The big de Rham-Witt complex

Lars Hesselholt

Introduction

The big de Rham-Witt complex was introduced by the author and Madsen in [12] with the purpose of giving an algebraic description of the equivariant homotopy groups in low degrees of Bökstedt’s topological Hochschild spectrum of a commutative ring. This functorial algebraic description, in turn, is essential for understading algebraic K-theory by means of the cyclotomic trace map of Bökstedt-Hsiang-Madsen [2]; compare [13,11,8]. The original construction, which relied on the adjoint functor theorem, was very indirect and a direct construction has been lacking. In this paper, we give a new and explicit construction of the big de Rham-Witt complex and we also correct the 2-torsion which was not quite correct in the original construction. The new construction is based on a theory, which is developed first, of modules and derivations over a λ-ring. The main result of this first part of the paper is that the universal derivation of a λ-ring is given by the universal derivation of the underlying ring together with an additional structure that depends directly on the λ-ring structure in question. In the case of the universal λ-ring, which is given by the ring of big Witt vectors, this additional structure consists in divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes possible the new direct construction of the big de Rham-Witt complex. This is carried out in the second part of the paper, where we also show that the big de Rham-Witt complex behaves well with respect to étale morphisms. Finally, we explicitly evaluate the big de Rham-Witt complex of the ring of integers.

In more detail, let A be a ring, always assumed to be commutative and unital. The ring 𝕎(A) of big Witt vectors in A is equipped with a natural action by the multiplicative monoid ℕ of positive integers with the action by n ∈ ℕ given by the nth Frobenius map

F_{n}: 𝕎(A) → 𝕎(A).

Lars Hesselholt Nagoya University, Nagoya, Japan E-mail: larsh@math.nagoya-u.ac.jp

The Frobenius maps give rise to a natural ring homomorphism

Δ: 𝕎(A) → 𝕎(𝕎(A))

whose Witt components Δ_{e}: 𝕎(A) → 𝕎(A) are characterized by the formula

F_{n}(a) = ∑_{e|n}eΔ_{e}(a)^{n/e}.

The triple (𝕎(−),Δ,ϵ) is a comonad on the category of rings and a λ-ring in the sense of Grothendieck [9] is precisely a coalgebra (A,λ_{A}) over this comonad.

Recently, Borger [5] has proposed that a λ-ring structure λ_{A}: A → 𝕎(A) on the ring A be considered as descent data for the extension of ℤ over a deeper base 𝔽_{1}. This begs the question as to the natural notions of modules and derivations over (A,λ_{A}). We show that the general approach of Quillen [20] leads to the following answer. Recall that if (A,λ_{A}) is a λ-ring, then the ring A is equipped with an ℕ-action with the action by n ∈ ℕ given by the nth associated Adams operation

ψ_{A,n}: A → A

that is defined by the formula

ψ_{A,n}(a) = ∑_{e|n}eλ_{A,e}(a)^{n/e}

where λ_{A,e}: A → A is the eth Witt component of λ_{A}: A → 𝕎(A). Now, the category of (A,λ_{A})-modules is identified with the category of left modules over the twisted monoid algebra A^{ψ} [ℕ] whose product is defined by the formula

n · a = ψ_{A,n}(a) · n.

Hence, an (A,λ_{A})-module is a pair (M,λ_{M}) that consists of an A-module M and an ℕ-indexed family of maps λ_{M,n}: M → M such that λ_{M,n} is ψ_{A,n}-linear, λ_{M,1} = id_{M}, and λ_{M,m}λ_{M,n}= λ_{M,mn}. Moreover, we identify the derivations

D: (A,λ_{A}) → (M,λ_{M})

with the derivations D: A → M that satisfy

λ_{M,n}(Da) = ∑_{e|n}λ_{A,e}(a)^{(n/e)−1} Dλ_{A,e}(a).

It is now easy to show that there is a universal derivation

d: (A,λ_{A}) → (Ω_{(A,λ_{A})},λ_{Ω_{(A,λ_{A})}} ).

We prove the following result.

Theorem A. For every λ-ring (A,λ_{A}), the canonical map

Ω_{A}→ Ω_{(A,λ_{A})}

is an isomorphism of A-modules.

It follows that for a λ-ring (A,λ_{A}), the A-module of differentials Ω_{A} carries the richer structure of an (A,λ_{A})-module. In the case of (𝕎(A),Δ_{A}), this implies that there are natural F_{n}-linear maps F_{n}: Ω_{𝕎(A)}→ Ω_{𝕎(A)}defined by

F_{n}(da) = ∑_{e|n}Δ_{e}(a)^{(n/e)−1} dΔ_{e}(a)

such that F_{1} = id, F_{m}F_{n} = F_{mn}, dF_{n}(a) = nF_{n}(da), and F_{n}(d[a]) = [a]^{n−1} d[a]. The p-typical analog of F_{p} was also constructed by Borger and Wieland in [6, 12.8].

The construction of the de Rham-Witt complex begins with the following variant of the de Rham complex. The ring 𝕎(ℤ) contains exactly the four units ±[±1] all of which are square roots of [1] and the 2-torsion element

d log[−1] = [−1]^{−1} d[−1] = [−1]d[−1] ∈ Ω_{𝕎(A)}

plays a special role. We define the graded 𝕎(A)-algebra

Ω̂_{𝕎(A)}^{·} = T_{𝕎(A)}^{·}Ω_{𝕎(A)}/J^{·}

to be the quotient of the tensor algebra of the 𝕎(A)-module Ω_{𝕎(A)} by the graded ideal J^{·}generated by all elements of the form

da ⊗ da − d log[−1] ⊗ F_{2}da

with a ∈ 𝕎(A). It is an anti-symmetric graded ring which carries a unique graded derivation d that extends d: 𝕎(A) → Ω_{𝕎(A)}and satisfies

ddω = d log[−1] · dω.

Moreover, the maps F_{n}: 𝕎(A) → 𝕎(A) and F_{n}: Ω_{𝕎(A)} → Ω_{𝕎(A)} extend uniquely to a map of graded rings F_{n}: Ω_{𝕎(A)}^{·}→ Ω_{𝕎(A)}^{·} which satisfies dF_{n} = nF_{n}d. Next, we show that the maps d and F_{n} both descend to the further quotient

Ω̌_{𝕎(A)}^{·} = Ω̂_{𝕎(A)}^{·} /K^{·}

by the graded ideal generated by all elements of the form

F_{p}dV_{p}(a) − da − (p − 1)d log[−1] · a

with p a prime number and a ∈ 𝕎(A). We now recall the Verschiebung maps

V_{n}: 𝕎(A) → 𝕎(A)

which are additive and satisfy the projection formula

aV_{n}(b) = V_{n}(F_{n}(a)b).

These maps, however, do not extend to Ω̌_{𝕎(A)}^{·}, and the de Rham-Witt complex is roughly speaking the largest quotient

η: Ω̌_{𝕎(A)}^{·} → 𝕎Ω_{A}^{·}

such that the Verschiebung maps extend to 𝕎Ω_{A}^{·} and such that the extended maps F_{n} and V_{n} satisfy the projection formula. The precise definition given in Section 4 below is by recursion with respect to the quotients 𝕎_{S}(A) of 𝕎(A) where S ranges over the finite subsets S ⊂ ℕ that are stable under division. We further prove the following result to the effect that the de Rham-Witt complex may be characterized as the universal example of an algebraic structure called a Witt complex, the precise definition of which is given in Definition 4.1.

Theorem B. There exists an initial Witt complex S ↦ 𝕎_{S}Ω_{A}^{·} over the ring A. In addition, the canonical maps

η_{S}: Ω̌_{𝕎_{S}(A)}^{q} → 𝕎_{S}Ω_{A}^{q}

are surjective, and the diagrams

\xymatrix{Ω̌_{𝕎_{S}(A)}^{q} \ar[r]^{η_{S}} \ar[d]^{R_{T}^{S}} & 𝕎_{S}Ω_{A}^{q} \ar[d]^{R_{T}^{S}} & Ω̌_{𝕎_{S}(A)}^{q} \ar[r]^{η_{S}} \ar[d]^{d} & 𝕎_{S}Ω_{A}^{q} \ar[d]^{d} & Ω̌_{𝕎_{S}(A)}^{q} \ar[r]^{η_{S}} \ar[d]^{F_{m}} & 𝕎_{S}Ω_{A}^{q} \ar[d]^{F_{m}} \\ Ω̌_{𝕎_{T}(A)}^{q} \ar[r]^{η_{T}} & 𝕎_{T}Ω_{A}^{q} & Ω̌_{𝕎_{S}(A)}^{q+1} \ar[r]^{η_{S}} & 𝕎_{S}Ω_{A}^{q+1} & Ω̌_{𝕎_{S/m}(A)}^{q} \ar[r]^{η_{S/m}} & 𝕎_{S/m}Ω_{A}^{q}}

commute.

If A is an 𝔽_{p}-algebra and S = {1,p,...,p^{n−1} }, then 𝕎_{S}Ω_{A}^{·} agrees with the original p-typical de Rham-Witt complex W_{n}Ω_{A}^{·}of Bloch-Deligne-Illusie [16]. More generally, if A is a ℤ_{(p)}-algebra and S = {1,p,...,p^{n−1} }, then 𝕎_{S}Ω_{A}^{·} agrees with the p-typical de Rham-Witt complex W_{n}Ω_{A}^{·} constructed by the author and Madsen [14] for p odd and by Costeanu [7] for p = 2. Finally, if 2 is either invertible or zero in A and S is arbitrary, then 𝕎_{S}Ω_{A}^{·}agrees with the big de Rham-Witt complex introduced by the author and Madsen [12]. We also note that if f: R → A is a map of ℤ_{(p)}- algebras and S = {1,p,...,p^{n−1} }, then the relative p-typical de Rham-Witt complex W_{n}Ω_{A/R}^{·}of Langer-Zink [18] agrees with the quotient of 𝕎_{S}Ω_{A}^{·} by the differential graded ideal generated by the image of 𝕎_{S}Ω_{R}^{1}→ 𝕎_{S}Ω_{A}^{1}.

We recall that van der Kallen [22, Theorem 2.4] and Borger [3, Theorem B] have proved independently that for every étale morphism f: A → B and every finite subset S ⊂ ℕ stable under division, the induced morphism

𝕎_{S}(f): 𝕎_{S}(A) → 𝕎_{S}(B)

again is étale. Based on this theorem, we prove the following result.

Theorem C. Let f: A → B be an étale map and let S ⊂ ℕ be a finite subset stable under division. Then the induced map

𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}→ 𝕎_{S}Ω_{B}^{q}

is an isomorphism.

To prove Theorem C, we verify that the left-hand terms form a Witt complex over the ring B and use Theorem B to obtain the inverse of the map in the statement. The verification of the Witt complex axioms, in turn, is significantly simplified by the existence of the divided Frobenius on Ω_{𝕎(A)}as follows from Theorem A.

Finally, we evaluate the de Rham-Witt complex of ℤ. The result is that 𝕎Ω_{ℤ}^{q} is non-zero for q ⩽ 1 only. Moreover, we may consider 𝕎Ω_{ℤ}^{·} as the quotient

Ω_{𝕎(ℤ)}^{·} → 𝕎Ω_{ℤ}^{·}

of the de Rham complex of 𝕎(ℤ) by a differential graded ideal generated by elements of degree 1. Hence, following Borger [5], we may interpret 𝕎Ω_{ℤ}^{·} as the complex of differentials along the leaves of a foliation of Spec(ℤ) considered as an 𝔽_{1}-space. Moreover, this foliation is compatible with the canonical action of the multiplicative monoid ℕ on the 𝔽_{1}-space Spec(ℤ).

As mentioned earlier, the big de Rham-Witt complex was introduced in [12] with the purpose of giving an algebraic description of the equivariant homotopy groups

TR_{q}^{r}(A) = [S^{q}∧ (𝕋/C_{r})_{+} ,T (A)]_{𝕋}

of the topological Hochschild 𝕋-spectrum T (A) of the ring A. Here 𝕋 = ℝ/ℤ is the circle group, C_{r} ⊂ 𝕋 is the subgroup of order r, and [−,−]_{𝕋} is the abelian group of maps in the homotopy category of orthogonal 𝕋-spectra. We proved in [10, Section 1] that the groups TR_{q}^{r}(A) give rise to a Witt complex over the ring A in the sense of Definition 4.1 below. Therefore, by Theorem B, there is a unique map

𝕎_{⟨r⟩}Ω_{A}^{q}→ TR_{q}^{r}(A)

of Witt complexes over A, where ⟨r⟩ denotes the set of divisors of r. We will show elsewhere that this map is an isomorphism for all r and all q ⩽ 1.

1 Witt vectors

We begin with a brief review of Witt vectors and λ-rings. In the approach taken here, all necessary congruences are isolated in Dwork’s lemma. The reader is also referred to the very readable account by Bergman [19, Appendix] and to the more modern and general exposition by Borger [3].

Let ℕ be the set of positive integers. We say that S ⊂ ℕ is a truncation set if whenever n ∈ S and d is a divisor in n, then d ∈ S. The big Witt ring 𝕎_{S}(A) is defined to be the set A^{S} equipped with a ring structure such that the ghost map

w: 𝕎_{S}(A) → A^{S}

that takes the vector (a_{n} | n ∈ S) to the sequence ⟨w_{n} | n ∈ S⟩ with

w_{n} = ∑_{d|n}da_{d}^{n/d}

is a natural transformation of functors from the category of rings to itself. Here the target A^{S} is considered a ring with componentwise addition and multiplication. To prove that there exists a unique ring structure on 𝕎_{S}(A) characterized in this way, we first prove the following result. We write v_{p}(n) for the p-adic valuation of n.

Lemma 1.1 (Dwork). Suppose that for every prime number p, there exists a ring homomorphism ϕ_{p}: A → A with the property that ϕ_{p}(a) ≡ a^{p} modulo pA. Then the sequence ⟨x_{n} | n ∈ S⟩ is in the image of the ghost map

w: 𝕎_{S}(A) → A^{S}

if and only if x_{n} ≡ ϕ_{p}(x_{n/p}) modulo p^{v_{p}(n)}A for every prime number p every n ∈ S with v_{p}(n) ⩾ 1.

Proof. We first show that if a ≡ b modulo pA, then a^{p^{v−1}} ≡ b^{p^{v−1}} modulo p^{v}A. If we write a = b + pϵ, then

a^{p^{v−1}} = b^{p^{v−1}} + ∑_{1⩽i⩽p^{v−1}} \binom{p^{v−1}}{i} b^{p^{v−1} −i} p^{i}ϵ^{i}.

In general, the p-adic valuation of the binomial coefficient \binom{m+n}{n} is equal to the number of carrys in the addition of m and n in base p. So

v_{p}( \binom{p^{v−1}}{i} p^{i}) = v − 1 + i − v_{p}(i) ⩾ v

which proves the claim. Now, since ϕ_{p} is a ring-homomorphism,

ϕ_{p}(w_{n/p}(a)) = ∑_{d|(n/p)}dϕ_{p}(a_{d}^{n/pd})

which is congruent to ∑_{d|(n/p)}da_{d}^{n/d} modulo p^{v_{p}(n)}A. If d divides n but not n/p, then v_{p}(d) = v_{p}(n) and hence this sum is congruent to ∑_{d|n}da_{d}^{n/d} = w_{n}(a) modulo p^{v_{p}(n)}A as stated. Conversely, if ⟨x_{n} | n ∈ S⟩ is a sequence such that x_{n} ≡ ϕ_{p}(x_{n/p}) modulo p^{v_{p}(n)}A, we find a vector a = (a_{n} | n ∈ S) with w_{n}(a) = x_{n} as follows. We let a_{1} = x_{1} and assume, inductively, that a_{d} has been chosen, for all d that divides n, such that w_{d}(a) = x_{d}. The calculation above shows that the difference

x_{n}− ∑_{d|n,d≠n}da_{d}^{n/d}

is congruent to zero modulo p^{v_{p}(n)}A. Hence, we can find a_{n} ∈ A such that na_{n} is equal to this difference.

Proposition 1.2. There exists a unique ring structure such that the ghost map

w: 𝕎_{S}(A) → A^{S}

is a natural transformation of functors from rings to rings.

Proof. Let A be the polynomial ring ℤ[a_{n},b_{n}| n ∈ S]. The unique ring homomorphism

ϕ_{p}: A → A

that maps a_{n} to a_{n}^{p} and b_{n} to b_{n}^{p} satisfies ϕ_{p}(f) = f^{p} modulo pA. Let a and b be the vectors (a_{n} | n ∈ S) and (b_{n} | n ∈ S). Then Lemma 1.1 shows that the sequences w(a) + w(b), w(a) · w(b), and −w(a) are in the image of the ghost map. Hence, there are sequences of polynomials s = (s_{n} | n ∈ S), p = (p_{n} | n ∈ S), and ι = (ι_{n} | n ∈ S) such that w(s) = w(a) + w(b), w(p) = w(a) · w(b), and w(ι) = −w(a). Moreover, as A is torsion free, the ghost map is injective, so these polynomials are unique.

Let now A′ be any ring, and let a′= (a′_{n} | n ∈ S) and b′= (b′_{n} | n ∈ S) be two vectors in 𝕎_{S}(A′). Then there is a unique ring homomorphism f: A → A′such that 𝕎_{S}(f)(a) = a′and 𝕎_{S}(f)(b) = b′. We define a′+b′= 𝕎_{S}(f)(s), a·b = 𝕎_{S}(f)(p), and −a = 𝕎_{S}(f)(ι). It remains to prove that the ring axioms are verified. Suppose first that A′is torsion free. Then the ghost map is injective, and hence, the ring axioms are satisfied in this case. In general, we choose a surjective ring homomorphism g: A′′→ A′from a torsion free ring A′′. Then

𝕎_{S}(g): 𝕎_{S}(A′′) → 𝕎_{S}(A′)

is again surjective, and since the ring axioms are satisfied on the left-hand side, they are satisfied on the right-hand side.

If T ⊂ S are two truncation sets, then the forgetful map

R_{T}^{S}: 𝕎_{S}(A) → 𝕎_{T}(A)

is a natural ring homomorphism called the restriction from S to T. If n ∈ ℕ and S ⊂ ℕ is a truncation set, then S/n = {d ∈ ℕ | nd ∈ S} again is a truncation set.

Lemma 1.3. The nth Verschiebung map V_{n}: 𝕎_{S/n}(A) → 𝕎_{S}(A), whose mth component is a_{d}, if m = nd, and 0, otherwise, is an additive map.

Proof. The following diagram, where V_{n}^{w} takes the sequence ⟨x_{d} | d ∈ S/n⟩ to the sequence whose mth component is nx_{d}, if m = nd, and 0, otherwise, commutes.

\xymatrix{𝕎_{S/n}(A) \ar[r]^{w} \ar[d]^{V_{n}} & A^{S/n} \ar[d]^{V_{n}^{w}} \\ 𝕎_{S}(A) \ar[r]^{w} & A^{S}}

Since V_{n}^{w} is additive, so is V_{n}. Indeed, if A is torsion free, the horizontal maps are both injective, and hence V_{n} is additive in this case. In general, we choose a surjective ring homomorphism g: A′→ A and argue as in the proof of Proposition 1.2.

Lemma 1.4. There exists a unique natural ring homomorphism

F_{n}: 𝕎_{S}(A) → 𝕎_{S/n}(A)

called the nth Frobenius that makes the following diagram, where the map F_{n}^{w} takes the sequence ⟨x_{m}| m ∈ S⟩ to the sequence whose dth component is x_{nd}, commute.

\xymatrix{𝕎_{S}(A) \ar[r]^{w} \ar[d]^{F_{n}} & A^{S} \ar[d]^{F_{n}^{w}} \\ 𝕎_{S/n}(A) \ar[r]^{w} & A^{S/n}}

Proof. The construction of the map F_{n} is similar to the proof of Proposition 1.2. We let A = ℤ[a_{n} | n ∈ S] and let a be the vector (a_{n} | n ∈ S). Then Lemma 1.1 shows that the sequence F_{n}^{w}(w(a)) ∈ A^{S/n} is the image of a (unique) vector

F_{n}(a) = (f_{n,d} | d ∈ S/n) ∈ 𝕎_{S/n}(A)

by the ghost map. If A′is any ring and a′= (a′_{n} | n ∈ S) a vector in 𝕎_{S}(A′), then we define F_{n}(a′) = 𝕎_{S/n}(g)(F_{n}(a)) where g: A → A′is the unique ring homomorphism that maps a to a′. Finally, since F_{n}^{w} is a ring homomorphism, an argument similar to the proof of Lemma 1.3 shows that also F_{n} is a ring homomorphism.

The Teichmüller representative is the map

[−]_{S}: A → 𝕎_{S}(A)

whose nth component is a for n = 1, and 0 for n > 1. It is a multiplicative map. Indeed, the following diagram, where [a]_{S}^{w} is the sequence with nth component a^{n}, commutes, and [−]_{S}^{w} is a multiplicative map.

[figure: A; [−]_{S}; A; [−]_{S}^{w}; 𝕎_{S}(A); w; A^{S}]

Lemma 1.5. The following relations holds.

(i) a = ∑_{n∈S} V_{n}([a_{n}]_{S/n}).

(ii) F_{n}V_{n}(a) = na.

(iii) aV_{n}(a′) = V_{n}(F_{n}(a)a′).

(iv) F_{m}V_{n} = V_{n}F_{m} if (m,n) = 1.

(v) F_{n}([a]_{S}) = [a^{n}]_{S/n}.

Proof. One easily verifies that both sides of each equation have the same image by the ghost map. This shows that the relations hold, if A is torsion free, and hence, in general.

Proposition 1.6. The ring of Witt vectors in ℤ is equal to the product

𝕎_{S}(ℤ) = ∏_{n∈S}ℤ · V_{n}([1]_{S/n})

and the product given by V_{m}([1]_{S/m}) · V_{n}([1]_{S/n}) = c · V_{e}([1]_{S/e}) with c = (m,n) the greatest common divisor and e = [m,n] the least common multiple.

Proof. The formula for the product follows from Lemma 1.5 (ii)-(iv). For finite S, we prove the statement by induction beginning from the case S = ∅ which is trivial. So suppose that S is non-empty, let m ∈ S be maximal, and let T = S ∖ {m}. The sequence of abelian groups

0 → 𝕎_{{1}}(ℤ) −→^{V_{m}} 𝕎_{S}(ℤ) −→^{R_{T}^{S}} 𝕎_{T}(ℤ) → 0

is exact, and we wish to show that it is equal to the sequence

0 → ℤ · [1]_{{1}} −→^{V_{m}} ∏_{n∈S}ℤ · V_{n}([1]_{S/n}) −→^{R_{T}^{S}} ∏_{n∈T}ℤ · V_{n}([1]_{T /n}) → 0.

The latter sequence is a sub-sequence of the former, and by induction, the right-hand terms of the two sequences are equal. Since also the left-hand terms are equal, so are the middle terms. This completes the proof for S finite. Finally, every truncation set S is the union of its finite sub-truncation sets S_{α} ⊂ S and 𝕎_{S}(ℤ) = lim_{α} 𝕎_{S_{α}}(ℤ).

The values of the restriction, Frobenius, and Verschiebung maps on the generators V_{n}([1]_{S/n}) are readily evaluated by using Lemma 1.5 (ii)–(iv). To give a formula for the Teichmüller representative, we recall the Möbius inversion formula. Let g: ℕ → ℤ be a function and define the function f: ℕ → ℤ by f(n) = ∑_{d|n}g(d). Then the original function is given by g(n) = ∑_{d|n}μ(d)f(n/d), where μ: ℕ → {−1,0,1} is the Möbius function defined by μ(d) = (−1)^{r}, if d is a product of r ⩾ 0 distinct prime numbers, and μ(d) = 0, otherwise.

Addendum 1.7. Let m be an integer. Then

[m]_{S} = ∑_{n∈S} \frac{1}{n}( ∑_{d|n}μ(d)m^{n/d})V_{n}([1]_{S/n}).

If m = q is a prime power, the coefficient of V_{n}([1]_{S/n}) is equal to the number of monic irreducible polynomials of degree n over the finite field 𝔽_{q}.

Proof. It suffices to prove that the formula holds in 𝕎_{S}(ℤ). By Proposition 1.6, there are unique integers r_{d}, d ∈ S such that

[m]_{S} = ∑_{d∈S}r_{d}V_{d}([1]_{S/d}).

Evaluating the nth ghost component of this equation, we find that

m^{n} = ∑_{d|n}dr_{d}.

The Möbius inversion formula completes the proof.

Lemma 1.8. If A is an 𝔽_{p}-algebra, then

F_{p} = R_{S/p}^{S}∘ 𝕎_{S}(φ): 𝕎_{S}(A) → 𝕎_{S/p}(A),

where φ: A → A is the Frobenius endomorphism.

Proof. We let A = ℤ[a_{n} | n ∈ S] and recall from the proof of Lemma 1.4 the family of polynomials f_{p,n} ∈ A, n ∈ S, defined by the following system of equations.

∑_{d|n}df_{p,d}^{n/d} = ∑_{d|pn}da_{d}^{pn/d}

The lemma is equivalent to the statement that for all n ∈ S/p, f_{p,n} ≡ a_{n}^{p} modulo pA. We proceed by induction on n ∈ S. Since f_{p,1} = a_{1}^{p} + pa_{p}, the statement holds for n = 1. So let n > 1 and assume that for all proper divisors d of n, f_{p,d} ≡ a_{d}^{p} modulo pA. As in the proof of Lemma 1.1, it follows that df_{p,d}^{n/d} ≡ da_{d}^{pn/d} modulo p^{v_{p}(n)+1}A. We now rewrite the defining system of equations as

∑_{d|n}df_{p,d}^{n/d} = ∑_{d|n}da_{d}^{pn/d}+ ∑_{d|pn,d∤n}da_{d}^{pn/d}

and note that if d | pn and d ∤ n, then v_{p}(d) = v_{p}(n) + 1. Therefore, we may conclude that nf_{p,n} ≡ na_{n}^{p} modulo p^{v_{p}(n)+1}A. But A is torsion free, so f_{p,n} ≡ a_{n}^{p} modulo pA as desired. This completes the proof.

Lemma 1.9. Suppose that the integer m is invertible (resp. a non-zero-divisor) in A. Then m is invertible (resp. a non-zero-divisor) in 𝕎_{S}(A).

Proof. As in the proof of Proposition 1.6, we may assume that S is finite. We proceed by induction on S beginning from the trivial case S = ∅. So let S be non-empty and assume the statement for all proper sub-truncation sets of S. We let n ∈ S be maximal, and let T = S ∖ {n}. Then we have the exact sequence

0 → 𝕎_{{1}}(A) ⟶^{V_{n}} 𝕎_{S}(A) −→^{R_{T}^{S}} 𝕎_{T}(A) → 0

from which the induction step readily follows.

We consider the truncation set P = {1,p,p^{2},...} ⊂ ℕ that consists of all powers of a fixed prime number p. The proper non-empty sub-truncation sets of P all are of the form {1,p,...,p^{n−1} }, for some positive integer n. The rings

W (A) = 𝕎_{P}(A), W_{n}(A) = 𝕎_{{1,p,...,p^{n−1} }}(A)

are called the ring of p-typical Witt vectors in A and the ring of p-typical Witt vectors of length n in A, respectively. We have the following p-typical decomposition.

Proposition 1.10. Let p be a prime number, S a truncation set, and I(S) the set of k ∈ S not divisible by p. If A is a ring in which every k ∈ I(S) is invertible, then the ring of Witt vectors 𝕎_{S}(A) admits the idempotent decomposition

𝕎_{S}(A) = ∏_{k∈I(S)}𝕎_{S}(A)e_{k}, e_{k} = ∏_{l∈I(S)∖\{1\}}( \frac{1}{k}V_{k}([1]_{S/k}) − \frac{1}{kl}V_{kl}([1]_{S/kl})).

Moreover, the composite map

𝕎_{S}(A)e_{k} ↪ 𝕎_{S}(A) ⟶^{F_{k}} 𝕎_{S/k}(A) −−⟶^{R_{S/k∩P}^{S/k}} 𝕎_{S/k∩P}(A)

is an isomorphism of rings.

Proof. We first show that e_{k}, k ∈ I(S), are orthogonal idempotents in 𝕎_{S}(A). It suffices to consider A = ℤ[I(S)^{−1} ], and since the ghost map is injective in this case, it further suffices to show that w(e_{k}), k ∈ I(S), are orthogonal idempotents in A^{S}. This, in turn, follows immediately from the calculation that

w_{n}(e_{k}) = \{ \begin{matrix}1 & if n ∈ S ∩ kP \\ 0 & otherwise. \end{matrix}

Finally, to prove the second part of the statement, we consider the composite map

𝕎_{S/k∩P}(A) ⟶^{σ} 𝕎_{S/k}(A) ⟶^{\frac{1}{k}V_{k}} 𝕎_{S}(A) ⟶^{pr} 𝕎_{S}(A)e_{k},

where σ is any set theoretic section of R_{S/k∩P}^{S/k} and pr the canonical projection. We claim that this map is the inverse of the map in the statement. Indeed, this is readily verified by evaluating the induced map in ghost coordinates.

Example 1.11. If S = {1,2,...,n}, then S/k ∩ P = {1,p,...,p^{s−1} } where s = s(n,k) is the unique integer with p^{s−1} k ⩽ n < p^{s}k. Hence, if every integer 1 ⩽ k ⩽ n not divisible by p is invertible in A, then Proposition 1.10 shows that

𝕎_{{1,2,...,n}}(A) ⟶^{∼} ∏ W_{s}(A)

where the product ranges over integers 1 ⩽ k ⩽ n not divisible by p and s = s(n,k). We consider the rings W_{n}(A) in more detail. The p-typical ghost map

w: W_{n}(A) → A^{n}

takes the vector (a_{0},...,a_{n−1} ) to the sequence ⟨w_{0},...,w_{n−1} ⟩ with

w_{i} = a_{0}^{p^{i}} + pa_{1}^{p^{i−1}} + ··· + p^{i}a_{i}.

If A is equipped with a ring endomorphism ϕ: A → A with ϕ(a) ≡ a^{p} modulo pA, then Lemma 1.1 identifies the image of the ghost map with the subring of sequences ⟨x_{0},...x_{n−1} ⟩ such that x_{i} ≡ ϕ(x_{i−1} ) modulo p^{i}A, for all 1 ⩽ i ⩽ n − 1. We write

[−]_{n}: A → W_{n}(A)

for the Teichmüller representative and

\begin{matrix}F : & W_{n}(A) → W_{n−1} (A) \\ V : & W_{n−1} (A) → W_{n}(A) \end{matrix}

for the pth Frobenius and pth Verschiebung.

Lemma 1.12. If A is an 𝔽_{p}-algebra, then V F = p · id.

Proof. By Lemma 1.5 (iii), V F = V ([1]_{n−1} ) · id. We show by induction on n that for an 𝔽_{p}-algebra A, V ([1]_{n−1} ) = p · [1]_{n}; the case n = 1 is trivial. The exact sequences

\xymatrix{0 \ar[r] & A \ar[r]^{V^{n−1}} & W_{n}(A) \ar[r]^{R} & W_{n−1} (A) \ar[r] & 0}

furnish an induction argument which shows that W_{n}(A) is annihilated by p^{n}. In particular, V ([1]_{n−1} ) is annihilated by p^{n−1} . Now, Addendum 1.7 shows that

[p]_{n} = p[1]_{n}+ ∑_{0<s<n} \frac{p^{p^{s}} − p^{p^{s−1}}}{p^{s}} V^{s}([1]_{n−s}).

Here [p]_{n}= 0 since A is an 𝔽_{p}-algebra, and V^{s}([1]_{n−s}) = p^{s}V ([1]_{n−1} ) by the inductive hypothesis. Therefore, the formula becomes

0 = p[1]_{n}+ (p^{p^{n−1} −1} − 1)V ([1]_{n−1} ).

But p^{n−1} − 1 ⩾ n − 1, so the induction step follows.

Let A be a p-torsion free ring equipped with a ring homomorphism ϕ: A → A such that ϕ(a) ≡ a^{p} modulo pA. By Lemma 1.1, there is a unique ring homomorphism

λ_{ϕ} : A → W (A)

whose nth ghost component is ϕ^{n}. We define τ_{ϕ} : A → W (A/pA) to be the composite of λ_{ϕ} and the map induced by the canonical projection of A onto A/pA. We recall that A/pA is said to be perfect, if the Frobenius φ: A/pA → A/pA is an automorphism.

Proposition 1.13. Let A be a p-torsion free ring equipped with a ring homomorphism ϕ: A → A such that ϕ(a) ≡ a^{p} modulo pA, and suppose that A/pA is perfect. Then for every positive integer n, the map τ_{ϕ} induces an isomorphism

τ_{ϕ} : A/p^{n}A ⟶^{∼} W_{n}(A/pA).

Proof. The map τ_{ϕ} induces a map as stated since

V^{n}W (A/pA) = V^{n}W (ϕ^{n}(A/pA)) = V^{n}F^{n}W (A/pA) = p^{n}W (A/pA);

compare Lemmas 1.8 and 1.12. The proof is completed by an induction argument based on the following commutative diagram.

\xymatrix{0 \ar[r] & A/pA \ar[r]^{p^{n−1}} \ar[d]^{φ^{n−1}} & A/p^{n}A \ar[r]^{pr} \ar[d]^{τ_{ϕ}} & A/p^{n−1} A \ar[r] \ar[d]^{τ_{ϕ}} & 0 \\ 0 \ar[r] & A/pA \ar[r]^{V^{n−1}} & W_{n}(A/pA) \ar[r]^{R} & W_{n−1} (A/pA) \ar[r] & 0.}

Here the top horizontal sequence is exact, since A is p-torsion free, and the left-hand vertical map is an isomorphism, since A/pA is perfect.

We return to the ring of big Witt vectors. We write (1+tA[[t]])^{∗} for the multiplicative group of power series over A with constant term 1.

Proposition 1.14. There is a natural commutative diagram

\xymatrix{𝕎(A) \ar[r]^{γ} \ar[d]^{w} & (1 + tA[[t]])^{∗} \ar[d]^{t \frac{d}{dt}log} \\ A^{ℕ} \ar[r]^{γ^{w}} & tA[[t]]}

where the horizontal maps are defined by

γ(a_{1},a_{2},...) = ∏_{n⩾1}(1 − a_{n}t^{n})^{−1} , γ^{w}⟨x_{1},x_{2},...⟩ = ∑_{n⩾1}x_{n}t^{n}.

The maps γ and γ^{w} are isomorphisms of abelian groups.

Proof. It is clear that γ^{w} is an isomorphism of abelian groups. We show that the map γ is a bijection. To this end, we write

∏_{n⩾1}(1 − a_{n}t^{n})^{−1} = (1 + b_{1}t + b_{2}t^{2}+ ...)^{−1}

where the coefficient b_{n} is given by the sum

b_{n} = ∑ (−1)^{r}a_{i_{1}} ...a_{i_{r}}

that ranges over all 1 ⩽ i_{1} < ···< i_{r} ⩽ n such that i_{1}+ 2i_{2}+ ··· + ri_{r} = n. It follows that the coefficients a_{n} are determined recursively by the coefficients b_{n}, and hence, that γ is a bijection. It remains to prove that γ is a homomorphism from the additive group 𝕎(A) to the multiplicative group (1+tA[[t]])^{∗}. We may assume that A is torsion free. Then the vertical maps in the diagram of the statement are both injective, and it suffices to show that this diagram commutes. The calculation

\begin{aligned}t & \frac{d}{dt}log(∏_{d⩾1}(1 − a_{d}t^{d})^{−1} ) = − ∑_{d⩾1}t \frac{d}{dt}log(1 − a_{d}t^{d}) = ∑_{d⩾1} \frac{da_{d}t^{d}}{1 − a_{d}t^{d}} \\ & = ∑_{d⩾1}∑_{s⩾0}da_{d}t^{d}· a_{d}^{s}t^{sd} = ∑_{d⩾1} ∑_{q⩾1}da_{d}^{q}t^{qd} = ∑_{n⩾1}( ∑_{d|n}da_{d}^{n/d})t^{n} \end{aligned}

completes the proof.

Addendum 1.15. The map γ induces an isomorphism of abelian groups

γ_{S}: 𝕎_{S}(A) ⟶^{∼} Γ_{S}(A)

where Γ_{S}(A) is the quotient of the multiplicative group Γ (A) = (1 + tA[[t]])^{∗} by the subgroup I_{S}(A) of all power series of the form ∏ _{n∈ℕ∖S}(1 − a_{n}t^{n})^{−1} .

Proof. The kernel of the restriction map R_{S}^{ℕ}: 𝕎(A) → 𝕎_{S}(A) is equal to the subset of all vectors a = (a_{n} | n ∈ ℕ) such that a_{n} = 0, if n ∈ S. The image of this subset by the map γ is the subset I_{S}(A) ⊂ Γ .

Example 1.16. If S = {1,2,...,n}, then I_{S}(A) = (1+t^{n+1} A[[t]])^{∗}. Hence, in this case, Addendum 1.15 gives an isomorphism of abelian groups

γ_{S}: 𝕎_{{1,2,...,n}}(A) ⟶^{∼} (1 + tA[[t]])^{∗}/(1 + t^{n+1} A[[t]])^{∗}.

For A a ℤ_{(p)}-algebra, the structure of this group was examined in Example 1.11.

Lemma 1.17. Let p be a prime number, and let A be any ring. Then the ring homomorphism F_{p}: 𝕎(A) → 𝕎(A) satisfies that F_{p}(a) ≡ a^{p} modulo p𝕎(A).

Proof. We first let A = ℤ[a_{1},a_{2},...] and a = (a_{1},a_{2},...). If suffices to show that there exists b ∈ 𝕎(A) such that F_{p}(a)−a^{p} = pb. By Lemma 1.9, the element is necessarily unique; we use Lemma 1.1 to prove that it exists. We have

w_{n}(F_{p}(a) − a^{p}) = ∑_{d|pn}da_{d}^{pn/d}− ( ∑_{d|n}da_{d}^{n/d})^{p}

which is clearly congruent to zero modulo pA. So let x = ⟨x_{n} | n ∈ ℕ⟩ with

x_{n} = \frac{1}{p}(F_{p}(a) − a^{p}).

We wish to show that x = w(b), for some b ∈ 𝕎(A). The unique ring homomorphism ϕ_{ℓ}: A → A that maps a_{n} to a_{n}^{ℓ} satisfies that ϕ_{ℓ}(f) = f^{ℓ} modulo ℓA, and hence, Lemma 1.1 shows that x is in the image of the ghost map if and only if

x_{n} ≡ ϕ_{ℓ}(x_{n/ℓ})

modulo ℓ^{v_{ℓ}(n)}A, for all primes ℓ and all n ∈ ℓℕ. This is equivalent to showing that

w_{n}(F_{p}(a) − a^{p}) ≡ ϕ_{ℓ}(w_{n/p}(F_{p}(a) − a^{p}))

modulo ℓ^{v_{ℓ}(n)}A, if ℓ ≠ p and n ∈ ℓℕ, and modulo ℓ^{v_{ℓ}(n)+1}A, if ℓ = p and n ∈ ℓℕ. If ℓ ≠ p, the statement follows from Lemma 1.1, and if ℓ = p and n ∈ ℓℕ, we calculate

\begin{matrix}w_{n}(F_{p}(a) − a^{p}) − ϕ_{p}(w_{n/p}(F_{p}(a) − a^{p})) \\ = ∑_{d|pn,d∤n}da_{d}^{pn/d}− ( ∑_{d|n}da_{d}^{n/d})^{p} + ( ∑_{d|(n/p)}da_{d}^{n/d})^{p}. \end{matrix}

If d | pn and d ∤ n, then v_{p}(d) = v_{p}(n) + 1, so the first summand is congruent to zero modulo p^{v_{p}(n)+1}A. Similarly, if d | n and d ∤ (n/p), then v_{p}(d) = v_{p}(n), and hence,

∑_{d|n}da_{d}^{n/d} ≡ ∑_{d|(n/p)}da_{d}^{n/d}

modulo p^{v_{p}(n)}A. But then

( ∑_{d|n}da_{d}^{n/d})^{p} ≡ ( ∑_{d|(n/p)}da_{d}^{n/d})^{p}

modulo p^{v_{p}(n)+1}A; compare the proof of Lemma 1.1. This completes the proof.

We write ϵ = ϵ_{A}: 𝕎(A) → A for the natural ring homomorphism w_{1}: 𝕎(A) → A.

Proposition 1.18. There exists a unique natural ring homomorphism

Δ = Δ_{A}: 𝕎(A) → 𝕎(𝕎(A))

such that w_{n}(Δ(a)) = F_{n}(a) for all positive integers n. Moreover, the diagrams

[figure: 𝕎(A); 𝕎(𝕎(A)); 𝕎(ϵ_{A}); ϵ_{𝕎(A)}; 𝕎(A); 𝕎(𝕎(𝕎(A))); 𝕎(𝕎(A)); Δ_{𝕎(A)}; 𝕎(A); Δ_{A}; 𝕎(𝕎(A)); 𝕎(Δ_{A}); 𝕎(A); Δ_{A}; Δ_{A}]

commute.

Proof. By naturality, we may assume that A is torsion free. Lemma 1.9 shows that also 𝕎(A) is torsion free such that the ghost map

w: 𝕎(𝕎(A)) → 𝕎(A)^{ℕ}

is injective. Lemma 1.17 and Lemma 1.1 show that the sequence ⟨F_{n}(a) | a ∈ ℕ⟩ is in the image of the ghost map. Hence, the natural ring homomorphism Δ exists. The commutativity of the diagrams is readily verified by evaluating the corresponding maps in ghost coordinates.

Remark 1.19. The map Δ_{n}: 𝕎(A) → 𝕎(A) given by the nth Witt component of the map Δ is generally not a ring homomorphism. For example, for a prime number p, the map Δ_{p} is the unique natural solution to the equation F_{p}(a) = a^{p} + pΔ_{p}(a).

The definition of a λ-ring due to Grothendieck [9] may be stated as follows.

Definition 1.20. A λ-ring is a pair (A,λ_{A}) of a ring A and a ring homomorphism λ_{A}: A → 𝕎(A) that the following diagrams commute.

[figure: A; 𝕎(A); ϵ_{A}; 𝕎(𝕎(A)); 𝕎(A); Δ_{A}; A; λ_{A}; 𝕎(A); 𝕎(λ_{A}); A; λ_{A}; λ_{A}]

A morphism of λ-rings f: (A,λ_{A}) → (B,λ_{B}) is a ring homomorphism f: A → B with the property that λ_{B}∘ f = 𝕎(f) ∘ λ_{A}.

Remark 1.21. The commutative diagrams in Proposition 1.18 and Definition 1.20 express that (𝕎(−),Δ,ϵ) is a comonad on the category of rings and that (A,λ_{A}) is a coalgebra over this comonad, respectively. The composite ring homomorphism

\xymatrix{A \ar[r]^{λ_{A}} & 𝕎(A) \ar[r]^{w_{n}} & A}

is called the nth associated Adams operation. We write ψ_{A,n}: A → A for this map. We further write λ_{A,n}: A → A for the nth component of the map λ_{A}: A → 𝕎(A). It is related to the traditional λ-operation λ_{A}^{n}: A → A [1, Exposé V] by the formula

∏_{n⩾1}(1 − λ_{A,n}(a)t^{n}) = 1 + ∑_{n⩾1}(−1)^{n}λ_{A}^{n}(a)t^{n}.

We note that λ_{A,n}: A → A and λ_{A}^{n}: A → A are generally not ring homomorphisms.

Finally, we recall the following general theorem which was proved independently by Borger [3, Theorem B], [4, Corollary 15.4] and van der Kallen [22, Theorem 2.4].

Theorem 1.22. Let f: A → B be an étale morphism, let S be a finite truncation set, and let n be a positive integer. Then the morphism

𝕎_{S}(f): 𝕎_{S}(A) → 𝕎_{S}(B)

is étale and the diagram

\xymatrix{𝕎_{S}(A) \ar[r]^{𝕎_{S}(f)} \ar[d]^{F_{n}} & 𝕎_{S}(B) \ar[d]^{F_{n}} \\ 𝕎_{S/n}(A) \ar[r]^{𝕎_{S/n}(f)} & 𝕎_{S/n}(B)}

is cartesian.

We remark that in loc. cit., the theorem above is stated only for those finite truncation sets that consist of all divisors of a given positive integer. These truncation sets, however, include all finite p-typical truncation sets, and therefore, the general case follows immediately by applying Proposition 1.10. Indeed, since S is finite, the canonical maps 𝕎_{S}(A_{(p)}) → 𝕎_{S}(A_{(p)})_{(p)} ← 𝕎_{S}(A)_{(p)} are isomorphisms.

2 Modules and derivations over λ-rings

In general, given a category 𝒞 in which finite limits exist and an object X of 𝒞, Quillen [20] defines the category of X-modules to be the category (𝒞 /X)_{ab}of abelian group objects in the category over X. Quillen further defines the set of derivations from X to the X-module (Y /X,+_{Y},0_{Y},−_{Y}) to be the set of morphisms

Der(X,(Y /X,+_{Y},0_{Y},−_{Y})) = Hom_{𝒞 /X} (X/X,Y /X)

in the category 𝒞 /X. This set inherits an abelian group structure from the abelian group object structure on Y /X. In this section, we identify and study these notions in the case of the category 𝒜_{λ} of λ-rings.

To begin, we first recall the case of the category 𝒜 of rings, which we always assume to be commutative and unital. Let A be a ring and let ℳ (A) be the usual category of A-modules. We define an equivalence of categories

\xymatrix{(𝒜 /A)_{ab} \ar[r]^{F} & ℳ (A) \ar[l]^{G}}

as follows. Let f: B → A be an object of 𝒜 /A, and let

[figure: B ×_{A}B; +; B; f; A; 0; B; f; B; −; f; B; f; A; A; A; A; A; A]

be abelian group object structure maps. Then the functor F associates to the abelian group object (f,+,0,−) the A-module M defined by the kernel of the ring homomorphism f with the A-module structure map given by a · x = 0(a)x. Conversely, the functor G associates to the A-module M the abelian group object (f,+,0,−) where f: A ⊕ M → A is the canonical projection from the ring given by the direct sum equipped with the multiplication (a,x) · (a′,x′) = (aa′,ax′+ a′x) together with the abelian group object structure maps

[figure: A ⊕ M ×_{A}A ⊕ M; +; A ⊕ M; f; A; 0; A ⊕ M; f; A ⊕ M; −; f; A ⊕ M; f; A; A; A; A; A; A]

defined by (a,x)+(a,x′) = (a,x+x′), 0(a) = (a,0), and −(a,x) = (a,−x). We claim that the natural transformations η: id⇒ F G and ϵ: GF ⇒ idgiven by

\begin{aligned}η(x) & = (0,x) \\ ϵ(a,x) & = 0(a) + x \end{aligned}

are well-defined and isomorphisms. This is clear for the map η. It is also clear that the map ϵ: A ⊕ M → B is an isomorphism of abelian groups; we must show that it is a ring homomorphism and a map of abelian group objects. Since +_{B}: B ×_{A}B → B is a ring homomorphism, the compositions +_{B} and + on M are mutually distributive. They further have the common unit 0 = 0(0), and hence, agree. Therefore,

\begin{matrix}ϵ(a,x) +_{B}ϵ(a,x′) = (0(a) + x) +_{B}(0(a) + x′) \\ = (0(a) +_{B}0(a)) + (x +_{B}x′) = 0(a) + (x + x′) = ϵ(a,x + x′). \end{matrix}

It remains to prove that ϵ is multiplicative, or equivalently, that M ⊂ B is a square-zero ideal. Since +_{B}: B ×_{A}B → B is a ring homomorphism, we have

ϵ(a,x)ϵ(b,y) +_{B}ϵ(a,x′)ϵ(b,y′) = (ϵ(a,x) +_{B}ϵ(a,x′))(ϵ(b,y′) +_{B}ϵ(b,y′))

which, by the formula just proved, becomes

\begin{matrix}0(ab) + 0(a)y + 0(b)x + xy + 0(a)y′+ 0(b)x′+ x′y′ \\ = 0(ab) + 0(a)(y + y′) + 0(b)(x + x′) + (x + x′)(y + y′). \end{matrix}

This shows that ϵ: A ⊕ M → B is a ring homomorphism as desired.

We next consider the ring of Witt vectors in the ring A ⊕ M. We note that the polynomials s_{n}, p_{n}, and ι_{n} that defined the sum, product, and opposite in the ring of Witt vectors have zero constant terms. Therefore, the ring of Witt vectors is defined also for non-unital rings. Since M has zero multiplication, it follows that 𝕎(M) has zero multiplication and that the addition is componentwise addition.

Lemma 2.1. The canonical inclusions induce a ring isomorphism

in_{1∗}+in_{2∗}: 𝕎(A) ⊕ 𝕎(M) ⟶^{∼} 𝕎(A ⊕ M)

where, on the left-hand side, the ring structure is induced from the 𝕎(A)-module structure on 𝕎(M) defined by (ax)_{n} = w_{n}(a)x_{n}.

Proof. In the following diagram of rings and ring homomorphisms, the sequence of map directed from left to right is an exact sequence of additive abelian groups.

\xymatrix{0 \ar[r] & M \ar[r]^{^{in^{2}}} & A ⊕ M \ar[r]_{pr_{1}} & A \ar[r] \ar[l]_{^{in^{1}}} & 0}

It induces a diagram of rings and ring homomorphisms where again the sequence of maps directed from left to right is an exact sequence of abelian groups.

\xymatrix{0 \ar[r] & 𝕎(M) \ar[r]^{^{in^{2∗}}} & 𝕎(A ⊕ M) \ar[r]_{pr_{1∗}} & 𝕎(A) \ar[r] \ar[l]_{^{in^{1∗}}} & 0}

Hence, the map in_{1∗}+in_{2∗} is an isomorphism of abelian groups. Moreover, since 𝕎(M) has zero multiplication, this map is a ring isomorphism for the ring structure on the domain induced from the 𝕎(A)-module structure on 𝕎(M) defined by the formula in_{2∗}(ax) = in_{1∗}(a)in_{2∗}(x). Finally, we wish to show that ax is equal to the Witt vector y with nth component w_{n}(a)x_{n}. Since every ring admits a surjective ring homomorphism from a torsion free ring, we may further assume that A and M are both torsion free. Since the ghost map is injective in this case, it will suffice to show that w_{n}(ax) = w_{n}(y), or equivalently, that in_{2}(w_{n}(ax)) = in_{2}(w_{n}(y)), for all n ⩾ 1. Now, since w_{n} is a natural ring homomorphism, we find that for all n ⩾ 1,

in_{2}(w_{n}(ax))= w_{n}(in_{2∗}(ax)) = w_{n}(in_{1∗}(a)in_{2∗}(x)) = w_{n}(in_{1∗}(a))w_{n}(in_{2∗}(x))
= in_{1}(w_{n}(a))in_{2}(w_{n}(x)) = in_{2}(w_{n}(a)w_{n}(x)) = in_{2}(nw_{n}(a)x_{n})
= in_{2}(ny_{n}) = in_{2}(w_{n}(y))

as desired. Here the fifth equality follows from the definition of the multiplication on the direct sum ring A ⊕ M.

We give an explicit recursive formula for the ring isomorphism in_{1∗}+in_{2∗}.

Lemma 2.2. Let b = in_{1∗}(a) + in_{2∗}(x) and write b_{n} = (a_{n},y_{n}). Then for all n ⩾ 1,

∑_{d|n}a_{d}^{(n/d)−1}y_{d} = x_{n}.

Proof. As before, we may assume that A and M are torsion free. We now calculate w_{n}(b) in two different ways. First, since w_{n} is a natural ring homomorphism, we have

w_{n}(b)= w_{n}(in_{1∗}(a)) + w_{n}(in_{2∗}(x)) = in_{1}(w_{n}(a)) + in_{2}(w_{n}(x))

= (w_{n}(a),w_{n}(x)) = (∑_{d|n}da_{d}^{n/d},nx_{n}).

Second, by the definition of the multiplication in A ⊕ M, we have

w_{n}(b) = ∑_{d|n}db_{d}^{n/d} = ∑_{d|n}d(a_{d},y_{d})^{n/d} = (∑_{d|n}da_{d}^{n/d},∑_{d|n}na_{d}^{(n/d)−1}y_{d}).

The stated formula follows as M was assumed to be torsion free.

Example 2.3. Let p be a prime number. Then y_{p} = x_{p} − a_{1}^{p−1} x_{1}.

We proceed to identify the category (𝒜_{λ}/(A,λ_{A}))_{ab} of abelian group objects in the category of λ-rings over (A,λ_{A}). By category theory, there is an adjunction

\xymatrix{𝒜_{λ} \ar[r]^{u} & 𝒜 \ar[l]^{R}}

with the left adjoint functor defined by u(A,λ_{A}) = A and with the right adjoint functor defined by R(A) = (𝕎(A),Δ_{A}). The unit and counit of the adjunction are defined to be the maps λ_{A}: (A,λ_{A}) → (𝕎(A),Δ_{A}) and ϵ_{A}: 𝕎(A) → A, respectively. Since 𝕎(−) preserves limits, u creates limits. In particular, the category 𝒜_{λ} has all small limits. It follows, that for (A,λ_{A}) an object of 𝒜_{λ}, there is an adjoint pair of functors

\xymatrix{𝒜_{λ}/(A,λ_{A}) \ar[r]^{u_{(A,λ_{A})}} & 𝒜 /A \ar[l]^{R_{(A,λ_{A})}}}

where the left adjoint functor u_{(A,λ_{A})} maps f: (B,λ_{B}) → (A,λ_{A}) to f: B → A, and where the right adjoint functor R_{(A,λ_{A})} maps f: B → A to pr_{2}: (C,λ_{C}) → (A,λ_{A}) defined by the following pull-back diagram.

\xymatrix{(C,λ_{C}) \ar[r]^{pr_{1}} \ar[d]^{pr_{2}} & (𝕎(B),Δ_{B}) \ar[d]^{𝕎(f)} \\ (A,λ_{A}) \ar[r]^{λ_{A}} & (𝕎(A),Δ_{A})}

The unit and counit maps are defined to be the maps (λ_{B},f), and ϵ_{B}∘pr_{1}, respectively.

If M is an A-module, we view 𝕎(M) as a 𝕎(A)-module as in Lemma 2.1. We now make the following definition of the category ℳ (A,λ_{A}) of (A,λ_{A})-modules.

Definition 2.4. Let (A,λ_{A}) be a λ-ring. An (A,λ_{A})-module is a pair (M,λ_{M}) of an A-module M and a λ_{A}-linear map

λ_{M}: M → 𝕎(M)

with the property that the diagrams

[figure: M; 𝕎(M); ϵ_{M}; 𝕎(𝕎(M)); 𝕎(M); Δ_{M}; M; λ_{M}; 𝕎(M); 𝕎(λ_{M}); M; λ_{M}; λ_{M}]

commute. A morphism h: (M,λ_{M}) → (N,λ_{N}) of (A,λ_{A})-modules is an A-linear map h: M → N such that λ_{N}∘ h = 𝕎(h) ∘ λ_{M}.

Remark 2.5. Let (A,λ_{A}) be a λ-ring, M an A-module, and λ_{M}: M → 𝕎(M) a map. Then (M,λ_{M}) is an (A,λ_{A})-module if and only if the components λ_{M,n}: M → M of the map λ_{M} satisfy the following conditions: The map λ_{M,n} is linear with respect to the nth Adams operation ψ_{A,n}= w_{n}∘ λ_{A}; the map λ_{M,1} is the identity map; and the composite map λ_{M,m}∘ λ_{M,n}is equal to the map λ_{M,mn}.

Example 2.6. Let (A,λ_{A}) be a λ-ring. The functor that to an (A,λ_{A})-module (M,λ_{M}) assigns the underlying set of M has a left adjoint functor that to the set S assigns the free (A,λ_{A})-module (F (S),λ_{F (S)}) defined as follows. The A-module F (S) is defined to be the free A-module generated by the symbols λ_{F (S),n}(s), where s ∈ S and n ∈ ℕ, and λ_{F (S)}: F (S) → 𝕎(F (S)) is defined to be the map with mth component

λ_{F (S),m}(∑ a_{s,n}λ_{F (S),n}(s)) = ∑ ψ_{A,m}(a_{s,n})λ_{F (S),mn}(s).

It follows from Remark 2.5 that the pair (F (S),λ_{F (S)}) is an (A,λ_{A})-module. The unit of the adjunction maps s ∈ S to λ_{F (S),1}(s) ∈ F (S), and the counit of the adjunction maps ∑ a_{x,n}λ_{F (M),n}(x) ∈ F (M) to ∑ a_{x,n}λ_{M,n}(x) ∈ M.

Example 2.7. Let (A,λ_{A}) be a λ-ring. Then (A,λ_{A}) is not an (A,λ_{A})-module except in trivial cases. However, the Adams operations ψ_{A,n}: A → A define an (A,λ_{A})-module structure on A considered as an A-module. The (A,λ_{A})-module defined in this way is not a free module except in trivial cases.

Example 2.8. Let A be a ring, unital and commutative as usual, and let K_{∗}(A) be the graded ring given by the Quillen K-groups. Then the ring K_{0}(A) has a canonical λ-ring structure defined by Grothendieck [9], and for all q ⩾ 1, the group K_{q}(A) has a canonical structure of a module over this λ-ring defined by Kratzer [17] and Quillen [15]. The (K_{0}(A),λ_{K_{0}(A)})-module structure maps are determined by the maps

λ_{K_{q}(A)}^{n}: K_{q}(A) → K_{q}(A)

defined in [17, Théorème 5.1]. We remark that the corresponding Adams operation is given by ψ_{K_{q}(A),n}= (−1)^{n−1} nλ_{K_{q}(A)}^{n} ; compare Remark 1.21.

We show that the categories ℳ (A,λ_{A}) and (𝒜_{λ}/(A,λ_{A}))_{ab} are equivalent. First note that since the adjoint functors u_{(A,λ_{A})}and R_{(A,λ_{A})}considered earlier both preserve limits, they give rise to the following adjoint pair of functors.

\xymatrix{(𝒜_{λ}/(A,λ_{A}))_{ab} \ar[r]^{u_{(A,λ_{A})}^{ab}} & (𝒜 /A)_{ab} \ar[l]^{R_{(A,λ_{A})}^{ab}}}

Next, we define a pair of functors

\xymatrix{(𝒜_{λ}/(A,λ_{A}))_{ab} \ar[r]^{F_{λ}} & ℳ (A,λ_{A}) \ar[l]^{G_{λ}}}

as follows. If f: (B,λ_{B}) → (A,λ_{A}) is an object of 𝒜_{λ}/(A,λ_{A}) and if

[figure: (B,λ_{B}) ×_{(A,λ_{A})}(B,λ_{B}); +; (B,λ_{B}); f; (A,λ_{A}); (A,λ_{A})]

[figure: (A,λ_{A}); 0; (B,λ_{B}); f; (B,λ_{B}); −; f; (B,λ_{B}); f; (A,λ_{A}); (A,λ_{A}); (A,λ_{A}); (A,λ_{A})]

are abelian group object structure maps, then the functor F^{λ} associates to the abelian group object (f,+,0,−) the (A,λ_{A})-module (M,λ_{M}), where M = ker(f) considered as an A-module via 0: A → B, and where λ_{M}: M → 𝕎(M) is the induced map of kernels of the vertical maps in the following diagram.

\xymatrix{B \ar[r]^{λ_{B}} \ar[d]^{f} & 𝕎(B) \ar[d]^{𝕎(f)} \\ A \ar[r]^{λ_{A}} & 𝕎(A)}

Conversely, if (M,λ_{M}) is an (A,λ_{A})-module, then G^{λ}(M,λ_{M}) is the unique abelian group object such that u_{(A,λ_{A})}^{ab}(G^{λ}(M,λ_{M})) = G(M) and such that the λ-ring structure on the ring A ⊕ M is given by the map λ_{A⊕M} defined by the composition

\xymatrix{A ⊕ M \ar[r]^{λ_{A}⊕λ_{M}} & 𝕎(A) ⊕ 𝕎(M) \ar[r]^{in_{1∗}+in_{2∗}} & 𝕎(A ⊕ M).}

Finally, it follows from what was proved earlier that there are well-defined natural isomorphisms η^{λ} : id⇒ F^{λ}G^{λ} and ϵ^{λ} : G^{λ}F^{λ} ⇒ idgiven by

\begin{aligned}η^{λ}(x) & = (0,x) \\ ϵ^{λ}(a,x) & = 0(a) + x. \end{aligned}

This shows that the categories (𝒜_{λ}/(A,λ_{A}))_{ab} and ℳ (A,λ_{A}) are equivalent.

We summarize with the following diagram of categories and functors, where the horizontal pairs of functors are equivalences of categories, and where the vertical pairs of functors are adjoint pairs of functors.

\xymatrix{(𝒜 /A)_{ab} \ar[r]^{F} \ar[d]^{R_{(A,λ_{A})}^{ab}} & ℳ (A) \ar[d]^{R′} \ar[l]^{G} \\ (𝒜_{λ}/(A,λ_{A}))_{ab} \ar[r]^{F^{λ}} \ar[u]^{u_{(A,λ_{A})}^{ab}} & ℳ (A,λ_{A}) \ar[l]^{G^{λ}} \ar[u]^{u′}}

The right-hand adjunction is defined as follows. The left adjoint functor u′takes the (A,λ_{A})-module (M,λ_{M}) to the A-module M, and the right adjoint functor R′ takes the A-module N to the (A,λ_{A})-module (λ_{A}^{∗}𝕎(N),Δ_{N}), where λ_{A}^{∗}𝕎(N) denotes the 𝕎(A)-module 𝕎(N) considered as an A-module via λ_{A}: A → 𝕎(A). The unit and counit of the adjunction are given by the maps λ_{M}: (M,λ_{M}) → (λ_{A}^{∗}𝕎(M),Δ_{M}) and ϵ_{N}: λ_{A}^{∗}𝕎(N) → N, respectively.

Proposition 2.9. The category ℳ (A,λ_{A}) of modules over the λ-ring (A,λ_{A}) is an abelian category. Moreover, the forgetful functor u′: ℳ (A,λ_{A}) → ℳ (A) creates all small limits and colimits.

Proof. We may identify ℳ (A,λ_{A}) with the category ℳ (A^{ψ} [ℕ]) of left modules over the twisted monoid algebra A^{ψ} [ℕ] with multiplication given by n·a = ψ_{A,n}(a)·n.

Remark 2.10. A map of λ-rings f: (B,λ_{B}) → (A,λ_{A}) gives rise to a functor

f^{∗}: ℳ (A,λ_{A}) → ℳ (B,λ_{B})

defined by viewing an (A,λ_{A})-module (N,λ_{N}) as a (B,λ_{B})-module f^{∗}(N,λ_{N}) via the map f. The functor f^{∗} has a left adjoint functor that to the (B,λ_{B})-module (M,λ_{M}) associates the (A,λ_{A})-module defined by

(A,λ_{A}) ⊗_{(B,λ_{B})}(M,λ_{M}) = (A ⊗_{B}M,λ_{A⊗_{B}M})

where λ_{A⊗_{B}M} is given by the composition of λ_{A}⊗_{λ_{B}} λ_{M} and the map

𝕎(A) ⊗_{𝕎(B)}𝕎(M) → 𝕎(A ⊗_{B}M)

that to a ⊗ x associates the vector whose nth Witt component is w_{n}(a) ⊗ x_{n}.

Using Lemma 2.2, one similarly verifies that Quillen’s notion of a derivation of the λ-ring (A,λ_{A}) into the (A,λ_{A})-module may be characterized as follows.

Definition 2.11. Let (A,λ_{A}) be a λ-ring and let (M,λ_{M}) be an (A,λ_{A})-module. A derivation from (A,λ_{M}) to (M,λ_{M}) is a map

D: (A,λ_{A}) → (M,λ_{M})

such that the following (1)–(3) hold.

(1) For all a,b ∈ A, D(a + b) = D(a) + D(b).

(2) For all a,b ∈ A, D(ab) = bD(a) + aD(b).

(3) For all a ∈ A and n ∈ ℕ, λ_{M,n}(D(a)) = ∑_{e|n}λ_{A,e}(a)^{(n/e)−1} D(λ_{A,e}(a)).

The set of derivations from (A,λ_{A}) to (M,λ_{M}) is denoted by Der((A,λ_{A}),(M,λ_{M})).

Lemma 2.12. Let (A,λ_{A}) be a λ-ring. There exists a derivation

d: (A,λ_{A}) → (Ω_{(A,λ_{A})},λ_{Ω_{(A,λ_{A})}} )

with the property that for every (A,λ_{A})-module (M,λ_{M}), the map

δ: Hom_{(A,λ_{A})}((Ω_{(A,λ_{A})},λ_{Ω_{(A,λ_{A})}} ),(M,λ_{M})) → Der((A,λ_{A}),(M,λ_{M}))

defined by δ(f)(a) = f(d(a)) is a bijection.

Proof. We define the target of the map d to be the quotient of the free (A,λ_{A})-module (F,λ_{F}) generated by {d(a) | a ∈ A} by the sub-(A,λ_{A})-module (R,λ_{R}) ⊂ (F,λ_{F}) generated by d(a+b)−d(a)−d(b) with a,b ∈ a; by d(ab)−bd(a)−ad(b) with a,b ∈ A; and by λ_{F,n}(da) − ∑_{e|n}λ_{A,e}(a)^{(n/e)−1} d(λ_{A,e}(a)) with a ∈ A and n ∈ ℕ. The map d takes a ∈ A to the class of d(a) in Ω_{(A,λ_{A})}. By construction, the map d is a derivation and the map δ is a bijection.

The inclusion of Der((A,λ_{A}),(M,λ_{M})) in Der(A,M) in general is strict. It is represented by a map of A-modules Ω_{A} → Ω_{(A,λ_{A})} that we call the canonical map. We now prove Theorem A which states that this map is an isomorphism.

Proof of Theorem A. We consider the following diagram of adjunctions.

\xymatrix{𝒜 /A \ar[r]^{(−)_{ab}} \ar[d]^{R_{(A,λ_{A})}} & (𝒜 /A)_{ab} \ar[l]^{i} \ar[d]^{R_{(A,λ_{A})}^{ab}} \ar[r]^{F} & ℳ (A) \ar[l]^{G} \ar[d]^{R′} \\ 𝒜_{λ}/(A,λ_{A}) \ar[r]^{(−)_{ab}^{λ}} \ar[u]^{u_{(A,λ_{A})}} & (𝒜_{λ}/(A,λ_{A}))_{ab} \ar[l]^{i^{λ}} \ar[u]^{u_{(A,λ_{A})}^{ab}} \ar[r]^{F^{λ}} & ℳ (A,λ_{A}) \ar[l]^{G^{λ}} \ar[u]^{u′}}

Here the functors i and i^{λ} are the forgetful functors that forget the abelian group object structure maps, and the functors (−)_{ab}and (−)_{ab}^{λ}are the respective left adjoint functors which we now define. It suffices to define the left adjoint functors F ∘ (−)_{ab} and F^{λ} ∘(−)_{ab}^{λ}of the composite functors i∘G and i^{λ} ∘G^{λ}. The functor i∘G takes the A-module M to the canonical projection f: A ⊕ M → A, and the left adjoint functor F ∘ (−)_{ab} takes f: B → A to the A-module

F ((f: B → A)_{ab}) = A ⊗_{B}Ω_{B}.

The unit and counit maps of the adjunction are defined by η(b) = (f(b),1 ⊗ b) and ϵ(1 ⊗ d(a,x)) = x, respectively. Similarly, the functor i^{λ} ∘ G^{λ} takes the (A,λ_{A})- module (M,λ_{A}) to the canonical projection f: (A ⊕ M,λ_{A⊕M}) → (A,λ_{A}). Here, we recall, λ_{A⊕M} is defined to be the composite map (in_{1∗}+in_{2∗}) ∘ (λ_{A}⊕ λ_{M}). The left adjoint functor F^{λ} ∘ (−)_{ab}^{λ} takes f: (B,λ_{B}) → (A,λ_{A}) to the (A,λ_{A})-module

F^{λ}((f: (B,λ_{B}) → (A,λ_{A}))_{ab}) = (A,λ_{A}) ⊗_{(B,λ_{B})}Ω_{(B,λ_{B})}

and the unit and counit of the adjunction are defined by η(b) = (f(b),1 ⊗ db) and ϵ(1 ⊗ λ_{Ω_{(B,λ_{B})},n} (d(a,x))) = λ_{M,n}(x), respectively.

Now, in the diagram of adjunctions above, the diagram of right adjoint functors commutes up to natural isomorphism. Therefore, the diagram of left adjoint functors commutes up to natural isomorphisms. It follows that the natural transformation

A ⊗_{B}Ω_{B} → u′((A,λ_{A}) ⊗_{(B,λ_{B})}Ω_{(B,λ_{B})})

is an isomorphism. Taking (B,λ_{B}) = (A,λ_{A}), the theorem follows.

Theorem 2.13. There are natural F_{n}-linear maps F_{n}: Ω_{𝕎(A)}→ Ω_{𝕎(A)}defined by

F_{n}(da) = ∑_{e|n}Δ_{A,e}(a)^{(n/e)−1} dΔ_{A,e}(a).

Moreover, the following (1)–(3) hold.

(1) For all m,n ∈ ℕ, F_{m}F_{n} = F_{mn}, and F_{1} = id.

(2) For all n ∈ ℕ and a ∈ 𝕎(A), dF_{n}(a) = nF_{n}(da).

(3) For all n ∈ ℕ and a ∈ A, F_{n}d[a] = [a]^{n−1} d[a].

Proof. The canonical map Ω_{𝕎(A)} → Ω_{(𝕎(A),Δ_{A})} is an isomorphism by Theorem A, and the codomain is a (𝕎(A),Δ_{A})-module. We define

F_{n} = λ_{Ω_{(𝕎(A),Δ_{A})},n} : Ω_{(𝕎(A),Δ_{A})} → Ω_{(𝕎(A),Δ_{A})}

to be the nth Witt component of the (𝕎(A),Δ_{A})-module structure map. It follows from Remark 2.5 that it is an F_{n} = w_{n}∘ Δ_{A}-linear map and from Definition 2.11 (3) that it is given by the stated formula. Properties (1) and (2) follow immediately from the definition of a (𝕎(A),Δ_{A})-module and from the calculation

dF_{n}(a) = d(∑_{e|n}eΔ_{A,e}(a)^{n/e}) = ∑_{e|n}nΔ_{A/e}^{(n/e)−1} dΔ_{A,e}(a) = nF_{n}(da).

Finally, to prove property (3), it suffices to show that Δ_{A,e}([a]) is equal to [a] and 0, respectively, as e = 1 and e > 1, or equivalently, that Δ([a]) = [[a]]. We may assume that A = ℤ[a]. In this case, the ghost map is injective, and the calculation

w_{n}(Δ([a])) = F_{n}([a]) = [a]^{n} = w_{n}([[a]])

shows that Δ([a]) and [[a]] have the same image by the ghost map.

3 The anti-symmetric graded algebras Ω̂_{𝕎(A)}^{·} and Ω̌_{𝕎(A)}^{·}

We next introduce the anti-symmetric graded algebra Ω̂_{𝕎(A)}^{·} which agrees with the alternating algebra Ω_{𝕎(A)}^{·}= ⋀_{𝕎(A)}Ω_{𝕎(A)}^{1} if the element d log[−1] ∈ Ω_{𝕎(A)}^{1} is zero but which is different in general.

Definition 3.1. Let A be a ring. The graded 𝕎(A)-algebra

Ω̂_{𝕎(A)}^{·} = T_{𝕎(A)}^{·}Ω_{𝕎(A)}^{1}/J^{·}

is defined to be the quotient of the tensor algebra of the 𝕎(A)-module Ω_{𝕎(A)}^{1} by the graded ideal generated by all elements of the form

da ⊗ da − d log[−1] ⊗ F_{2}da

with a ∈ 𝕎(A).

We remark that the defining relation da·da = d log[−1]·F_{2}da is analogous to the relation {a,a} = {−1,a} in Milnor K-theory.

Lemma 3.2. The graded 𝕎(A)-algebra Ω̂_{𝕎(A)}^{·} is anti-symmetric.

Proof. It suffices to show that the sum da · db + db · da ∈ Ω̂_{𝕎(A)}^{2} is equal to zero for all a,b ∈ 𝕎(A). Now, on the one hand, we have

d(a + b) · d(a + b) = d log[−1] · F_{2}d(a + b) = d log[−1] · F_{2}da + d log[−1] · F_{2}db,

since F_{2}d is additive, and on the other hand, we have

\begin{matrix}d(a + b) · d(a + b) = da · da + da · db + db · da + db · db \\ = d log[−1] · F_{2}da + da · db + db · da + d log[−1] · F_{2}db \end{matrix}

This shows that da · db + db · da is zero as desired.

Proposition 3.3. There exists a unique graded derivation

d: Ω̂_{𝕎(A)}^{·} → Ω̂_{𝕎(A)}^{·}

that extends the derivation d: 𝕎(A) → Ω_{𝕎(A)}^{1}and satisfies the formula

ddω = d log[−1] · dω

for all ω ∈ Ω̂_{𝕎(A)}^{·}. Moreover, the element d log[−1] is a cycle.

Proof. The relation ddω = d log[−1] · dω implies that d log[−1] is a cycle for the desired derivation d. Indeed,

d(d log[−1])= d([−1]d[−1]) = d[−1] · d[−1] + [−1] · dd[−1]
= d log[−1] · F_{2}d[−1] + [−1]d log[−1] · d[−1]
= d log[−1] · [−1]d[−1] + d log[−1] · [−1]d[−1]

which is zero by Lemma 3.2. This proves that the desired derivation d is unique and given by the formula

d(a_{0}da_{1}...da_{q}) = da_{0}da_{1}··· da_{q}+ qd log[−1] · a_{0}da_{1}...da_{q}

where a_{0},a_{1},...,a_{q} ∈ 𝕎(A). To complete the proof, it remains to prove that the map d given by this formula is (a) well defined, (b) a graded derivation, and (c) satisfies ddω = d log[−1] · dω. First, we have

d(a_{0}da_{1}...da_{p} · b_{0}db_{1}...db_{q}) = d(a_{0}b_{0}da_{1}...da_{p}db_{1}...db_{q}) = d(a_{0}b_{0})da_{1}...da_{p}db_{1}...db_{q}+ (p + q)d log[−1] · a_{0}b_{0}da_{1}...da_{p}db_{1}...db_{q} = da_{0}da_{1}...da_{p} · b_{0}db_{1}...db_{q}+ pd log[−1] · a_{0}da_{1}...da_{p} · b_{0}db_{1}...db_{q}

+ (−1)^{p}(a_{0}da_{1}...da_{p} · db_{0}db_{1}...db_{q}+ a_{0}da_{1}...da_{p} · d log[−1] · b_{0}db_{1}...db_{q}) = d(a_{0}da_{1}...da_{p}) · b_{0}db_{1}...db_{q}+ (−1)^{p}a_{0}da_{1}...da_{p} · d(b_{0}db_{1}...db_{q})

which proves (b). Next, using that 2 divides q^{2}+ q, we find

dd(a_{0}da_{1}...da_{q}) = d(da_{0}da_{1}...da_{q}+ qd log[−1] · a_{0}da_{1}...da_{q})
= (q + 1)d log[−1] · da_{0}da_{1}...da_{q}− qd log[−1] · da_{0}da_{1}...da_{q}
− qd log[−1] · qd log[−1] · a_{0}da_{1}...da_{q}
= d log[−1] · (da_{0}da_{1}...da_{q}+ qd log[−1] · a_{0}da_{1}...da_{q})
= d log[−1] · d(a_{0}da_{1}...da_{q})

which proves (c). Finally, to prove (a), we must show that for all a,b ∈ 𝕎(A), the following two types of elements of Ω̂_{𝕎(A)}^{·} are zero.

\begin{matrix}d(d(ab) − bda − adb) \\ d(dada − d log[−1] · F_{2}da) \end{matrix}

First, using Lemma 3.2 together with (b) and (c), we find that

d(d(ab) − bda − adb) = dd(ab) − dbda − bdda − dadb − addb
= d log[−1] · d(ab) − d log[−1] · bda − d log[−1] · adb
= d log[−1] · (d(ab) − bda − adb)

which is zero, since d: 𝕎(A) → Ω̂_{𝕎(A)}^{1} is a derivation. This shows that the first type of elements are zero. Next, (b) and (c) show that

d(dada) = 2d log[−1] · dada

which is zero as is

d(d log[−1] · F_{2}da)= d log[−1] · dF_{2}da
= d log[−1] · d(ada + dΔ_{2}(a))
= d log[−1] · (dada + d log[−1] · F_{2}da)

by the definition of Ω̂_{𝕎(A)}^{2} . Hence also the second type of elements are zero. This completes the proof of (a) and hence of the proposition.

Remark 3.4. In general, there is no 𝕎(A)-algebra map f: Ω̂_{𝕎(A)}^{·} → Ω_{𝕎(A)}^{·} that is compatible with the derivations.

Proposition 3.5. Let A be a ring and let n be a positive integer. There is a unique homomorphism of graded rings

F_{n}: Ω̂_{𝕎(A)}^{·} → Ω̂_{𝕎(A)}^{·}

that is given by the maps F_{n}: 𝕎(A) → 𝕎(A) and F_{n}: Ω_{𝕎(A)}^{1}→ Ω_{𝕎(A)}^{1} in degrees 0 and 1, respectively. In addition, the following formula holds.

dF_{n} = nF_{n}d

Proof. The uniqueness statement is clear: The map F_{n} is necessarily given by

F_{n}(a_{0}da_{1}...da_{q}) = F_{n}(a_{0})F_{n}(da_{1})...F_{n}(da_{q})

where a_{0},...,a_{q} ∈ 𝕎(A). We show that this formula gives a well-defined map. To prove this, we must show that for every a ∈ 𝕎(A),

F_{n}(da)F_{n}(da) = F_{n}(d log[−1])F_{n}(F_{2}da).

It will suffice to let n = p be a prime number. In this case, we find that

F_{p}(da)F_{p}(da) = (a^{p−1} da + dΔ_{p}(a)) · (a^{p−1} da + dΔ_{p}(a))

= (a^{p−1} )^{2}da · da + dΔ_{p}(a) · dΔ_{p}(a) = d log[−1] · ((a^{p−1} )^{2}F_{2}da + F_{2}dΔ_{p}(a))

= d log[−1] · (F_{2}(a^{p−1} )F_{2}da + F_{2}dΔ_{p}(a)) = d log[−1] · F_{2}F_{p}da

= F_{p}(d log[−1] · F_{2}da)

where we have used that F_{2}(a) is congruent to a^{2} modulo 2𝕎(A). This shows that the map F_{n} is well-defined. It is a graded ring homomorphism by definition.

We next prove the formula dF_{n} = nF_{n}d. Again, we may assume that n = p is a prime number. We already know from the definition of F_{n}: Ω_{𝕎(A)}^{1}→ Ω_{𝕎(A)}^{1} that for all a ∈ 𝕎(A), dF_{p}(a) = pF_{p}(a). Now, for all a ∈ 𝕎(A),

dF_{p}(da) = d(a^{p−1} da + dΔ_{p}(a)) = (p − 1)a^{p−2} dada + d log[−1] · F_{p}da

which is equal to zero for p = 2, and equal to d log[−1] · F_{p}da for p odd. Hence, for every prime p and every a ∈ 𝕎(A), we have

dF_{p}(da) = pd log[−1] · F_{p}da = pF_{p}(d log[−1] · da) = pF_{p}d(da)

as desired. Now, let a_{0},...,a_{q} ∈ 𝕎(A). We find

dF_{p}(a_{0}da_{1}...da_{q}) = d(F_{p}(a_{0})F_{p}da_{1}...F_{p}da_{q})

= dF_{p}(a_{0})F_{p}da_{1}...F_{p}da_{q}+ ∑_{1⩽i⩽q}(−1)^{i−1} F_{p}(a_{0})F_{p}da_{1}...dF_{p}da_{i}...F_{p}da_{q}

= p(F_{p}da_{0}F_{p}da_{1}...F_{p}da_{q}+ ∑_{1⩽i⩽q}(−1)^{i−1} F_{p}(a_{0})F_{p}da_{1}...F_{p}dda_{i}...F_{p}da_{q})

= pF_{p}d(a_{0}da_{1}...da_{q})

as stated. This completes the proof.

We next define the quotient graded algebra Ω̌_{𝕎(A)}^{·} of the graded algebra Ω̂_{𝕎(A)}^{·} and show that the Frobenius F_{n} and derivation d descend to this quotient.

Definition 3.6. Let A be a ring. The graded 𝕎(A)-algebra

Ω̌_{𝕎(A)}^{·} = Ω̂_{𝕎(A)}^{·} /K^{·}

is defined to be the quotient by the graded ideal K^{·}generated by the elements

F_{p}dV_{p}(a) − da − (p − 1)d log[−1] · a

where p ranges over all prime numbers and a over all elements of 𝕎(A).

We remark that if the prime number p is invertible in A and hence in 𝕎(A), then the element F_{p}dV_{p}(a) − da − (p − 1)d log[−1] · a already is zero in Ω̂_{𝕎(A)}^{1} .

Lemma 3.7. The Frobenius F_{n}: Ω̂_{𝕎(A)}^{·} → Ω̂_{𝕎(A)}^{·} induces a map of graded algebras

F_{n}: Ω̌_{𝕎(A)}^{·} → Ω̌_{𝕎(A)}^{·} .

Proof. It will suffice to show let n = ℓ be a prime number and show that for all prime numbers p and all a ∈ 𝕎(A), the element

F_{ℓ}(F_{p}dV_{p}(a) − da − (p − 1)d log[−1] · a) ∈ Ω_{𝕎(A)}^{1}

maps to zero in Ω̂_{𝕎(A)}^{1} . Suppose first that p = ℓ. For p odd, we find

\begin{matrix}F_{p}(F_{p}dV_{p}(a) − da) = F_{p}(p^{p−2} V_{p}(a^{p−1} )dV_{p}(a) − p^{p−2} dV_{p}(a^{p})) \\ = p^{p−1} a^{p−1} F_{p}dV_{p}(a) − p^{p−2} F_{p}dV_{p}(a^{p}) \end{matrix}

whose image in Ω̌_{𝕎(A)}^{1} is zero since F_{p}dV_{p} = d: 𝕎(A) → Ω̌_{𝕎(A)}^{1} and the common map is a derivation. Similarly, for p = 2, we find

F_{2}(F_{2}dV_{2}(a) − da − d log[−1] · a) = F_{2}(V_{2}(a)dV_{2}(a) − dV_{2}(a^{2}) − d log[−1] · a)

= 2aF_{2}dV_{2}(a) − F_{2}dV_{2}(a^{2}) − d log[−1] · F_{2}(a)

whose image in Ω̌_{𝕎(A)}^{1} is

2ada − d(a^{2}) − d log[−1] · (a^{2}+ F_{2}(a))

which is zero since d is a derivation and since a^{2}+F_{2}(a) divisible by 2. Suppose next that p ≠ ℓ. In this case, we recall, F_{ℓ}F_{p} = F_{p}F_{ℓ} and ℓ divides p^{ℓ−1} −1. Now, if both p and ℓ are different from 2, we find

F_{ℓ}(F_{p}dV_{p}(a) − da) = F_{p}(F_{ℓ}dV_{p}(a)) − F_{ℓ}da

= F_{p}(V_{p}(a)^{ℓ−1}dV_{p}(a) + d(\frac{F_{ℓ}V_{p}(a) − V_{p}(a)^{ℓ}}{ℓ})) − a^{ℓ−1}da − d(\frac{F_{ℓ}(a) − a^{ℓ}}{ℓ})

= F_{p}(p^{ℓ−2}V_{p}(a^{ℓ−1})dV_{p}(a) + dV_{p}(\frac{F_{ℓ}(a) − p^{ℓ−1}a^{ℓ}}{ℓ} )) − a^{ℓ−1}da − d(\frac{F_{ℓ}(a) − a^{ℓ}}{ℓ})

whose image in Ω̌_{𝕎(A)}^{1} is

(p^{ℓ−1} − 1)a^{ℓ−1}da − \frac{p^{ℓ−1} − 1}{ℓ} d(a^{ℓ})

which is zero since d is a derivation. Similarly, if p = 2 and ℓ ≠ p, we find F_{ℓ}(F_{2}dV_{2}(a) − da − d log[−1] · a) = F_{2}(F_{ℓ}dV_{2}(a)) − F_{ℓ}da − d log[−1] · F_{ℓ}(a)

= F_{2}(V_{2}(a)^{ℓ−1}dV_{2}(a) + d(\frac{F_{ℓ}V_{2}(a) − V_{2}(a)^{ℓ}}{ℓ})

− a^{ℓ−1}da − d(\frac{F_{ℓ}(a) − a^{ℓ}}{ℓ}) − d log[−1] · F_{ℓ}(a)

= 2^{ℓ−1}a^{ℓ−1}F_{2}dV_{2}(a) + F_{2}dV_{2}(\frac{F_{ℓ}(a) − 2^{ℓ−1}a^{ℓ}}{ℓ})

− a^{ℓ−1}da − d(\frac{F_{ℓ}(a) − a^{ℓ}}{ℓ}) − d log[−1] · F_{ℓ}(a)

whose image in Ω̌_{𝕎(A)}^{1} is

(2^{ℓ−1} − 1)a^{ℓ−1}da − \frac{2^{ℓ−1} − 1}{ℓ} d(a^{ℓ}) + d log[−1] · (\frac{F_{ℓ}(a) − 2^{ℓ−1}a^{ℓ}}{ℓ} − F_{ℓ}(a))

which is zero since d is a derivation and since ℓ is congruent to 1 modulo 2. Finally, if ℓ = 2 and p ≠ ℓ, we find

F_{2}(F_{p}dV_{p}(a) − da) = F_{p}(F_{2}dV_{p}(a)) − F_{2}da

= F_{p}(V_{p}(a)dV_{p}(a) + d(\frac{F_{2}V_{p}(a) − V_{p}(a)^{2}}{2})) − ada − d(\frac{F^{2}(a) − a^{2}}{2} )

= paF_{p}dV_{p}(a) + F_{p}dV_{p}(\frac{F_{2}(a) − pa^{2}}{2} ) − ada − d(\frac{F^{2}(a) − a^{2}}{2} )

whose image in Ω̌_{𝕎(A)}^{1} is

(p − 1)ada − \frac{p − 1}{2} d(a^{2})

which is zero since d is a derivation. Lemma 3.8. For all positive integers n and a ∈ 𝕎(A), the relation

F_{n}dV_{n}(a) = da + (n − 1)d log[−1] · a

holds in Ω̌_{𝕎(A)}^{1} .

Proof. We argue by induction on the number r of prime factors in n that the stated relation holds for all a ∈ 𝕎(A). The case r = 1 follows from Definition 3.6. So we let n be a positive integer with r > 1 prime factors and assume that the relation has been proved for all positive integers with less than r prime factors. We write n = pm with p a prime number. Then

F_{n}dV_{n}(a)= F_{p}F_{m}dV_{m}V_{p}(a) = F_{p}(dV_{p}(a) − (m − 1)d log[−1] · V_{p}(a))
= F_{p}dV_{p}(a) − (m − 1)d log[−1] · F_{p}V_{p}(a)
= da − (p − 1)d log[−1] · a − p(m − 1)d log[−1] · a
= da − (n − 1)d log[−1] · a

which proves the induction step.

Lemma 3.9. The graded derivation d: Ω̂_{𝕎(A)}^{·} → Ω̂_{𝕎(A)}^{·} induces a graded derivation

d: Ω̌_{𝕎(A)}^{·} → Ω̌_{𝕎(A)}^{·} .

Proof. We must show that for all prime numbers p and a ∈ 𝕎(A), the element

d(F_{p}dV_{p}(a) − da − (p − 1)d log[−1] · a) ∈ Ω̂_{𝕎(A)}^{2}

maps to zero in Ω̌_{𝕎(A)}^{2} . First, for p = 2, we have

d(F_{2}dV_{2}(a) − da − d log[−1] · a) = dF_{2}dV_{2}(a) − dda + d log[−1] · da
= F_{2}ddV_{2}(a) = 2F_{2}ddV_{2}(a) = 2d log[−1] · F_{2}dV_{2}(a)

which is even zero in Ω̂_{𝕎(A)}^{2} . For p odd, we find

d(F_{p}dV_{p}(a) − da) = d(V_{p}(a)^{p−1} dV_{p}(a) + dΔ_{p}(a)) − dda
= (p − 1)V_{p}(a)^{p−2} dV_{p}(a)dV_{p}(a) + d log[−1] · F_{p}dV_{p}(a) − d log[−1] · da
= (p − 1)V_{p}(a)^{p−2} d log[−1] · F_{2}dV_{p}(a) + d log[−1] · (F_{p}dV_{p}(a) − da)
= d log[−1] · (F_{p}dV_{p}(a) − da)

which maps to zero in Ω̌_{𝕎(A)}^{2} as desired. Here we have used that p − 1 is even.

Definition 3.10. Let A be a ring, let S ⊂ ℕ be a truncation set, and let I_{S}(A) ⊂ 𝕎(A) be the kernel of the restriction map R_{S}^{ℕ}: 𝕎(A) → 𝕎_{S}(A). The maps

\begin{aligned}R_{S}^{ℕ}: Ω̂_{𝕎(A)}^{·} & → Ω̂_{𝕎_{S}(A)}^{·} \\ R_{S}^{ℕ}: Ω̌_{𝕎(A)}^{·} & → Ω̌_{𝕎_{S}(A)}^{·} \end{aligned}

are defined to be the quotient maps that annihilate the respective graded ideals generated by I_{S}(A) and dI_{S}(A).

Remark 3.11. The kernel K_{S}^{·} of the canonical projection Ω̂_{𝕎_{S}(A)}^{·} → Ω̌_{𝕎_{S}(A)}^{·} is equal to the graded ideal generated by the elements

V_{p}(R_{S/p}^{S}(a)^{p−1} )dV_{p}R_{S/p}^{S}(a) − p^{p−2} dV_{p}R_{S/p}^{S}(a^{p}) − (p − 1)d log[−1]_{S}· a

with p a prime number and a ∈ 𝕎_{S}(A). Remark 3.12. Let p be a prime number and let S be a p-typical truncation set. Then the ideal V_{p}𝕎_{S/p}(A) ⊂ 𝕎_{S}(A) has a canonical divided power structure defined by

V_{p}(a)^{[n]}= \frac{p^{n−1}}{n!} V_{p}(a^{n}).

Suppose, in addition, that the prime number p is odd and that A is a ℤ_{(p)}-algebra. Then d: 𝕎_{S}(A) → Ω̌_{𝕎_{S}(A)}^{1} is a divided power derivation in the sense that

d(V_{p}(a)^{[n]}) = V_{p}(a)^{[n−1]}dV_{p}(a)

and it is universal with this property; see [18, Lemma 1.2].

Lemma 3.13. The derivation, restriction, and Frobenius induce maps

d: Ω̂_{𝕎_{S}(A)}^{·} → Ω̂_{𝕎_{S}(A)}^{·}  (resp. d: Ω̌_{𝕎_{S}(A)}^{·} → Ω̌_{𝕎_{S}(A)}^{·})
R_{T}^{S}: Ω̂_{𝕎_{S}(A)}^{·} → Ω̂_{𝕎_{T}(A)}^{·}  (resp. R_{T}^{S}: Ω̌_{𝕎_{S}(A)}^{·} → Ω̌_{𝕎_{T}(A)}^{·})
F_{n}: Ω̂_{𝕎_{S}(A)}^{·} → Ω̂_{𝕎_{S/n}(A)}^{·}  (resp. F_{n}: Ω̌_{𝕎_{S}(A)}^{·} → Ω̌_{𝕎_{S/n}(A)}^{·})

Proof. To prove the statement for d, we note that as d is a derivation, it suffices to show that R_{S}^{ℕ}(ddI_{S}(A)) ⊂ Ω̂_{𝕎_{S}(A)}^{2} is zero. But if x ∈ I_{S}(A), then

R_{S}^{ℕ}(ddx) = R_{S}^{ℕ}(d log[−1] · dx) = R_{S}^{ℕ}(d log[−1]) · R_{S}^{ℕ}(dx)

which is zero as desired. It follows that R_{T}^{ℕ}(dI_{S}(A)) = dR_{T}^{ℕ}(I_{S}(A). Hence, also the statement for R_{T}^{S} follows as R_{T}^{ℕ}(I_{S}(A) is trivially zero. Finally, to prove the statement for F_{n}, we show that both R_{S/n}^{ℕ}(F_{n}(I_{S}(A))) and R_{S/n}^{ℕ}(F_{n}(dI_{S}(A))) are zero. For the former, this follows immediately from Lemma 1.4, and for the latter, it will suffice to show that for divisors e of n, Δ_{e}(I_{S}(A)) ⊂ I_{S/e}(A). Moreover, to prove this, we may assume that A is torsion free. So let e be a divisor of n and assume that for all proper divisors d of e, Δ_{d}(I_{S}(A)) ⊂ I_{S/d}(A). Since F_{e}(I_{S}(A)) ⊂ I_{S/e}(A), the formula

F_{e}(a) = ∑_{d|e}dΔ_{d}(a)^{e/d}

shows that eΔ_{e}(I_{S}(A)) ⊂ I_{S/e}(A). The lemma follows.

Lemma 3.14. For every ring A, the differential graded algebras Ω̂_{A}^{·} and Ω_{A}^{·} are equal and the canonical projection Ω̂_{A}^{·} → Ω̌_{A}^{·} is an isomorphism.

Proof. Since d log[−1]_{{1}} is zero, Ω̂_{A}^{·} = Ω_{A}^{·} as stated. Moreover, Remark 3.11 shows that the kernel K_{{1}} of the canonical projection Ω̂_{A}^{·} → Ω̌_{A}^{·} is zero.

4 The big de Rham-Witt complex

In this section, we construct the big de Rham-Witt complex. The set J of truncation sets is partially ordered under inclusion. We consider J as a category with one morphism from T to S if T ⊂ S. If A is a ring, the assignment S ↦ 𝕎_{S}(A) defines a contravariant functor from the category J to the category of rings, and this functor takes colimits in the category J to limits in the category of rings. Moreover, the assignment S ↦ S/n is an endo-functor on the category J, and the ring homomorphism

F_{n}: 𝕎_{S}(A) → 𝕎_{S/n}(A)

and the abelian group homomorphism

V_{n}: 𝕎_{S/n}(A) → 𝕎_{S}(A)

are natural transformations with respect to the variable S.

We proceed to define the notion of a Witt complex over A. The original definition given in [12, Definition 1.1.1] is not quite correct unless the prime 2 is either invertible or zero in A. The correct definition of a 2-typical Witt complex was given first by Costeanu [7, Definition 1.1]. The definition given below was also inspired by [21].

Definition 4.1. A Witt complex over A is a contravariant functor

S ↦ E_{S}^{·}

that to every truncation set S ⊂ ℕ assigns an anti-symmetric graded ring E_{S}^{·} and that takes colimits to limits together with a natural ring homomorphism

η_{S}: 𝕎_{S}(A) → E_{S}^{0}

and natural maps of graded abelian groups

\begin{aligned}d & : E_{S}^{q} → E_{S}^{q+1} \\ F_{n} & : E_{S}^{q} → E_{S/n}^{q} (n ∈ ℕ) \\ V_{n} & : E_{S/n}^{q}→ E_{S}^{q} (n ∈ ℕ) \end{aligned}

such that the following (i)–(v) hold.

(i) For all x ∈ E_{S}^{q} and x′∈ E_{S}^{q′},

\begin{matrix}d(x · x′) = d(x) · x′+ (−1)^{q}x · d(x′), \\ d(d(x)) = d logη_{S}([−1]_{S}) · d(x). \end{matrix}

(ii) For all positive integers m and n,

F_{1} = V_{1} = id, F_{m}F_{n} = F_{mn}, V_{n}V_{m}= V_{mn},
F_{n}V_{n} = n · id, F_{m}V_{n} = V_{n}F_{m} if (m,n) = 1,
F_{n}η_{S} = η_{S/n}F_{n}, η_{S}V_{n} = V_{n}η_{S/n}.

(iii) For all positive integers n, the map F_{n} is a ring homomorphism and the maps F_{n} and V_{n} satisfy the projection formula that for all x ∈ E_{S}^{q} and y ∈ E_{S/n}^{q′′},

x · V_{n}(y) = V_{n}(F_{n}(x)y).

(iv) For all positive integers n and all y ∈ E_{S/n}^{q},

F_{n}dV_{n}(y) = d(y) + (n − 1)d logη_{S/n}([−1]_{S/n}) · y.

(v) For all positive integers n and a ∈ A,

F_{n}dη_{S}([a]_{S}) = η_{S/n}([a]_{S/n}^{n−1})dη_{S/n}([a]_{S/n}).

A map of Witt complexes is a natural map of graded rings

f: E_{S}^{·} → E′_{S}^{·}

such that fη = η′f, fd = d′f, fF_{n} = F′_{n}f, and fV_{n} = V′_{n}f.

Remark 4.2. (a) For T ⊂ S a pair of truncation sets, we write R_{T}^{S}: E_{S}^{·} → E_{T}^{·} for the map of graded rings that is part of the structure of a Witt complex.

(b) Every Witt functor is determined, up to canonical isomorphism, by its value on finite truncation sets.

(c) The element d logη_{S}([−1]_{S}) is annihilated by 2. Indeed, since d is a derivation,

2d logη_{S}([−1]_{S}) = d logη_{S}([1]_{S}) = 0.

Therefore, d logη_{S}([−1]_{S}) is zero if 2 is invertible in A and hence in 𝕎_{S}(A). It is also zero if 2 = 0 in A since, in this case, [−1]_{S} = [1]_{S}. Finally, the general formula

[−1]_{S} = −[1]_{S}+ V_{2}([1]_{S/2})

shows that d logη_{S}([−1]_{S}) is zero if every n ∈ S is odd.

Lemma 4.3. Let m and n be positive integers, let c = (m,n) be the greatest common divisor, and let i and j be any pair of integers such that mi + nj = c. The following relations hold in every Witt complex.

dF_{n} = nF_{n}d, V_{n}d = ndV_{n},
F_{m}dV_{n} = idF_{m/c}V_{n/c}+ jF_{m/c}V_{n/c}d + (c − 1)d logη_{S/m}([−1]_{S/m}) · F_{m/c}V_{n/c},
d logη_{S}([−1]_{S}) = ∑_{r⩾1}2^{r−1} dV_{2^{r}}η_{S/2^{r}}([1]_{S/2^{r}}),
d logη_{S}([−1]_{S}) · d logη_{S}([−1]_{S}) = 0, dd logη_{S}([−1]_{S}) = 0,
F_{n}(d logη_{S}([−1]_{S})) = d logη_{S/n}([−1]_{S/n}),

Proof. The following calculation verifies the first two relations.

dF_{n}(x)= F_{n}dV_{n}F_{n}(x) − (n − 1)d logη[−1] · F_{n}(x)
= F_{n}d(V_{n}η([1]) · x) − (n − 1)d logη([−1]) · F_{n}(x)
= F_{n}(dV_{n}η([1]) · x + V_{n}η([1]) · dx) − (n − 1)d logη([−1]) · F_{n}(x)

= F_{n}dV_{n}η([1]) · F_{n}(x) + F_{n}V_{n}η([1]) · F_{n}d(x) − (n − 1)d logη([−1]) · F_{n}(x)

= (n − 1)d logη([−1]) · F_{n}(x) + nF_{n}d(x) − (n − 1)d logη([−1]) · F_{n}(x)
= nF_{n}d(x)

V_{n}d(x)= V_{n}(F_{n}dV_{n}(x) − (n − 1)d logη([−1]) · x) = V_{n}η([1])dV_{n}(x) − (n − 1)V_{n}(d logη([−1]) · x)

= d(V_{n}η([1])V_{n}(x)) − dV_{n}η([1])V_{n}(x) − (n − 1)V_{n}(d logη([−1]) · x)

= dV_{n}(F_{n}V_{n}η([1])x) − V_{n}(F_{n}dV_{n}η([1])x) − (n − 1)V_{n}(d logη([−1]) · x)

= ndV_{n}(x) − 2V_{n}(d logη([−1]) · x)
= ndV_{n}(x)

Next, the last formula follows from the calculation

F_{m}(d logη([−1]_{S}))= F_{m}(η([−1]^{−1} )dη([−1])) = F_{m}η([−1]^{−1} )F_{m}dη([−1])
= η([−1]^{−m})η([−1]^{m−1})dη([−1])
= η([−1]^{−1} )dη([−1]) = d logη([−1]).

Using the three relations proved thus far together with the projection formula, we find F_{m}dV_{n}(x)= F_{m/c}F_{c}dV_{c}V_{n/c}(x)

= F_{m/c}dV_{n/c}(x) + (c − 1)d logη([−1]) · F_{m/c}V_{n/c}(x)

= ((m/c)i + (n/c)j)F_{m/c}dV_{n/c}(x) + (c − 1)d logη([−1]) · F_{m/c}V_{n/c}(x) = idF_{m/c}V_{n/c}(x) + jF_{m/c}V_{n/c}d(x) + (c − 1)d logη([−1]) · F_{m/c}V_{n/c}(x).

To prove the formula for d logη([−1]_{S}), we use the formula

[−1]_{S} = −[1]_{S}+ V_{2}([1]_{S/2})

which is easily verified by evaluating ghost coordinates. We find

d logη([−1]_{S}) = η([−1]_{S})dη([−1]_{S}) = η(−[1]_{S}+ V_{2}([1]_{S/2})dη(−[1]_{S}+ V_{2}([1]_{S/2})
= −dV_{2}η([1]_{S/2}) + V_{2}(F_{2}dV_{2}η([1]_{S/2}))
= −dV_{2}η([1]_{S/2}) + V_{2}(d logη([−1]_{S/2}))
= dV_{2}η([1]_{S/2}) + V_{2}(d logη([−1]_{S/2}))

where the last equality uses that 2dV_{2}η([1]_{S/2}) = V dη([1]_{S/2}) = 0. The stated formula follows by easy induction. Using this formula, we find

\begin{aligned}dV_{2}d logη([−1]_{S/2}) & = ∑_{r⩾1}2^{r}ddV_{2^{r+1}} η([1]_{S/2^{r+1}} ) \\ & = ∑_{r⩾1}2^{r}d logη([−1]_{S}) · dV_{2^{r+1}} η([1]_{S/2^{r+1}} ) \end{aligned}

which is zero, since 2d logη([−1]_{S}) = 0. The calculation

(d logη([−1]_{S}))^{2} = (dη([−1]_{S}))^{2} = (dV_{2}η([1]_{S/2}))^{2}

= d(V_{2}η([1]_{S/2})dV_{2}η([1]_{S/2})) − V_{2}η([1]_{S/2})ddV_{2}η([1]_{S/2})

= dV_{2}d logη([−1]_{S/2}) − V_{2}η([1]_{S/2})dV_{2}d logη([−1]_{S/2})
= dV_{2}d logη([−1]_{S/2})η([1]_{S}− V_{2}([1]_{S/2}))

then shows that (d logη([−1]_{S}))^{2} is zero as stated. This, in turn, shows that

dd logη([−1]_{S}) = dη([−1]_{S}dη([−1]_{S}))
= dη([−1]_{S})dη([−1]_{S}) + η([−1]_{S})ddη([−1]_{S})
= η([−1]_{S})ddη([−1]_{S})
= η([−1]_{S})d logη([−1]_{S})dη([−1]_{S})
= dη([−1]_{S})dη([−1]_{S}) = 0.

This completes the proof.

Proposition 4.4. For every Witt complex and every positive integer m, the following diagram, where the horizontal maps take a_{0}da_{1} to η(a_{0})dη(a_{1}), commutes.

\xymatrix{Ω_{𝕎_{S}(A)}^{1} \ar[r]^{η_{S}} \ar[d]^{F_{m}} & E_{S}^{1} \ar[d]^{F_{m}} \\ Ω_{𝕎_{S/m}(A)}^{1} \ar[r]^{η_{S/m}} & E_{S/m}^{1}}

Proof. We may assume that m = p is a prime number and that S = ℕ. Moreover, by Lemma 1.5 (i), it will suffice to show that for every n ∈ ℕ and a ∈ A,

F_{p}dV_{n}η_{ℕ}([a]_{ℕ}) = η_{ℕ}F_{p}dV_{p}([a]_{ℕ}).

To ease notation, we suppress the subscript ℕ. We first suppose n is prime to p and set k = (1 − n^{p−1} )/p and ℓ = n^{p−2} . Since kp + ℓn = 1, Lemma 4.3 shows that

F_{p}dV_{n}η([a])= k · dV_{n}F_{p}η([a]) + ℓ · V_{n}F_{p}dη([a])
= k · dV_{n}η([a]^{p}) + ℓ · V_{n}(η([a])^{p−1} dη([a]))
= k · dV_{n}η([a]^{p}) + ℓ · V_{n}η([a]^{p−1} d[a])

Moreover, by Theorem 2.13 and Remark 1.19, we have

ηF_{p}dV_{n}([a])= η(V_{n}([a])^{p−1} dV_{n}([a]) + dΔ_{p}V_{n}([a]))
= η(ℓ · V_{n}([a]^{p−1} )dV_{n}([a]) + k · dV_{n}([a]^{p}))
= ℓ · V_{n}η([a]^{p−1} )dV_{n}η([a]) + k · dV_{n}η([a]^{p})
= ℓ · V_{n}η([a]^{p−1} F_{n}dV_{n}η([a])) + k · dV_{n}η([a]^{p})
= ℓ · V_{n}η([a]^{p−1} d[a]) + k · dV_{n}η([a]^{p})

where the last equality uses that n^{p−2} (n − 1)d logη([−1]) is zero. This proves that the desired equality holds if p does not divide n. Suppose next that p divides n and write n = pr. We consider the cases p = 2 and p odd separately. First, for p odd,

F_{p}dV_{n}η([a])= dV_{r}η([a])

ηF_{p}dV_{n}([a])= η(V_{n}([a])^{p−1} dV_{n}([a]) + dV_{r}([a]) − p^{p−2} r^{p−1} dV_{n}([a]^{p}))

= V_{n}η([a])^{p−1} dV_{n}η([a]) + dV_{r}η([a]) − p^{p−2} r^{p−1} dV_{n}(η([a])^{p})

and the desired equality follows from the calculation

p^{p−2} r^{p−1} dV_{n}(η([a])^{p})= p^{p−3} r^{p−2} V_{n}d(η([a])^{p})
= p^{p−2} r^{p−2} V_{n}(η([a])^{p−1} dη([a]))
= n^{p−2} V_{n}(η([a])^{p−1} )dV_{n}η([a])
= V_{n}η([a])^{p−1} dV_{n}η([a]).

Finally, if p = 2, then

F_{2}dV_{n}η([a])= dV_{r}η([a]) + d logη([−1]) · V_{r}η([a])
ηF_{2}dV_{n}([a])= η(V_{n}([a])dV_{n}([a]) + dV_{r}([a]) − rdV_{n}([a]^{2}))
= V_{n}η([a])dV_{n}η([a]) + dV_{r}η([a]) − rdV_{n}(η([a])^{2})

and must show that

d logη([−1]) · V_{r}η([a]) = V_{n}η([a])dV_{n}η([a]) − rdV_{n}(η([a])^{2}).

Suppose first that r = 1. Since [−1] = −[1] + V_{2}([1]), we have

d logη([−1]) = V_{2}η([1])dV_{2}η([1]) − dV_{2}η([1]),

and hence,

d logη([−1]) · η([a]) = V_{2}(η([a])^{2})dV_{2}η([1]) − η([a])dV_{2}η([1])
= V_{2}(η([a]^{2})F_{2}dV_{2}η([1])) − dV_{2}(η([a])^{2}) + dη([a]) · V_{2}η([1])
= V_{2}(η([a])^{2}d logη([−1])) − dV_{2}(η([a])^{2}) + V_{2}(η([a])dη([a]))
= −dV_{2}(η([a])^{2}) + V_{2}(η([a]))dV_{2}η([a])

as desired. In general, we apply V_{r} to the equation just proved. This gives

d logη([−1]) · V_{r}η([a]) = −rdV_{n}(η([a])^{2}) + V_{r}(V_{2}η([a])dV_{2}η([a]))
= −rdV_{n}(η([a])^{2}) + V_{n}(η([a])F_{2}dV_{2}η([a]))
= −rdV_{n}(η([a])^{2}) + V_{n}(η([a])F_{n}dV_{n}η([a]))
= −rdV_{n}(η([a])^{2}) + V_{n}η([a])dV_{n}η([a])

where F_{2}dV_{2} = F_{n}dV_{n} since n is even. This completes the proof.

Remark 4.5. For every Witt complex over the ring A, E_{ℕ}^{1} is a (𝕎(A),Δ)-module in the sense of Definition 2.4 with structure maps λ_{E,n}= F_{n}: E_{ℕ}^{1} → E_{ℕ}^{1}. Now, Proposition 4.4 states that d is a derivation in the sense of Definition 2.11

d: (𝕎(A),Δ_{A}) → (E_{ℕ}^{1},λ_{E}).

Moreover, in Definition 4.1, we may substitute this statement for axiom (v).

Corollary 4.6. Let E_{S}^{·} be a Witt complex over the ring A. There is a unique natural homomorphism of graded rings

η_{S}: Ω̌_{𝕎_{S}(A)}^{·} → E_{S}^{·}

that extends the natural ring homomorphism η_{S}: 𝕎_{S}(A) → E_{S}^{0} and commutes with the derivations. In addition, for every positive integer m, the diagram

\xymatrix{Ω̌_{𝕎_{S}(A)}^{·} \ar[r]^{η_{S}} \ar[d]^{F_{m}} & E_{S}^{·} \ar[d]^{F_{m}} \\ Ω̌_{𝕎_{S/m}(A)}^{·} \ar[r]^{η_{S/m}} & E_{S/m}^{·}}

commutes.

Proof. The map η_{S} necessarily is given by

η_{S}(a_{0}da_{1}...da_{q}) = η_{S}(a_{0})dη_{S}(a_{1})...dη_{S}(a_{q}).

We show that this formula gives a well-defined map. First, from Proposition 4.4, we find that for all a ∈ 𝕎(A),

F_{2}dη_{ℕ}(a) = η_{ℕ}F_{2}d(a) = η_{ℕ}(ada + dΔ_{2}(a)) = η_{ℕ}(a)dη_{N}(a) + dη_{ℕ}Δ_{2}(a).

Applying d to this equation, the left-hand and right-hand sides becomes

dF_{2}dη_{ℕ}(a) = 2F_{2}ddη_{N}(a) = 0

and

\begin{matrix}dη_{N}(a)dη_{ℕ}(a) + d logη_{ℕ}([−1])_{ℕ} · (η_{ℕ}(a)dη_{ℕ}(a) + dη_{ℕ}Δ_{2}(a)) \\ = dη_{N}(a)dη_{ℕ}(a) + d logη_{ℕ}([−1])_{ℕ} · F_{2}dη_{ℕ}(a), \end{matrix}

respectively. This shows that the formula above gives a well-defined map of graded rings from Ω̂_{𝕎_{S}(A)}^{·} to E_{S}^{·}, and by axiom (iv) in Definition 4.1, this map factors through the projection onto Ω̌_{𝕎_{S}(A)}^{·}. Finally, Proposition 4.4 shows that the diagram in the statement commutes.

Proof of Theorem B. We define quotient maps of graded rings

η_{S}: Ω̌_{𝕎_{S}(A)}^{·} → 𝕎_{S}Ω_{A}^{·}

together with maps R_{T}^{S} and F_{n} of graded rings and a graded derivation d that make the diagrams of the statement commute and a map of graded abelian groups

V_{n}: 𝕎_{S/n}Ω_{A}^{·}→ 𝕎_{S}Ω_{A}^{·}

that make the following diagrams commute.

\xymatrix{𝕎_{S/n}(A) \ar[r]^{η_{S/n}} \ar[d]^{V_{n}} & 𝕎_{S/n}Ω_{A}^{0} \ar[d]^{V_{n}} & 𝕎_{S/n}Ω_{A}^{·} \ar[r]^{V_{n}} \ar[d]^{R_{T /n}^{S/n}} & 𝕎_{S}Ω_{A}^{·} \ar[d]^{R_{T}^{S}} \\ 𝕎_{S}(A) \ar[r]^{η_{S}} & 𝕎_{S}Ω_{A}^{0} & 𝕎_{T /n}Ω_{A}^{·} \ar[r]^{V_{n}} & 𝕎_{T}Ω_{A}^{·}}

[figure: 𝕎_{S/n}Ω_{A}^{·}⊗ 𝕎_{S/n}Ω_{A}^{·}; μ; 𝕎_{S/n}Ω_{A}^{·}⊗ 𝕎_{S}Ω_{A}^{·}; id⊗F_{n}; V_{n}⊗id; 𝕎_{S}Ω_{A}^{·}⊗ 𝕎_{S}Ω_{A}^{·}; μ; 𝕎_{S/n}Ω_{A}^{·}; V_{n}; 𝕎_{S}Ω_{A}^{·}]

It will suffice to define the above structure for finite truncation sets S. We proceed recursively starting from 𝕎_{∅}Ω_{A}^{·} that we define to be the zero graded ring. So let S ≠ ∅ be a finite truncation set and assume, inductively, that the maps η_{T}, R_{U}^{T}, F_{n}, d, and V_{n} have been defined for all proper truncation sets T ⊂ S, all truncation sets U ⊂ T, and all positive integers n such that the diagrams in the statement and the diagrams above commute. We then define

η_{S}: Ω̌_{𝕎_{S}(A)}^{·} → 𝕎_{S}Ω_{A}^{·}

to be the quotient map that annihilates the graded ideal N_{S}^{·}generated by all sums

\begin{matrix}∑_{α} & V_{n}(x_{α})dy_{1,α} ...dy_{q,α} \\ d(∑_{α} & V_{n}(x_{α})dy_{1,α} ...dy_{q,α}) \end{matrix}

for which the Witt vectors x_{α} ∈ 𝕎_{S/n}(A) and y_{1,α} ,...,y_{q,α} ∈ 𝕎_{S}(A) and the integers n ⩾ 2 and q ⩾ 1 satisfy that the sum

η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

is zero in 𝕎_{S/n}Ω_{A}^{q}. For every positive integer n, we now define

V_{n}: 𝕎_{S/n}Ω_{A}^{·}→ 𝕎_{S}Ω_{A}^{·}

to be the map of graded abelian groups given by

V_{n}η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) = η_{S}(V_{n}(x)dy_{1}...dy_{q}).

Here we use that every element of 𝕎_{S/n}Ω_{A}^{q} can be written as a sum of elements of the form η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) with x ∈ 𝕎_{S/n}(A) and y_{1},...,y_{q} ∈ 𝕎_{S}(A). Indeed,

dx = F_{n}dV_{n}(x) − (n − 1)d log[−1]_{S/n}· x = F_{n}dV_{n}(x) − (n − 1)xF_{n}d([−1]_{S}).

By definition, the map V_{1} is the identity map, and that

V_{mn}= V_{m}V_{n}: 𝕎_{S/mn}Ω_{A}^{q}→ 𝕎_{S}Ω_{A}^{q}

for all integers m,n ⩾ 2 and q ⩾ 0 follows from the calculation

V_{mn}η_{S/mn}(xF_{mn}dy_{1}...F_{mn}dy_{q}) = η_{S}(V_{mn}(x)dy_{1}...dy_{q})
= η_{S}(V_{m}(V_{n}(x))dy_{1}...dy_{q}) = V_{m}η_{S/m}(V_{n}(x)F_{m}dy_{1}...F_{m}dy_{q})
= V_{m}(V_{n}(η_{S/mn}(x))F_{m}dη_{S/m}y_{1}...F_{m}dη_{S/m}y_{q})
= V_{m}V_{n}η_{S/mn}(xF_{mn}dy_{1}...F_{mn}dy_{q})

where the last equality uses the inductive hypothesis.

We proceed to construct the right-hand vertical maps in the square diagrams in the statement. To this end, it will suffice to show that the left-hand vertical maps in these diagrams satisfy R_{T}^{S}(N_{S}^{q}) ⊂ N_{T}^{q}, d(N_{S}^{q}) ⊂ N_{S}^{q+1} , and F_{n}(N_{S}^{q}) ⊂ N_{S/n}^{q}, respectively. We therefore fix a positive integer n and an element

ω = ∑_{α} V_{n}(x_{α})dy_{1,α} ...dy_{q,α} ∈ Ω̌_{𝕎_{S}(A)}^{q}

such that

η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α}) ∈ 𝕎_{S/n}Ω_{A}^{q}

is zero and show that η_{T}R_{T}^{S}(ω), η_{S}(ddω), η_{S/m}F_{m}(ω), and η_{S/m}F_{m}(dω) are all zero. First, in order to show that

η_{T}R_{T}^{S}(ω) = η_{T}(∑_{α} V_{n}R_{T /n}^{S/n}(x_{α})dR_{T}^{S}(y_{1,α})...dR_{T}^{S}(y_{q,α}))

is zero, it suffices by the definition of the ideal N_{T}^{·} to show that

η_{T /n}(∑_{α} R_{T /n}^{S/n}(x_{α})F_{n}dR_{T}^{S}(y_{1,α})...F_{n}dR_{T}^{S}(y_{q,α}))

is zero. But this element is equal to

η_{T /n}R_{T /n}^{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which, by the inductive hypothesis, is equal to

R_{T /n}^{S/n}η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which we assumed to be zero. Similarly, we have

\begin{aligned}η_{S}(ddω) & = η_{S}(d logη_{S}([−1]_{S}) · dω) = η_{S}d(d logη_{S}([−1]_{S}) · ω) \\ & = η_{S}d(∑_{α} V_{n}(x_{α} [−1]_{S/n}^{n})d[−1]_{S}dy_{1,α} ...dy_{q,α}), \end{aligned}

and by the definition of N_{S}^{·}, this is zero since

\begin{matrix}η_{S/n}(∑_{α} x_{α} η_{S/n}([−1]_{S/n}^{n})F_{n}dη_{S}([−1]_{S})F_{n}dy_{1,α} ...F_{n}dy_{q,α}) \\ = d logη_{S/n}([−1]_{S/n}) · η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α}) \end{matrix}

is zero. Next, to prove that η_{S/m}F_{m}(ω) and η_{S/m}F_{m}(dω) are zero, it suffices to consider the three cases n = km, m = kn, and (m,n) = 1. First, if n = km, then

η_{S/m}F_{m}(ω) = η_{S/m}(∑_{α} F_{m}V_{n}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α})

= mη_{S/m}(∑_{α} V_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α})

= mV_{k}η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which is zero as desired. Here the last equality uses the inductive hypothesis. Similarly, using Proposition 3.3, we find

\begin{aligned}η_{S/m}F_{m}(dω) & = η_{S/m}F_{m}(∑_{α} dV_{n}(x_{α})dy_{1,α} ...dy_{q,α}) \\ & + qd logη_{S/m}([−1]_{S/m}) · η_{S/m}F_{m}(ω). \end{aligned}

The second summand on the right-hand side is zero, by what was just proved, and by Lemma 3.8, we may rewrite the first summand as the sum

\begin{matrix}η_{S/m}(∑_{α} dV_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α}) \\ + (m − 1)d logη_{S/m}([−1]_{S/m}) · η_{S/m}(∑_{α} V_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α}) \end{matrix}

where again the second summand is zero by what was proved above. We further rewrite the first summand as

\begin{matrix}η_{S/m}d(∑_{α} V_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α}) \\ − qd logη_{S/m}([−1]_{S/m}) · η_{S/m}(∑_{α} V_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α}) \end{matrix}

where the second summand is zero as before. Finally, using the inductive hypothesis, we rewrite the first summand as

dη_{S/m}(∑_{α} V_{k}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α}) = dV_{k}η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which is zero as desired. Next, suppose that m = kn. Then

\begin{aligned}η_{S/m}F_{m}(ω) & = nη_{S/m}F_{k}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α}) \\ & = nF_{k}η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α}) \end{aligned}

which is zero as desired. Here the last equality uses the inductive hypothesis. Using that η_{S/m}F_{m}(ω) is zero, we further find that

η_{S/m}F_{m}(dω) = η_{S/m}F_{m}(∑_{α} dV_{n}(x_{α})dy_{1,α} ...dy_{q,α})

= η_{S/m}F_{k}(∑_{α} dx_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

= η_{S/m}F_{k}d(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

= η_{S/k}F_{k}dη_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which is zero as desired. Here, again, the last equality uses the inductive hypothesis. Finally, we consider the case where m and n are relatively prime. We have

η_{S/m}F_{m}(ω) = η_{S/m}(∑_{α} V_{n}F_{m}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α})

= ∑_{α} V_{n}F_{m}η_{S/n}(x_{α})F_{m}dη_{S}y_{1,α} ...F_{m}dη_{S}y_{q,α}

= ∑_{α} V_{n}(F_{m}η_{S/n}(x_{α})F_{n}F_{m}dη_{S}y_{1,α} ...F_{n}F_{m}dη_{S}y_{q,α})

= V_{n}F_{m}η_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which is zero. Here the second and third equality uses the inductive hypothesis. To prove that also η_{S/m}F_{m}dω is zero, it will suffice to show that both mη_{S/m}F_{m}dω and nη_{S/n}F_{m}dω are zero. First, since η_{S/m}F_{m}(ω) is zero, so is

mη_{S/m}F_{m}dω = η_{S/m}dF_{m}ω = dη_{S/m}F_{m}(ω).

We next use the equality

\begin{matrix}ndV_{n}(x) \\ = dV_{n}F_{n}V_{n}(x) = d(V_{n}([1]_{S/n}) · V_{n}(x)) \end{matrix} ; = V_{n}(x)dV_{n}([1]_{S/n}) + V_{n}([1]_{S/n})dV_{n}(x)

together with the fact that η_{S/m}F_{m}(ω) is zero to see that

nη_{S/m}F_{m}(dω) = η_{S/m}F_{m}(∑_{α} ndV_{n}(x_{α})dy_{1,α} ...dy_{q,α})

= η_{S/m}F_{m}(∑_{α} V_{n}(x_{α})dV_{n}([1]_{S/n})dy_{1,α} ...dy_{q,α})

+ η_{S/m}F_{m}(∑_{α} V_{n}([1]_{S/n})dV_{n}(x_{α})dy_{1,α} ...dy_{q,α})

= η_{S/m}(∑_{α} V_{n}F_{m}(x_{α})F_{m}dV_{n}([1]_{S/n})F_{m}dy_{1,α} ...F_{m}dy_{q,α})

+ η_{S/m}(∑_{α} V_{n}F_{m}([1]_{S/mn})F_{m}dV_{n}(x_{α})F_{m}dy_{1,α} ...F_{m}dy_{q,α})

= V_{n}η_{S/mn}F_{m}(F_{n}dV_{n}([1]_{S/n}) · ∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

+ V_{n}η_{S/mn}F_{m}(∑_{α} F_{n}dV_{n}(x_{α})F_{n}dy_{1,α} ...F_{n}dy_{q,α})

= V_{n}η_{S/mn}F_{m}d(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

= V_{n}F_{m}dη_{S/n}(∑_{α} x_{α} F_{n}dy_{1,α} ...F_{n}dy_{q,α})

which is zero as desired. Here, again, the third and the last equalites use the inductive hypothesis.

To complete the recursive definition, it remains only to prove that for all positive integers n, ω ∈ 𝕎_{S}Ω_{A}^{q}, and τ ∈ 𝕎_{S/n}Ω_{A}^{r}, the projection formula

V_{n}(ω)τ = V_{n}(ωF_{n}(τ))

holds. It suffices to let ω = η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) and τ = η_{S}(zdw_{1}...dw_{r}). Then

V_{n}(ω)τ= V_{n}η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) · η_{S}(zdw_{1}...dw_{r})
= η_{S}(V_{n}(x)dy_{1}...dy_{q}· zdw_{1}...dw_{r})
= η_{S}(V_{n}(xF_{n}(z))dy_{1}...dy_{q}dw_{1}...dw_{r})
= V_{n}(η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) · η_{S/n}F_{n}(zdw_{1}...dw_{r}))
= V_{n}(η_{S/n}(xF_{n}dy_{1}...F_{n}dy_{q}) · F_{n}η_{S}(zdw_{1}...dw_{r}))
= V_{n}(ωF_{n}(z))

as desired. This completes the proof.

Definition 4.7. The initial Witt complex 𝕎_{S}Ω_{A}^{·} over the ring A is called the big de Rham-Witt complex of A.

Addendum 4.8. (i) For all q, the map η_{{1}}: Ω_{A}^{q}→ 𝕎_{{1}}Ω_{A}^{q} is an isomorphism. (ii) For all S, the map η_{S}: 𝕎_{S}(A) → 𝕎_{S}Ω_{A}^{0} is an isomorphism.

Proof. This follows immediately from the proof of Theorem B and Lemma 3.14.

5 Étale morphisms

The functor that to the ring A associates the 𝕎_{S}(A)-module 𝕎_{S}Ω_{A}^{q}defines a presheaf of 𝕎_{S}(𝒪)-modules on the category of affine schemes. In this section, we use the theorem of Borger [3] and van der Kallen [22] which we recalled as Theorem 1.22 to show that for S finite, this presheaf is a quasi-coherent sheaf of 𝕎_{S}(𝒪)-modules for the étale topology. This is the statement of Theorem C which we now prove.

Proof of Theorem C. Let α denote the map of the statement. We define a structure of Witt complex over B on the domain of α. By Theorem 1.22, the map

𝕎_{S}(f): 𝕎_{S}(A) → 𝕎_{S}(B)

is étale. Hence, the derivation d: 𝕎_{s}Ω_{A}^{q}→ 𝕎_{S}Ω_{A}^{q+1} extends uniquely to a derivation

d: 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}→ 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q+1}

defined by the formula

d(b ⊗ x) = (db)x + b ⊗ dx

where db is the image of b by the composition

𝕎_{S}(B) ⟶^{d} Ω_{𝕎_{S}(B)}^{1}⟵^{∼} 𝕎_{S} (B) ⊗_{𝕎_{S}(A)}Ω_{𝕎_{S}(A)}^{1}→ 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{1}.

Here the middle map and the right-hand map are the canonical isomorphism and the canonical projection, respectively. We further define R_{T}^{S} = R_{T}^{S}⊗R_{T}^{S} and F_{n} = F_{n}⊗F_{n}. To define the map

V_{n}: 𝕎_{S/n}(B) ⊗_{𝕎_{S/n}(A)}𝕎_{S/n}Ω_{A}^{q}→: 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}

we use that the square in the statement of Theorem 1.22 is cocartesian. It follows that the map

F_{n}⊗ id: 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S/n}Ω_{A}^{q}→ 𝕎_{S/n}(B) ⊗_{𝕎_{S/n}(A)}𝕎_{S/n}Ω_{A}^{q}

is an isomorphism, and we then define V_{n} to be the composition of the inverse of this isomorphism and the map

id⊗V_{n}: 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S/n}Ω_{A}^{q}→ 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}.

Finally, we define the map η_{S} to be the composition

𝕎_{S}(B) → 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}(A) −⟶^{id⊗η} 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{0}

of the canonical isomorphism and the map id⊗η_{S}. We proceed to show that this defines a Witt complex over B. The axioms (i)–(iii) of Definition 4.1 follow immediately from the definitions. For example, the identity dd(x) = d logη_{S}([−1]_{S})d(x) holds as the two sides are derivations which agree on 𝕎_{S}Ω_{A}^{q}. It remains to verify axioms (iv)–(v) of Definition 4.1.

To prove axiom (iv), we first show that for all b ∈ 𝕎_{S/n}(B),

F_{n}dV_{n}(b) + (n − 1)d logη_{S/n}([−1]_{S/n}) · b = d(b).

The right-hand side is the unique extension to 𝕎_{S/n}(B) of the derivation

d: 𝕎_{S/n}(A) → 𝕎_{S/n}Ω_{A}^{1}.

The left-hand side, D_{n}(b), is also an extension of this map since 𝕎_{S}Ω_{A}^{q} is a Witt complex over A. Hence, it will suffice to show that D_{n} is a derivation. Moreover, since D_{n} is an additive function of b, and since the square of rings in Theorem 1.22 is cocartesion, it is enough to consider elements of the form F_{n}(b)a with a ∈ 𝕎_{S/n}(A) and b ∈ 𝕎_{S}(B). Now,

D_{n}(F_{n}(b)aF_{n}(b′)a′) = D_{n}(F_{n}(bb′)aa′)

= F_{n}d(bb′V_{n}(aa′)) + (n − 1)d logη([−1])F_{n}(bb′)aa′

= F_{n}d(bb′)F_{n}V_{n}(aa′) + F_{n}(bb′)F_{n}dV_{n}(aa′) + (n − 1)d logη([−1])F_{n}(bb′)aa′ = F_{n}d(bb′)paa′+ F_{n}(bb′)d(aa′) = (F_{n}d(b)F_{n}(b′) + F_{n}(b)F_{n}d(b′))paa′+ F_{n}(bb′)(d(a)a′+ ad(a′)) = (F_{n}d(b)pa + F_{n}(b)d(a))F_{n}(b′)a′+ F_{n}(b)a(F_{n}d(b′)pa + F_{n}(b)d(a′)) = D_{n}(F_{n}(b)a)F_{n}(b′)a′+ F_{n}(b)aD_{n}(F_{n}(b′)a′)

which shows that D_{n} is a derivation, and hence, equal to the derivation d. We note that the special case of axiom (iv) proved thus far implies that

dF_{n} = nF_{n}d: 𝕎_{S}(B) → 𝕎_{S/n}(B) ⊗_{𝕎_{S/n}(A)}𝕎_{S/n}Ω_{A}^{1}.

To prove axiom (iv) in general, we must show that

F_{n}dV_{n}(b ⊗ x) + (n − 1)d logη_{S/n}([−1]_{S/n}) · b ⊗ x = d(b ⊗ x)

for all b ∈ 𝕎_{S/n}(B) and x ∈ 𝕎_{S/n}Ω_{A}^{q} . As before, this follows once we prove that the left-hand side, D_{n}(b⊗x), is a derivation. Moreover, since D_{n} is additive and since the square of rings in Theorem 1.22 is cocartesian, it will suffice to consider elements of the form F_{n}(b) ⊗ x with b ∈ 𝕎_{S}(B) and x ∈ 𝕎_{S/n}Ω_{A}^{q} . Finally, to prove that D_{n} is a derivation on elements of this form, it suffices to show that

D_{n}(F_{n}(b) ⊗ x) = D_{n}(F_{n}(b))x + F_{n}(b) ⊗ D_{n}(x).

Now, by the definition of V_{n} and by the case of axiom (iv) proved above, we find

D_{n}(F_{n}(b) ⊗ x) = F_{n}dV_{n}(F_{n}(b) ⊗ x) + (n − 1)d logη([−1]) · F_{n}(b) ⊗ x
= F_{n}d(b ⊗ V_{n}(x)) + (n − 1)d logη([−1]) · F_{n}(b) ⊗ x
= F_{n}d(b) · F_{n}V_{n}(x) + F_{n}(b) ⊗ F_{n}dV_{n}(x) + (n − 1)d logη([−1]) · F_{n}(b) ⊗ x
= nF_{n}d(b) · x + F_{n}(b) ⊗ D_{n}(x) = dF_{n}(b) · x + F_{n}(b) ⊗ D_{n}(x)
= D_{n}(F_{n}(b)) · x + F_{n}(b) ⊗ D_{n}(x)

as desired. This completes the proof of axiom (iv).

To prove axiom (v), we consider the following diagram where the left-hand horizontal maps are the canonical isomorphisms.

\xymatrix{Ω_{𝕎_{S}(B)}^{1} \ar[d]^{F_{n}} & 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}Ω_{𝕎_{S}(A)}^{1} \ar[l] \ar[r]^{id⊗η_{S}} \ar[d]^{F_{n}⊗F_{n}} & 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{1} \ar[d]^{F_{n}⊗F_{n}} \\ Ω_{𝕎_{S/n}(B)}^{1} & 𝕎_{S/n}(B) ⊗_{𝕎_{S/n}(A)}Ω_{𝕎_{S/n}(A)}^{1} \ar[l] \ar[r]^{id⊗η_{S/n}} & 𝕎_{S/n}(B) ⊗_{𝕎_{S/n}(A)}𝕎_{S/n}Ω_{A}^{1}}

The left-hand square commutes since F_{n}: Ω_{𝕎(B)}^{1}→ Ω_{𝕎(B)}^{1} is F_{n}-linear, and the righthand square commutes since 𝕎_{S}Ω_{A}^{q} is a Witt complex. Hence, also the outer square commutes and this immediately implies axiom (v); compare Remark 4.5.

We have proved that the domains of the canonical map α at the beginning of the proof form a Witt complex over B. Therefore, there exists a unique map

β: 𝕎_{S}Ω_{B}^{q} → 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}

of Witt complexes over B. The composition α ∘ β is a selfmap of the initial object 𝕎_{S}Ω_{B}^{q}, and therefore, is the identity map. The composition β ∘ α is a map of Witt complexes over B. In particular, it is a map of 𝕎_{S}(B)-modules, and therefore, is determined by the composition with the map

ι: 𝕎_{S}Ω_{A}^{q}→ 𝕎_{S}(B) ⊗_{𝕎_{S}(A)}𝕎_{S}Ω_{A}^{q}

that takes x to [1]_{S}⊗ x. But ι and β ∘ α ∘ ι both are maps of Witt complexes over A with domain the initial Witt complex over A. Therefore, the two maps are equal, and hence, also β ∘ α is the identity map. This completes the proof.

6 The big de Rham-Witt complex of the ring of integers

We finally evaluate the absolute de Rham-Witt complex of the ring of integers. If m and n are positive integers, we write (m,n) and [m,n] for the greatest common divisor and least common multiple of m and n, respectively. We define (m,n] to be the unique integer modulo [m,n] such that (m,n] ≡ 0 modulo m and (m,n] ≡ (m,n) modulo n, and define {m,n} to the unique integer modulo 2 that is non-zero if and only if both m and n are even. We note that (m,n] + (n,m] ≡ (m,n) modulo [m,n].

Theorem 6.1. The big de Rham-Witt complex of ℤ is given as follows.

\begin{aligned}𝕎_{S}Ω_{ℤ}^{0} & = ∏_{n∈S}ℤ · V_{n}η([1]_{S/n}) \\ 𝕎_{S}Ω_{ℤ}^{1} & = ∏_{n∈S}ℤ/nℤ · dV_{n}η([1]_{S/n}) \end{aligned}

and the groups in degrees q ⩾ 2 are zero. The multiplication is given by

V_{m}η([1]_{S/m}) · V_{n}η([1]_{S/n}) = (m,n) · V_{[m,n]}η([1]_{S/[m,n]})

V_{m}η([1]_{S/m}) · dV_{n}η([1]_{S/n}) = (m,n] · dV_{[m,n]}η([1]_{S/[m,n]})

+ {m,n} ∑_{r⩾1}2^{r−1} [m,n] · dV_{2^{r}[m,n]}η([1]_{S/2^{r}[m,n]})

and the mth Frobenius and Verschiebung maps are given by

F_{m}V_{n}η([1]_{S/n})= (m,n) · V_{n/(m,n)}η([1]_{S/(m,n)})
F_{m}dV_{n}η([1]_{S/n})= (m,n]/m · dV_{n/(m,n)}η([1]_{S/[m,n]})

+ {m,n} ∑_{r⩾1}(2^{r−1} n/(m,n)) · dV_{2^{r}n/(m,n)}η([1]_{S/2^{r}[m,n]})

V_{m}(V_{n}η([1]_{S/mn}))= V_{mn}η([1]_{S/mn})

V_{m}(dV_{n}η([1]_{S/mn}))= m · dV_{mn}η([1]_{S/mn}).

Proof. We begin by showing that the groups E_{S}^{q} and structure maps listed in the statement form a Witt complex over ℤ. For notational convenience we suppress the subscript S. We first show that the product is associative. The formula for the product shows in particular that

d logη([−1]) = (−η([1]) + V_{2}η([1]))dV_{2}η([1]) = ∑_{r⩾1}2^{r−1} dV_{2^{r}}η([1]).

We claim that

V_{m}η([1]) · dV_{n}η([1]) = (m,n] · dV_{[m,n]}η([1]) + {m,n} · V_{[m,n]}η([1])d logη([−1]).

Indeed, by the formula for d logη([−1]) which we just proved,

\begin{matrix}V_{e}η([1]) · d logη([−1]) = ∑_{r⩾1}2^{r−1} V_{e}η([1])dV_{2^{r}}η([1]) \\ = ∑_{r⩾1}2^{r−1} (e,2^{r}]dV_{[e,2^{r}]}η([1]) + \{e,2\} ∑_{s⩾1}2^{s−1} [e,2]dV_{2^{s}[e,2]}η([1]) \end{matrix}

But 2^{r−1} (m,2^{r}] is congruent to 2^{r−1} m modulo [m,2^{r}], and therefore, if e is odd, the lower left-hand summand is equal to ∑_{r⩾1}2^{r−1} edV_{2^{r}e}η([1]) and the lower right-hand summand is zero, and if e is even, the lower left-hand summand is zero and the lower right-hand summand is equal to ∑_{s⩾1}2^{s−1} edV_{2^{s}e}η([1]). The claim follows. A similar calculation shows that for all positive integers a and b,

V_{a}η([1]) · (V_{b}η([1]) · d logη([−1])) = (V_{a}η([1]) · V_{b}η([1])) · d logη([−1]).

We now show that the product is associative. On the one hand,

\begin{matrix}V_{l}η([1]) · (V_{m}([1]) · dV_{n}η([1])) = (l,[m,n]](m,n] · dV_{[l,[m,n]]}η([1]) \\ + (\{l,[m,n]\}(m,n] + (l,[m,n])\{m,n\}) · V_{[l,[m,n]]}η([1])d logη([−1]) \end{matrix}

and on the other hand,

\begin{matrix}(V_{l}η([1]) · V_{m}([1])) · dV_{n}η([1]) = (l,m)([l,m],n] · dV_{[[l,m],n]}η([1]) \\ + (l,m)\{[l,m],n\} · V_{[[l,m],n]}η([1])d logη([1]). \end{matrix}

Here [l,[m,n]] = [[l,m],n] and to prove that (l,[m,n]](m,n] and (l,m)([l,m],n] are congruent modulo [l,[m,n]], we use that [l,[m,n]]ℤ is the kernel of the map

ℤ → ℤ/lℤ × ℤ/mℤ × ℤ/nℤ

that takes a to (a + lℤ,a + mℤ,a + nℤ). So it will suffice to verify that the desired congruence holds modulo l, m, and n, respectively. By definition, both numbers are zero modulo l and m, and the congruence modulo n follows from the identity

(l,[m,n]) · (m,n) = (l,m) · ([l,m],n)

which is readily verified by multiplying by [l,[m,n]] = [[l,m],n] on both sides. We also note that {l,[m,n]}(m,n] + (l,[m,n]){m,n} and (l,m){[l,m],n} are well-defined integers modulo 2 which are non-zero if and only if n and exactly one of l and m are even. This shows that the product is associative.

We proceed to verify the axioms (i)–(v) of Definition 4.1. First, we note that since the sum (m,n] + (n,m] is congruent to (m,n) modulo [m,n] = [n,m], we find

dV_{m}η([1]) · V_{n}η([1]) + V_{m}η([1]) · dV_{n}η([1])
= (n,m]dV_{[n,m]}η([1]) + {n,m}V_{[n,m]}η([1])d logη([−1])
+ (m,n]dV_{[m,n]}η([1]) + {m,n}V_{[m,n]}η([1])d logη([−1])
= (m,n)dV_{[m,n]}η([1])

which verifies axiom (i). Next, to verify axiom (iv) we note that

F_{m}(d logη([−1]))= ∑_{r⩾1}2^{r−1} F_{m}dV_{2^{r}}η([1])

= ∑_{r⩾1}2^{r−1} (m,2^{r}]/m · dV_{[m,2^{r}]/m}η([1])

+ {m,2}V_{[m,2]/m}η([1])d logη([−1])

= d logη([−1]),

where the last equality uses that 2^{r−1} (m,2^{r}] ≡ 2^{r−1} m modulo [m,2^{r}]. Therefore, we may rewrite the stated formula for the Frobenius in the form

F_{m}dV_{n}η([1]) = (m,n]/m · dV_{n/(m,n)}η([1]) + {m,n} · V_{n/(m,n)}η([1])d logη([−1])

which verifies axiom (iv). Similarly, to prove axiom (iii), we first note that

\begin{matrix}V_{m}(d logη([1])) & = V_{m}(∑_{r⩾1}2^{r−1} dV_{2^{r}}η([1])) = ∑_{r⩾1}2^{r−1} mdV_{2^{r}m}η([1]) \\ = V_{m}η([1])d logη([1]). \end{matrix}

Therefore, we find that

V_{m}F_{m}dV_{n}η([1]) = V_{m}((m,n]/m · dV_{[m,n]/m}η([1])
+ {m,n} · V_{[m,n]/m}η([1])d logη([1]))
= (m,n] · dV_{[m,n]}η([1]) + {m,n} · V_{[m,n]}η([1])d logη([−1])
= V_{m}η([1])dV_{n}η([1])

where we have used that [m,n]/m = n/(m,n). This proves that axiom (iii) holds. We next consider axiom (ii) and begin by proving that the identity F_{l}F_{m} = F_{lm} holds. Since (lm,n) = (l,n/(m,n))(m,n) we find on the one hand that

\begin{matrix}F_{l}(F_{m}dV_{n}η([1])) = (l,n/(m,n)]/l · (m,n]/m · dV_{n/(lm,n)}η([1]) \\ + (\{l,n/(m,n)\}(m,n]/m + (l,n/(m,n))\{m,n\}) · V_{n/(lm,n)}η([1])d logη([−1]) \end{matrix}

and on the other hand that

\begin{matrix}F_{lm}dV_{n}η([1]) = (lm,n]/lm · dV_{n/(lm,n)}η([1]) \\ + \{lm,n\} · V_{n/(lm,n)}η([1])d logη([−1]). \end{matrix}

The classes (lm,n] and (l,n/(m,n)](m,n] are congruent modulo [lm,n]. For both are congruent to 0 modulo lm and congruent to (lm,n) = (l,n/(m,n))(m,n) modulo n. It remains only to note that {lm,n} and {l,n/(m,n)}(m,n]/m + (l,n/(m,n)){m,n} are well-defined integers modulo 2 which are non-zero if and only if n and at least one of l and m are even. This shows that F_{l}F_{m} = F_{lm}. The formulas V_{l}V_{m} = V_{lm} and F_{m}V_{m} = m · id are readily verified, so we next show that F_{l}V_{m} = V_{m}F_{l} provided that l and m are relatively prime. Using what was proved earlier, we find that

F_{l}V_{m}dV_{n}η([1]) = m(l,mn]/l · dV_{mn/(l,mn)}η([1])
+ m{l,mn} · V_{mn/(l,mn)}η([1])d logη([−1])
V_{m}F_{l}dV_{n}η([1]) = m(l,n]/l · dV_{mn/(l,n)η([1])}
+ {l,n} · V_{mn/(l,n)}η([1])d logη([−1])

Since (l,m) = 1, we have (l,mn) = (l,n). Moreover, modulo n/(l,n), the congruence classes of (l,n]/l and l/(l,n) are inverse to each other, and modulo mn/(l,mn), the congruence classes of (l,mn]/l and l/(l,mn) are inverse to each other. It follows that the congruence classes of m(l,mn]/l and m(l,n]/l modulo mn/(l,n) are equal. Since also m{l,mn} = {l,n}, we find that F_{l}V_{m} = V_{m}F_{l} as desired. This proves axiom (ii). Finally, we consider axiom (v). Since the formula for [a]_{S} in Addendum 1.7 is quite complicated, it is an onerous task to verify this axiom directly. However, as noted in Remark 4.5, we may instead show that for every prime number p, the diagram

\xymatrix{Ω_{𝕎(A)}^{1} \ar[r]^{η} \ar[d]^{F_{p}} & E_{ℕ}^{1} \ar[d]^{F_{p}} \\ Ω_{𝕎(A)}^{1} \ar[r]^{η} & E_{ℕ}^{1}}

commutes. It further suffices to show that for every positive integer n, the image of the element dV_{n}([1]) by the two composites in the diagram are equal. We consider three cases separately. First, if p is odd and n = ps is divisible by p, then

F_{p}ηdV_{n}([1]) = F_{p}dV_{n}η([1]) = dV_{s}η([1])

which is equal to

ηF_{p}dV_{n}([1]) = η(dV_{s}([1]) + n^{p−2} (V_{n}([1])dV_{n}([1]) − sdV_{n}([1]))) = dV_{s}η([1])

as desired. Second, if p = 2 and n = 2s is even, then

F_{2}ηdV_{n}([1]) = F_{2}dV_{n}η([1]) = dV_{s}η([1]) + ∑_{r⩾1}2^{r−1} sdV_{2^{r}s}η([1])

while

\begin{matrix}ηF_{2}dV_{n}([1]) \\ = η(dV_{s}([1]) + V_{n}([1])dV_{n}([1]) − sdV_{n}([1])) \end{matrix} ; = dV_{s}η([1]) + ∑_{t⩾1}2^{t−1}ndV_{2^{t}n}η([1]) − sdV_{n}η([1]),

so the desired equality holds in this case, too. Third, if n is not divisible by p, we have ((1 − n^{p−1} )/p) · p + n^{p−2} · n = 1, and hence,

F_{p}ηdV_{n}([1]) = F_{p}dV_{n}η([1]) = \frac{1 − n^{p−1}}{p} dV_{n}η([1]).

We wish to prove that this equal to

ηF_{p}dV_{n}([1]) = η(n^{p−2} V_{n}([1]) + dV_{n}([1]) + \frac{1 − n^{p−1}}{p} dV_{n}([1])),

or equivalently, that n^{p−2} V_{n}η([1])dV_{n}η([1]) is zero. This is true for p odd, since ndV_{n}η([1]) is zero, and for p = 2, since V_{n}η([1])dV_{n}η([1]) is zero, for n odd.

We have now proved that the groups E_{S}^{q} and structure maps listed in the statement form a Witt complex over ℤ. It follows that there is unique map

𝕎_{S}Ω_{ℤ}^{q} → E_{S}^{q}

of Witt complexes over ℤ which one readily verifies to be an isomorphism for q ⩽ 1. Therefore, it remains only to prove that 𝕎_{S}Ω_{ℤ}^{q} is zero for q ⩾ 2. It will suffice to show that for S finite, the elements ddV_{n}η([1]_{S/n}) are zero for all n ∈ S. Now,

ddV_{n}η([1]_{S/n})= d logη([−1]_{S}) · dV_{n}η([1]_{S/n}) = d(d logη([−1]_{S}) · V_{n}η([1]_{S/n}))

= dV_{n}(d logη([−1]_{S/n})) = ∑_{r⩾1}2^{r−1} dV_{n}dV_{2^{r}}η([1]_{S/2^{r}n})

= nddV_{2n}η([1]_{S/2n}).

Since S is assumed to be finite, this furnishes an induction argument which shows that ddV_{n}η([1]_{S/n}) is zero. We conclude that 𝕎_{S}Ω_{ℤ}^{q} is zero for q ⩾ 2 as desired. This completes the proof.

Addendum 6.2. Let S be a finite truncation set. The kernel of the canonical map

η_{S}: Ω̂_{𝕎_{S}(Z)}^{·} → 𝕎_{S}Ω_{Z}^{·}

is equal to the graded ideal generated by the following elements (i)–(ii) together with their images by the derivation d.

(i) For all m,n ∈ S, the element

\begin{matrix}V_{m}([1]_{S/m})dV_{n}([1]_{S/n}) − (m,n]dV_{[m,n]}([1]_{S/[m,n]}) \\ − \{m,n\} ∑_{r⩾1}2^{r−1} [m,n]dV_{2^{r}[m,n]}([1]_{S/2^{r}[m,n]}) \end{matrix}

(ii) For all n ∈ S, the element ndV_{n}([1]_{S/n}).

Proof. This follows from the proof of Theorem 6.1.

We remark that in Addendum 6.2, the graded ring Ω̂_{𝕎_{S}(ℤ)}^{·} may be replaced by the graded ring Ω_{𝕎_{S}(ℤ)}^{·}.

Acknowledgements It is a pleasure to gratefully acknowledge the hospitality and financial support of the following institutions that I have visited while working on this paper: University of Copenhagen, University of Rennes, Australian National University, and the Lorentz Institute at Leiden. I would also like to acknowledge the mathematical debt of the work presented here to my collaboration with Ib Madsen and to conversations with Jim Borger.

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