A motivic construction of the de Rham-Witt complex

The theory of reciprocity sheaves due to Kahn-Saito-Yamazaki is a powerful framework to study invariants of smooth varieties via invariants of pairs $(X,D)$ of a variety $X$ and a divisor $D$. We develop a generalization of this theory where $D$ can be a $\mathbb{Q}$-divisor. As an application, we provide a motivic construction of the de Rham-Witt complex, which is analogous to the motivic construction of the Milnor $K$-theory due to Suslin-Voevodsky.


Introduction
In Voevodsky's theory of mixed motives, the notion of A 1 -invariant sheaf played a fundamental role (see [Voe00], [Voe98], [MVW06] etc.).An A 1 -invariant sheaf is a sheaf with transfers satisfying A 1 -invariance.A sheaf with transfers is a Nisnevich sheaf of abelian groups on the category of finite correspondences, denoted Cor k , where k is the fixed base field.The objects of Cor k are smooth schemes over k, and a morphism X → Y in Cor k is given by an algebraic cycle on X × Y whose components are finite surjective over a connected component of X.We say that a sheaf with transfers F is A 1 -invariant if pr * 1 : F (X) → F (X × A 1 ) is an isomorphism for any X ∈ Cor k .On the other hand, in a series of papers [KSY16], [KSY22], Kahn-Saito-Yamazaki developed the theory of reciprocity sheaves.This is a vast generalization of the theory of A 1 -invariant sheaves over a field, and it captures ramification-theoretic information of invariants of schemes (see e.g.[RS21]).The class of reciprocity sheaves includes many interesting examples that are not A 1 -invariant, such as the sheaf of differential forms, the Hodge-Witt sheaf, and all commutative algebraic groups.
Let us recall what reciprocity sheaves are.The key idea is to replace smooth schemes by proper modulus pairs.A proper modulus pair over k is a pair (X, D X ) of a proper k-scheme X and an effective Cartier divisor D X on X such that X \ |D X | is smooth over k.For example, the pair := (P 1 , [∞]) is a proper modulus pair, which we call the cube.We can define a category of proper modulus pairs MCor k similar to Cor k by taking into account the information of Cartier divisors.An additive presheaf F : (MCor k ) op → Ab is said to be cube invariant if for any modulus pair X = (X, D X ), the map ) is an isomorphism.A presheaf with transfers (resp.sheaf with transfers) F is said to be a reciprocity presheaf (resp.reciprocity sheaf) if it belongs to the essential image of cube invariant presheaves ֒→ PSh(MCor k ) where ω ! is the left Kan extension of (X, D X ) → X \ |D X | 1 .
In the first half of this paper, we will generalize the theory of modulus pairs and sheaves on them to allow the effective divisor D X to have rational coefficients (see applications below for the reason why we need this generalization).We will define the category of proper Q-modulus pairs MCor Q k over k (Definition 2.1).We can define the notion of cube invariant presheaf and the functor ω !: PSh(MCor Q k ) → PSh(Cor k ) as before.In fact, the resulting notion of Q-reciprocity presheaf coincides with the usual one.This allows us to use Q-modulus pairs in the theory of reciprocity sheaves.
In the latter half of this paper, we will give some applications of our theory.The main results include a motivic construction of the de Rham-Witt complex, which we will sketch below.
It is well-known that the multiplicative group G m , which is an important example of an A 1invariant sheaf, has a motivic presentation.That is, there exists a canonical isomorphism G m ≃ h A 1 0 (Z tr (A 1 \{0})/Z) in PSh(Cor aff k ), where Cor aff k is the full subcategory of Cor k consisting of affine schemes, Z tr : Cor k → PSh(Cor k ) is the Yoneda embedding, and h A 1 0 is the 0-th Suslin homology: Moreover, the group structure on G m is simply induced by the multiplication morphism (A 1 \{0})× (A 1 \ {0}) → A 1 \ {0}.Suslin-Voevodsky proved more generally that the sheaf of unramified Milnor K-theory admits a motivic construction [SV00, Theorem 3.4].These results are fundamental for various computations in the theory of mixed motives.
It is a natural idea to extend this to other invariants of schemes.We start with a motivic presentation of the ring of big Witt vectors W n (A) = 1 + tA[t]/(t n+1 ) of a ring A. For n ≥ 0, we define 1 Originally, the notion of reciprocity sheaf given in [KSY16] looks quite different from the above one.However, in [KSY22], it is shown that the above definition coincides with the original one, provided that the base field k is perfect.An advantage of the above description is that we can think of a reciprocity sheaf as the "shadow" of a cube invariant presheaf.Since the definition of cube invariant presheaves is very similar to that of A 1 -invariant presheaves, one can prove many important properties of cube invariant presheaves by following Voevodsky's classical methods, at least partly.
where Z tr : MCor Q k → PSh(MCor Q k ) is the Yoneda embedding.Then there are operations F s , V s on W + n for each s > 0 which are induced by the morphism t → t s on A 1 and its transpose.These operations are called the Frobenius and the Verschiebung.Moreover, the multiplication on A 1 induces a multiplication on W + n .The 0-th Suslin homology h A 1 0 generalizes to PSh(MCor Q k ): We write h 0 = ω !h 0 .First we prove the following result.
Theorem 1 (Theorem 5.10, Proposition 5.11).There is an isomorphism k ) which preserves the multiplication, Frobenius, and the Verschiebung.In the proof of this theorem, we use the fact that the group h 0 Z tr (P 1 , r[∞]) (r ∈ Q >0 ) depends only on ⌈r⌉.This is a consequence of the motivic Hasse-Arf theorem which we prove in Theorem 4.14.It is an analogue of the classical Hasse-Arf theorem which states that the upper ramification group G (r) of an abelian extension of a local field depends only on ⌈r⌉.
Theorem 1 has a non-trivial application to reciprocity presheaves.Imitating the construction in [Miy19], we define N F (X) = Ker(i * 0 : F (X × A 1 ) → F (X)) for a reciprocity presheaf F .Then we have N F = 0 if and only if F is A 1 -invariant.Using the above theorem, we can prove that W = lim ← −n≥0 W n acts on N F .As a consequence, we obtain a short proof of the following result of Binda-Cao-Kai-Sugiyama [Bin+17, Theorem 1.3].
Theorem 2 (Corollary 5.15).Let F be a reciprocity presheaf over k and assume that F is separated for the Zariski topology.
(1) If ch(k) = 0 and Next we give a motivic presentation of the ring of p-typical Witt vectors W n (A) of a Z (p) -algebra A. For n ≥ 0, we define As in the case of W + n , we have operations F s , V s for s > 0 and a multiplication on W + n .We define W + n to be the quotient of W + p n−1 ⊗ Z (p) by the images of the idempotents ℓ −1 V ℓ F ℓ for all prime numbers ℓ = p.The operations F p , V p descend to operations F, V on W + n called the Frobenius and the Verschiebung, and the multiplication also descends to W + n .We prove the following Theorem 3 (Corollary 5.12).There is an isomorphism which is compatible with the Frobenius, the Verschiebung, and the multiplication.
In order to treat the de Rham-Witt complex, we have to assume that k is perfect and p ≥ 3. We define Using the tensor structure on PSh(MCor Q k ) (see §2), we obtain an object W + n ⊗G +⊗q m of PSh(MCor Q k ).Our main result is the following: Theorem 4 (Theorem 6.21).Let k be a perfect field of characteristic p ≥ 3. Then there is an isomorphism , where a Nis denotes the Nisnevich sheafification.
The proof goes as follows: first we prove that the left hand side admits a Witt complex structure (Theorem 6.11).Since the de Rham-Witt complex is initial in the category of Witt complexes, we obtain a unique morphism from the right hand side to the left hand side.To prove that this morphism is an isomorphism, we construct an inverse by using the transfer structure on W n Ω q .
We note that a similar motivic presentation for Ω q is obtained by Rülling-Sugiyama-Yamazaki [RSY22] and the first author [Koi].Also, a presentation of the big de Rham-Witt complex W n Ω q using additive higher Chow groups is obtained by Rülling [R 07b] for fields, and by Krishna-Park [KP21] for regular semilocal algebras over a field.We expect that these results will be connected by some comparison isomorphism between Suslin homology and additive higher Chow groups.
The structure of the present paper is as follows.In §1, we prepare preliminary results on Q-Cartier divisors.In §2, we define Q-modulus pairs and prove some basic properties.In §3, we define and study the notion of cube invariant presheaf and the notion of reciprocity presheaf following [KSY22].In §4, we formulate and prove the motivic Hasse-Arf theorem, by comparing the 0-th Suslin homology group and the Chow group of relative 0-cycles of a modulus curve.This comparison can be seen as a generalization of the result in [RY16].In §5, we apply our machinery to give a motivic construction of the ring of Witt vectors and basic operations on it.As an application, we give a short proof of the result of Binda-Cao-Kai-Sugiyama [Bin+17] on torsion and divisibility of a reciprocity sheaf.In §6, we give a motivic construction of the de Rham-Witt complex.
Acknowledgements.We would like to thank Shuji Saito for his interest on this work.We also appreciate his valuable comments on earlier versions of this paper, which encouraged the authors to improve the main results.

Notations and conventions.
− For a scheme X and x ∈ X, we write k(x) for the residue field of X at x.We write X (d)  for the set of points on X of codimension d.An element of X (0) is called a generic point.− We say that a morphism of schemes f : X → Y is pseudo-dominant if it takes generic points to generic points, i.e., f (X (0) ) ⊂ Y (0) .− For an integral scheme X, we write k(X) for its function field.If X is a noetherian normal integral scheme and f ∈ k(X) × , we write div(f ) for the Weil divisor on X defined by f .− We write Sm k for the category of smooth separated k-schemes of finite type.We define Sm aff k to be the full subcategory of Sm k spanned by affine schemes.− For an additive category C, we write PSh(C) for the category of additive functors C op → Ab. − For a category C, we write Pro(C) for the category of pro-objects in C. A pro-functor 1. Preliminaries 1.1.Finite correspondences.First we recall Suslin-Voevodsky's category of finite correspondences Cor k .It is an additive category having the same objects as Sm k , and its group of morphisms Cor k (X, Y ) is the group of algebraic cycles on X ×Y whose components are finite pseudo-dominant over X.The fiber product over k gives a symmetric monoidal structure on Cor k .For any mor- 1.2.Q-Cartier divisors.We introduce the notion of Q-Cartier divisor over a general noetherian scheme, which will play a fundamental role in this paper.
Definition 1.1.For a noetherian scheme X, we write CDiv(X) for the group of Cartier divisors on X.We define the group of Q-Cartier divisors on X by CDiv Q (X) := CDiv(X) ⊗ Z Q.
If X is a noetherian normal scheme, then the group CDiv(X) is embedded into the free abelian group of Weil divisors on X.In particular, CDiv(X) is a free abelian group in this case.Let f : Y → X be a morphism of noetherian schemes and D be a Q-Cartier divisor on X.We say that the pullback f * D of D by f exists if there is an ordinary Cartier divisor E on X and r ∈ Q with D = rE such that the pullback f * E exists.In this situation, we define f * D := rf * E ∈ CDiv Q (Y ).This does not depend on the choice of E and r.
Let X be a noetherian scheme and D be a Q-Cartier divisor on X.We say that D is Q-effective if there is an ordinary effective Cartier divisor E on X and r If an ordinary Cartier divisor D is effective, then it is Q-effective.The converse is not true in general, but it is true for normal schemes: Lemma 1.2.Let X be a noetherian normal scheme and D be an ordinary Cartier divisor on X.Then D is effective if and only if D is Q-effective.
Proof.We may assume that X = Spec A with A a normal domain and D = div(a) with a ∈ Frac(A) × .If D is Q-effective, then there is some non-zero-divisor b ∈ A and n ≥ 1 such that div(a) = (1/n) div(b) in CDiv Q (X).Since X is normal, the map CDiv(X) → CDiv Q (X) is injective and hence div(a n ) = div(b) in CDiv(X).This implies that a n − ub = 0 for some u ∈ A × , so a ∈ A since A is normal.Therefore D is effective.
Corollary 1.3.Let X be a noetherian scheme and D, D ′ be ordinary Cartier divisors on X.Then we have Definition 1.4.Let X be a noetherian scheme and D be a Q-effective Q-Cartier divisor on X. Suppose that D = rE in CDiv Q (X) where E is an ordinary effective Cartier divisor on X and r ∈ Q >0 .Then the support |D| of D is defined to be |E|.This is well-defined since |E| = |nE| for any ordinary effective Cartier divisor E and n ≥ 1.
2. Q-modulus pairs 2.1.Q-Modulus pairs.Recall from [Kah+21] that a modulus pair (over k) is a pair X = (X, D X ) where X is a separated k-scheme of finite type and D X is an effective Cartier divisor on X such that X In order to avoid confusion, we use the word "Z-modulus pairs" to indicate modulus pairs in the sense of [Kah+21].In what follows, we fix Λ ∈ {Z, Q} and develop the theory of Λ-modulus pairs.
An ambient morphism of Λ-modulus pairs f : Let X, Y be Λ-modulus pairs over k and let , so the claim follows from the case Λ = Z.
We define MCor Λ k to be the category of Λ-modulus pairs over k, where the morphisms are given by MCor Λ k (X, Y).The canonical functor MCor Z k → MCor Q k is fully faithful by Corollary 1.3.We set The next lemma shows that this gives a symmetric monoidal structure on MCor Λ k : Proof.For Λ = Z, this is proved in [Kah+22, Lemma 2.1.3].For Λ = Q, we may reduce to the case Λ = Z as in the proof of Lemma 2.2.
We define MCor Λ k ⊂ MCor Λ k to be the full subcategory consisting of proper Λ-modulus pairs, i.e., X = (X, D X ) with X proper over k.There are natural functors τ * where τ * is the restriction functor and τ ! is the left Kan extension of τ .Similarly, the functor For Λ = Q, we may reduce to the case Λ = Z as in the proof of Lemma 2.2.By Lemma 2.4, we get a fully faithful symmetric monoidal functor Therefore the functor τ !can be written explicitly as Proposition 2.5.The following assertions hold: (1) The functor ω * is exact and fully faithful.
Proof.Both functors are clearly exact.Moreover, the above formula for ω !implies that ω !ω * ≃ id and hence ω * is fully faithful.Since ω is symmetric monoidal and ω ! is its extension by colimits, ω ! is also symmetric monoidal.

Cube invariance and reciprocity
Fix Λ ∈ {Z, Q}.Following [KSY22], we introduce a class of Λ-modulus presheaves called cube invariant presheaves which is an analogue of the class of A 1 -invariant presheaves used in the classical theory of motives.This leads to the notion of Λ-reciprocity presheaf, which is a presheaf with transfers that can be "lifted" to a cube invariant presheaf.We show that the notion of Q-reciprocity presheaf is actually the same as the notion of (Z-)reciprocity presheaf.

Cube invariance. The object :=
k is called the cube over k.We write π : → (Spec k, ∅) for the ambient morphism given by the canonical projection P 1 → Spec k, and i ε : (Spec k, ∅) → (ε = 0, 1) for the ambient morphism given by ε : Spec k In this case we write α 0 ∼ α 1 and call γ a cube homotopy between α 0 and α 1 .We say that α We call such β a cube homotopy inverse of α.
Lemma 3.6.The class of cube invariant objects in PSh(MCor Λ k ) is closed under taking subobjects, quotients and extensions.
Proof.The claim for subobjects follows from the equivalence of (1) and (2) in Lemma 3.5.The remaining assertions then follow by the five lemma.
There are two canonical ways to make a presheaf on MCor Λ k cube invariant: one is to take the maximal cube invariant quotient, and the other is to take the maximal cube invariant subobject.The former is called the cube-localization and the latter is called the cube invariant part.
There is a canonical epimorphism F ։ h 0 F .We write h 0 (X) for h 0 Z tr (X).
(1) For any If G is cube invariant, then the canonical homomorphism Hom(h 0 F, G) → Hom(F, G) is an isomorphism.In other words, h 0 F is the maximal cube invariant quotient of F .
Proof.(1) follows from the equivalence of (1) and (3) in Lemma 3.5.(2) The canonical morphism G → h 0 G is an isomorphism by Lemma 3.5.The claim follows from this and (1).(3) By the Yoneda lemma, it suffices to prove that if Lemma 3.10.The following assertions hold.
Remark 3.11.In [KSY22], h 0 and h 0, are defined to be functors taking values in the category of cube invariant presheaves.On the other hand, we define h 0 and h 0, as endofunctors of PSh(MCor Λ k ).This has the advantage that h 0, becomes right adjoint to h 0 .
Since G is cube invariant, Hom(F, G) is cube invariant when F is representable.Since Hom(−, G) turns colimits into limits, the same is true for a general F .
(2) For any H ∈ PSh(MCor Λ k ), we have Therefore we get the desired result by the Yoneda lemma.
Definition 3.13.We define two functors h 0 and h 0 as follows.
Remark 3.14.By Lemma 3.12, we have In particular, h 0 admits a canonical lax symmetric monoidal structure.Applying the exact symmetric monoidal functor ω !, we also see that there is a canonical epimorphism Remark 3.15.In [KSY22], a functor named ω CI is defined by the same formula as h 0 , but it is regarded as a functor taking values in the category of cube invariant presheaves rather than PSh(MCor Λ k ).
Proof.This follows from the definition of h 0 and the isomorphism The functors we have defined so far can be summarized as follows.
The following conditions are equivalent: (1) The counit morphism h 0 h 0 F → F is an isomorphism.
(2) For any X ∈ Sm k and any a ∈ F (X), there exists (3) Every section of F admits a Λ-modulus.
Proof.The equivalence of (1) and (2) follows from the formula The equivalence of (2) and (3) is clear from the definition.
Definition 3.19.We say that F has Λ-reciprocity or F is a Λ-reciprocity presheaf if it satisfies the equivalent conditions in Lemma 3.18.If F is moreover a Nisnevich sheaf, then we say that F is a reciprocity sheaf.We define RSC Λ k (resp.RSC Λ k,Nis ) to be the full subcategory of PSh(Cor k ) spanned by Λ-reciprocity presheaves (resp.Λ-reciprocity sheaves).

The category RSC Λ
k is closed under taking subobjects and quotients in PSh(Cor k ).In particular, RSC Λ k is an abelian category and the inclusion functor RSC Λ k → PSh(Cor k ) is exact.
Lemma 3.20.We have Then it is clear that every Z-modulus Y for a is also a Q-modulus for a, since the description of h 0 (Y) (Lemma 3.16) is the same for Λ = Z, Q. Conversely, if Y is a Q-modulus for a, then so is (Y, nD Y ) for n ∈ Z >0 , so there exists a Z-modulus for a.
We thus write RSC k for RSC Z k = RSC Q k and call its objects reciprocity presheaves over k.
The group h 0 F (X) can be thought of as the subgroup of ω * F (X) = F (X • ) consisting of elements whose "ramification" is bounded by D X .This actually recovers several classical notions in ramification theory such as the Artin conductor or the irregularity [RS21].For further developments of the ramification theory of reciprocity sheaves, see [RS21], [RS23a], [RS23b], and [RS22].

Modulus curves and motivic Hasse-Arf theorem
Fix Λ ∈ {Z, Q}.In this section, we describe h 0 (X) for a Λ-modulus curve (Definition 4.6) X over k, using the Chow group of relative 0-cycles (Definition 4.9).As a corollary, we prove a motivic analogue of the Hasse-Arf theorem.
(1) f is regular and invertible on U , and (2) div(f − 1) ≥ D X holds on U (including the case f = 1).We define Adm(X) ⊂ k(X) × to be the subset consisting of rational functions which are admissible with respect to D X .It follows from the identity k be a normal integral Λ-modulus pair.Then f ∈ k(X) × is admissible with respect to D X if and only if for any discrete valuation ring R with fraction field k(X) and any morphism ρ Proof.The "only if" part is easy.Let us prove the "if" part.Let x ∈ |D X |.Our assumption implies that for any discrete valuation ring R with fraction field k(X) dominating O X,x , we have f ∈ R × .Since O X,x is a noetherian normal domain, this shows that f ∈ O × X,x .Therefore f is invertible on some open neighborhood U of |D X |.The condition (1) in Definition 4.2 is satisfied for this U .Let us verify the condition (2).Let R be a valuation ring with fraction field k(X) and ρ : Spec R → U be a morphism extending Spec k(X) Lemma 4.4.Let (K, v) be a discrete valuation field and L/K be a finite extension.Let γ ∈ Q >0 , f ∈ L × and suppose that for every discrete valuation w : Proof.Let i denote the inseparable degree of L/K.Fix an algebraic closure K of K. Then v can be extended to a valuation on K. Let us fix such an extension v.We have where σ runs over the set of distinct K-embeddings of L into K.Now v • σ gives a valuation on L extending v for each σ, so we have v(σ(f − 1)) ≥ γ by our assumption.By the above expression for Nm L/K (f ), we get the desired inequality.
Lemma 4.5.Let X, C ∈ MCor Λ k be normal integral Λ-modulus pairs and q : X → C be an ambient morphism such that X → C is proper and generically finite.Then for any f ∈ Adm(X), we have Proof.By Lemma 4.3, it suffices to show that for any discrete valuation ring R with fraction field k(C) and any morphism ρ an arbitrary extension of v R and R ′ be its valuation ring.Since q : X → C is proper, there is a unique dashed arrow ϕ in the following diagram which makes it commute: , as was to be shown.

Modulus curves.
In this subsection, we fix S ∈ Sm k which is connected.
Definition 4.6.A Λ-modulus curve over S is a modulus pair C ∈ MCor Λ k equipped with a proper smooth morphism C → S of relative dimension 1 such that Proof.Since C → S is proper, V → S is also proper.For any x ∈ S, the fiber V x of V is a closed subscheme of C • x which is proper over Spec k(x).By our assumption that C • x is quasi-affine, there is an open immersion C • x ֒→ B into an affine scheme B. The properness of V x implies that the image of V x in B is closed.It follows that V x is finite over k(x) and hence V → S is quasi-finite.By Zariski's main theorem, we conclude that V → S is finite.
Lemma 4.8.Let C be a Λ-modulus curve over S. For an integral closed subscheme V ⊂ C • , the following conditions are equivalent: (1) V → S is finite surjective.
(2) V has codimension 1 and is closed in C. Proof.
(1) =⇒ (2): Suppose that V → S is finite surjective.Then V is finite over C and hence is closed in C. Let ξ be the generic point of V and η its image in S. Then the fiber dimension theorem for flat morphisms [GW10, Corollary 14.95] implies codim (2) =⇒ (1): Suppose that V has codimension 1 and is closed in C. Then V is finite over S by Lemma 4.7.We have codim C • (ξ) = 1 and codim C • η (ξ) = 1, so the fiber dimension theorem implies codim S (η) = 0. Therefore V → S is surjective.Proof.We may assume that C is connected.Then Adm(C) consists of rational functions f which is regular and invertible on some neighborhood of D C and f | DC = 1.On the other hand, Pic(C, D C ) is by definition the group of isomorphism classes of pairs (L, α) where L is a line bundle on C and α is a nowhere-vanishing section of L| DC .By our assumption that D C has an affine open neighborhood, any such α can be extended to a rational section of L which is regular and invertible on some neighborhood of |D C |.For two such extensions α 1 , α 2 , the quotient α 1 / α 2 is admissible with respect to D C .Therefore Pic(C, D C ) can be identified with the cokernel of div : Adm(C) → Q where Q is the group of Cartier divisors on C whose support is disjoint from |D C |.We have Q = Cor S (S, C • ) by Lemma 4.8, so the claim follows.
The following result generalizes [RY16, Theorem 1.1] to Λ = Q.Proof.We may assume that C is connected.Throughout this proof we identify with (P 1 , [1]) via the isomorphism t → t t−1 , and set := P 1 \ {1}.It suffices to prove that We will prove a stronger statement that there is a surjective homomorphism Φ : MCor Λ S ( ×S, C) ։ Adm(C) which makes the following diagram commutative: C).Let V denote the scheme-theoretic closure of V in P 1 × S × C and p : V → P 1 , q : V → C be the canonical projections.Then we have p Lemma 4.5 shows that this gives a homomorphism Φ : MCor Λ S ( × S, C) → Adm(C).Let us prove that (4.1) is commutative.Let K be the function field of S. Since the operations appearing in (4.1) are compatible with base change to Spec K and the map Cor S (S, C • ) → Cor K (Spec K, C • K ) is injective, we may assume that S = Spec K.In this case the claim follows from a standard computation of cycles.
It remains to show that Φ is surjective.Let f ∈ Adm(C).If f = 1, then Φ(0) = f .Otherwise, we take an open neighborhood U of |D C | satisfying the conditions (1) and (2) in Definition 4.2.Then f is regular and invertible on U .Define V ⊂ P 1 × S × C to be the closure of the graph of Applying Lemma 4.7 to the modulus curve × C over × S, we see that V is finite over × S. The admissibility of V follows from the assumption that div(f − 1) ≥ π * D C holds on U .Finally, we have Φ( Theorem 4.13.Let C be a Λ-modulus curve over k.Then for any connected S ∈ Sm k , the canonical surjection Moreover, if Λ = Z and S is affine, then we also have Proof.Applying Lemma 4.12 to the Λ-modulus curve S × C over S, we get CH 0 (C/S) ≃ Coker(MCor Λ S ( S , S × C) The last term is isomorphic to h 0 (C)(S) by Lemma 3.16.The second assertion follows from Lemma 4.11.

Motivic Hasse-Arf theorem.
Theorem 4.14.Let C be a Q-modulus curve over k.Then we have Proof.We want to prove that for any connected S ∈ Sm k , the canonical map h 0 (C, The following corollary can be seen as a motivic analogue of the Hasse-Arf theorem.
Corollary 4.15.Let C be a Q-modulus curve over k.Then for any F ∈ RSC k we have Proof.This follows from Theorem 4.14.
Remark 4.16.We have Z tr (X, D X ) = Z tr (X, ⌈D X ⌉) in general, so Corollary 4.15 is not obvious from the definition.Actually, Corollary 4.15 is false for modulus pairs of higher dimensions.
Lemma 5.3.The above definition of W n coincides with the usual one (e.g.[Hes15]).
Now the claim follows from the p-typical decomposition of W p n−1 (see [Hes15, Proposition 1.10]).
By Lemma 5.1, the homomorphisms U , ⋆, F p , and V p descend to and we have the following Lemma 5.4.The following assertions hold for W n : (1) ⋆ is commutative, associative, and unital with unit U . (
Definition 5.6.Let n ≥ 0. We define the unit, the multiplication, the Frobenius, and the Verschiebung on W + n as follows.
(1) We define U : Z → W + n to be the morphism induced by i 1 : Spec k → A 1 .(2) We define ⋆ : W + n ⊗ W + n → W + n to be the morphism induced by the multiplication map µ : (3) For s ≥ 1, we define F s : W + sn → W + n to be the morphism induced by ρ s : sn to be the morphism induced by t ρ s .Similarly, we define the unit, the multiplication, the Frobenius, and the Verschiebung on W + n .For X = Spec A ∈ Sm aff k and a ∈ A (resp. a ∈ A × ), we write [a] for the element of Lemma 5.7.The following assertions hold for both W + n and W + n : (1) ⋆ is commutative, associative, and unital with unit U . (2) Proof.(1) follows from the corresponding properties of µ : (2) follows from ρ 1 = id and ρ r • ρ s = ρ rs .The first assertion in (3) follows from ρ s t • ρ s = s • id.The second assertion follows from the fact that ρ s t • ρ r and t ρ r • ρ s are both represented by the cycle on A 2 defined by x s = y r .To prove (4), it suffices to show that the following diagram in Cor k is commutative: One can easily check that both compositions are represented by the cycle on A 3 defined by xy s = z s .Finally, (5) follows from Lemma 5.8.For any n ≥ 0, s ≥ 1 and 0 ≤ m ≤ sn, the morphism V s F s : The same also holds for W n .Proof.To prove the first statement, it suffices to show that the finite correspondence )) for every r ∈ Q >0 .This follows from the fact that t ρ s • ρ s is represented by the cycle on A 2 defined by x s = y s , which is symmetric in x and y.The second statement can be proved similarly.
Suppose that ch(k) = p > 0. For a prime number ℓ different from p, we have an endomorphism . By Lemma 5.7, ℓ −1 V ℓ F ℓ is an idempotent and hence defines a direct summand Im(ℓ −1 V ℓ F ℓ ) of ( W + n ) (p) .For n ≥ 1, we define (5.1) For X = Spec A ∈ Sm aff k and a ∈ A × , we write [a] for the image of [a] ∈ h 0 W + p n−1 (X) in h 0 W + n (X).By Lemma 5.7, the morphisms U , ⋆, F p , and V p descend to and we have the following Lemma 5.9.The following assertions hold for W + n : (1) ⋆ is commutative, associative, and unital with unit U . (2)
(1) There is an isomorphism ϕ : (2) There is an isomorphism ϕ : Proof.We prove only (1); the proof for (2) is similar.First we construct an isomorphism The last term can be easily identified with W n (X) ⊕ Z (see e.g.[Koi, Proposition 1.1]).Therefore we have . It remains to show that this isomorphism is compatible with transfers.Let X, Y ∈ Sm aff k and α ∈ Cor k (X, Y ).It suffices to prove that the following diagram is commutative: (5.2) We may assume that X, Y are connected.Let K be an algebraic closure of k(X), and consider the same problem for X K , Y K ∈ Sm aff k and α K ∈ Cor K (X K , Y K ): (5.3) Then there is a natural morphism of diagrams from (5.2) to (5.3).Since the composite is injective, we may assume that k is algebraically closed and X = Spec k.In this case, α can be written as a Z-linear combination of morphisms Spec k → Y , so the assertion is obvious.
Proposition 5.11.The isomorphisms ϕ : from Theorem 5.10 are compatible with the unit, the multiplication, the Frobenius, and the Verschiebung.
Proof.We prove only the statement for ϕ; the proof for ϕ is similar.It suffices to prove that for each X ∈ Sm aff k , the isomorphism ϕ X : is compatible with these operations.As in the proof of Theorem 5.10, we may assume that k is algebraically closed and X = Spec k.In this case, W n (k) is generated by elements of the form [a] with a ∈ k × .For a, b ∈ k × , we have . This proves the claim.Corollary 5.12.Suppose that ch(k) = p > 0. For n ≥ 1, there is an isomorphism which is compatible with the unit, the multiplication, the Frobenius, and the Verschiebung.If X = Spec A ∈ Sm aff k , then we have ϕ for a ∈ A × .5.4.Application to torsion and divisibility of reciprocity sheaves.In this subsection, we give an application of the motivic presentation of the ring of big Witt vectors to reciprocity sheaves.
Let F ∈ RSC k .Imitating the construction in [Miy19], we define N n F, N F ∈ PSh(Cor k ) by Here, we regard (X, ∅) as a pro-object in MCor Q k via the compactification functor.In other words, we define F ) by adjunction.Taking h 0 , we obtain an action h 0 W + n ⊗ N n F → N n F .Since we have an isomorphism of rings h 0 W + n (X) ≃ W n (X) for X ∈ Sm aff k by Theorem 5.10 and Proposition 5.11, we obtain the following Theorem 5.13.Let F ∈ RSC k and X ∈ Sm aff k .Then N n F (X) has a canonical structure of a W n (X)-module, which is natural in X and n.In particular, N F (X) has a canonical structure of a W(X)-module, where W(X) = lim ← −n≥0 W n (X).
Corollary 5.14.Let F ∈ RSC k , X ∈ Sm aff k , and let p be a prime number.(1) If p is invertible in k, then N n F (X) and N F (X) are uniquely p-divisible.
(2) If ch(k) = p > 0, then N n F (X) and N F (X) are p-groups.
Proof.This follows from the fact that if p is invertible (resp.nilpotent) in a ring A, then p is also invertible (resp.nilpotent) in W n (A); see [Hes15, Lemma 1.9 and Proposition 1.10].

Motivic construction of the de Rham-Witt complex
Throughout this section, we assume that k is perfect and ch(k) = p ≥ 3 2 .In this section, we construct the de Rham-Witt complex of smooth k-schemes using Q-modulus pairs.
2 Here, the perfectness of k is assumed in order to use the transfer structure on the de Rham-Witt complex.The assumption on ch(k) will be used in the proofs of Lemma 6.10 and Theorem 6.11.
6.1.De Rham-Witt complex.First we recall the definition of the de Rham-Witt complex.Here, we follow the axiomatization due to [HM04].Definition 6.1.Let A be a Z (p) -algebra.A Witt complex over A is a tuple (E * • , F, V, λ) where (1) n is a ring homomorphism compatible with R, F , and V , such that the following relations hold: The category of Witt complexes over A is known to have an initial object W n Ω * (A) and it is called the de Rham-Witt complex of A. We have W n Ω 0 (A) ≃ W n (A).It follows from the construction that W n Ω q (A) is a quotient of Ω q Wn(A) .The presheaf Spec A → W n Ω q (A) on Sm aff k extends to an étale sheaf W n Ω q on Sm k having global injectivity 3 .Moreover, since k is assumed to be perfect, W n Ω q can be regarded as an object of PSh(Cor k ) [KSY16, Theorem B.2.1].It is compatible with the trace maps defined in [R 07b, Theorem 2.6] (see [RS21, Section 7.9]).6.2.Motivic construction.Definition 6.2.We define where Z is viewed as a direct summand of Z tr (P 1 , ε Lemma 6.3.The following assertions hold. (1) Let F s : G + m → G + m be the morphism induced by ρ s .Then we have h 0 (F s ) = s • id.
(5) Let δ : Regarding the left hand side as a polynomial in y, the leading term and the constant term have invertible coefficients.Therefore Γ is finite locally free over (A 1 \ {0}) × A 1 and hence defines a finite correspondence 3 This means that for any X ∈ Sm k and a dense open subset U ⊂ X, the map WnΩ q (X) → WnΩ q (U ) is injective.
Therefore Γ induces a morphism G + m ⊗ Z tr ( ) → G + m whose restriction to t = 0 (resp.t = 1) is F s (resp.s • id).This shows that h Regarding the left hand side as a polynomial in y, the leading term and the constant term have invertible coefficients if s is odd.Therefore Γ is finite locally free over (A 1 \ {0}) × A 1 and hence defines a finite correspondence Regarding the left hand side as a polynomial in x, the leading term and the constant term have invertible coefficients.Therefore Γ is finite locally free over Spec A[t] and hence defines a finite correspondence One can check that Γ is finite locally free over (A 1 \ {0}) 2 × A 1 and hence defines a finite correspondence whose restriction to t = 0 (resp.t = 1) is id + τ (resp.0).This shows that h 0 (id + τ ) = 0 and hence h 0 (τ ) = −id.
(1) For q, r ≥ 0, we define the multiplication on For each X ∈ Sm aff k , this makes h 0 (W + n ⊗ G +⊗ * m )(X) into a graded ring.By Lemma 6.3 (4), this multiplication is graded commutative.Moreover, it is a Z (p) -algebra since h 0 W + n (X) ≃ W n (X) is so.(2) For q ≥ 0, we define the Frobenius by F := F ⊗ id : By Lemma 5.9, F is a graded ring homomorphism, and we have to be the inverse of the isomorphism ϕ (p) given in Corollary 5.12.This is a ring homomorphism compatible with F and V .Definition 6.6.Let Γ be the graph of the diagonal morphism ∆ : We define d : W + n → W + n ⊗ G + m to be the morphism induced by Γ.
Lemma 6.7.For any prime number ℓ different from p, the following diagram becomes commutative after applying h 0 : Proof.Recall from Remark 3.14 that we have a canonical epimorphism h 0 ( W . Therefore it suffices to show that the following diagram is commutative: This is further reduced to the commutativity of the following diagram in Cor k : Both compositions are given by the cycle on (A 1 \ {0}) 3 defined by x ℓ = y ℓ = z.Definition 6.8.Let n ≥ 1.We define the differential to be the morphism induced by d ⊗ id : via Lemma 6.7.
Lemma 6.9.The following diagram becomes commutative after applying h 0 : Proof.It suffices to show that the following diagram becomes commutative after composing with the canonical epimorphism Here, q 1 (resp.q 2 ) denotes the morphism (x, y) → (x, y, x) (resp.(x, y) → (x, y, y)).The two compositions are identified with where β is the morphism (x, y) → (x, y, x, y).Therefore it suffices to show that the two morphisms coincide after composing with the canonical epimorphism Z tr (A 1 \ {0}) ։ h 0 G + m .This follows from Lemma 6.3.Lemma 6.10.The following diagram becomes commutative after applying h 0 : Proof.By Lemma 6.3 (2) and our assumption that p is odd, we have h 0 (id⊗V p ) = id on h 0 ( W + p n−1 ⊗ G + m ).Therefore it suffices to show that the following diagram is commutative: This is further reduced to the commutativity of the following diagram in Cor k : Both compositions are given by the cycle on (A 1 \ {0}) 3 defined by x = y = z p .Theorem 6.11.Let X = Spec A ∈ Sm aff k .Then the tuple (a Nis h 0 (W + • ⊗ G +⊗ * m )(X), F, V, λ) is a Witt complex over A. In particular, we have a unique homomorphism of Witt complexes in Sh Nis (Sm k ).Proof.By Lemma 6.3 (5), we have 2d 2 = 0 and hence d 2 = 0.By Lemma 6.9 and Lemma 6.3 (4), we see that d satisfies the Leibniz rule.The relation F dV = d holds by Lemma 6.10.Let us show that the relation holds for any a ∈ A. We use the injectivity theorem for reciprocity sheaves [KSY16, Theorem 6 & 7]; since h 0 (W + n ⊗ G +⊗q m ) is a reciprocity presheaf, its Nisnevich sheafification has global injectivity.This allows us to assume that a is invertible in A. In this case, both sides of the claimed formula are represented by the morphism (a p , a) : Spec A → (A 1 \ {0}) 2 .6.3.Compatibility with transfers.In the last subsection, we have constructed a morphism θ : In this subsection, we prove that θ is compatible with transfers.Lemma 6.12 (Projection formula).Let X ∈ Sm k , Y ∈ Sm X and let π : Y → X be the structure morphism.Let a ∈ W n Ω q (X) and b ∈ W n Ω q (Y ).For any α ∈ Cor X (X, Y ), we have Proof.Since the transfer structure of W n Ω q is compatible with the trace maps defined in [R 07b, Theorem 2.6], the claim follows from the projection formula for the trace maps.Lemma 6.13.For any X = Spec A ∈ Sm k and α ∈ Cor X (X, A 1 X ), the following diagram is commutative: The following proof is inspired by the proof of [KP21, Proposition 5.19].
(1) If j 0 = • • • = j q = 0, then we can write ω as a Z-linear combination of elements of the form where c 0 , . . ., c q ∈ A. First we show that η is traceable.By the projection formula, it suffices to show that [c 0 t k0 ] is traceable.This follows from the case q = 0. Next we show that ξ is traceable.We write k 0 + 1 = mp e with p ∤ m.Then we have By the projection formula, it suffices to show that m −1 F e (d[t m ]) is traceable.This follows from the case q = 0.
Proof.The finite correspondence α can be factored as Therefore we may assume that Y ∈ Sm X and α ∈ Cor X (X, Y ).Since θ is a morphism of Nisnevich sheaves, we may assume that X = Spec A, Y = Spec B. Take a closed immersion Spec B → Spec A N where A N = A[t 1 , . . ., t N ].Since W n Ω(A N ) → W n Ω(B) is surjective, we may assume that B = A N , so the claim follows from Lemma 6.14.

Definition 4. 9 .
Let C be a Λ-modulus curve over S with C connected.Then for any f ∈ Adm(C), the components of the Weil divisor div(f ) on C • are closed in C and hence div(f ) ∈ Cor S (S, C • ) by Lemma 4.8.We define the Chow group of relative 0-cycles of C by CH 0 (C/S) := Coker(Adm(C) div − − → Cor S (S, C • )).If C is a Λ-modulus curve over S with C non-connected, then we define CH 0 (C/S) by taking a direct sum over connected components.Remark 4.10.When S = Spec k and Λ = Z, the above definition of CH 0 (C/S) coincides with the definition of the relative Chow group of 0-cycles C(C, D C ) from [KS16, Definition 1.6].The latter group is also defined for higher dimensional varieties, and it was used to establish ramified higher dimensional class field theory.When Λ = Z, we have the following comparison result between the Chow group of relative 0-cycles and the relative Picard group.Lemma 4.11.Let C be a Z-modulus curve over S such that D C is contained in some affine open subset of C. Then we have CH 0 (C/S) ≃ Pic(C, D C ).
is an isomorphism.By Theorem 4.13, it suffices to show thatCH 0 ((S × C, S × ⌈D C ⌉)/S) → CH 0 ((S × C, S × D C )/S)is an isomorphism.We have S × ⌈D C ⌉ = ⌈S × D C ⌉ since S is smooth over k.Now the claim follows from the fact that div(f − 1) has integral coefficients for any rational function f on S × C.
consisting of cycles whose components are left proper and admissible.For any ambient morphism f :X → Y over k, the graph of f • gives a morphism f : X → Y in MCor Λ k .If f is a proper minimal ambient morphism such that f • is finite pseudo-dominant, then the transpose of the graph of f • gives a morphism t f : Y → X in MCor Λ X , D Y , D Z by nD X , nD Y , nD Z for n ∈ Z >0 , so we may assume that X, Y, Z are Z-modulus pairs.Then we have MCor Z k