ∈ A n and F ≡ G mod Q (n−1) A n then δF ≡ δG mod Q (n) A n+1 .
Here is our main computation in characteristic zero.
Theorem 3.3.Assume C + has a p-triangular basis and let Q αβ and Q (n) ⊂ A n be as above.Then for n ≥ 1 and α ≤ β we have the congruences in the ring A n where Proof.Note that x s α ≡ p 2 [I] mod Q (0) for s ≥ 3. We get the following congruences mod Q (0) A 1 : Using the fact that δp ≡ 1 mod p we get the following congruences mod Q (1) A 2 : Using the fact that the 5 terms above are in A {1} ∩ I [p] , the fact that φδ = δφ, and the fact that δ((x ′ i ) p ), δ((x ′ i ) p (x ′ j ) p ) ∈ pA 2 ∩ I [p] ⊂ A {1} ∩ I [p] we get the following congruence mod Q (2) A 3 : Finally using the same kind of computation as for δ 3 Q αβ one proves by induction on n that for n ≥ 3 we have the following congruence mod Q (n−1) A n : )+[A {n−2} ∩I [p] ].
The fact that F αβn only depends on the variables indexed by γ ≥ α follows simply from the fact that for any pair α ≤ β we can make the computations above in the rings with variables indexed by indices γ ≥ α.In other words for n ≥ 2 factors through the complement of the identity component !
rather than through the identity component J n (W m (R)) π ; this makes the problem more subtle.
1 for all n.We denote by B * a prolongation sequence as above.A morphism of prolongation sequences B * = (B n ) and C * = (C n ) is by definition a sequence of morphisms B n → C n that commute, in the obvious sense, with the ϕs and the δs.Clearly, for any ring C, J * (C) := (J n (C)) is naturally a prolongation sequence.Moreover, for any prolongation sequence D * = (D n ) and any ring C we have Hom rings (C, D 0 ) ≃ Hom prol.seq.(J * (C), D * ).Finally consider the prolongation sequence R * = (R n ) where all R n are R = W (k), k a perfect field of characteristic p, and all ϕ are the identity.By a prolongation sequence over R we understand a morphism of prolongation sequences R * → B * ; we have a natural notion of morphism of prolongation sequences over R * .Clearly, for any ring C over R, J * (C) := (J n (C)) is naturally a prolongation sequence over R.Moreover, for any prolongation sequence D * = (D n ) over R and any ring C over R we have Hom rings/R (C, D 0 ) ≃ Hom prol.seq./R(J * (C), D * ).

Remark 4 . 7 .
It would be interesting to have an explicit understanding of the homomorphisms s :J n (W m (R)) → W m−n (R) in(2.16), or at least of their reduction mod p.This involves understanding the iterates of formula 4.2.Note however that by formula 4.2 it follows that s(π) = 1 − (1 − p p−2 ρ(v 1 )) ∈ pW m−1 (R).