Reciprocity sheaves and their ramification filtrations

We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an increasing and exhaustive filtration on $F(L)$, where $L$ is any henselian discrete valuation field of geometric type over the perfect ground field. We show if $F$ is a smooth group scheme, then the motivic conductor extends the Rosenlicht-Serre conductor; if $F$ assigns to $X$ the group of finite characters on the abelianized \'etale fundamental group of $X$, then the motivic conductor agrees with the Artin conductor defined by Kato-Matsuda; if $F$ assigns to $X$ the group of integrable rank one connections (in characteristic zero), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with {\em perfect} residue field can be uniquely extended to all such fields without any restriction on the residue field. For example the Kato-Matsuda Artin conductor is characterized as the canonical extension of the classical Artin conductor defined in the perfect residue field case.


Introduction 2
Part 1. The general theory 8 2. Presheaves with transfers on pro-smooth schemes 8 3. Review of reciprocity sheaves 11 The first author is supported by the DFG Heisenberg Grant RU 1412/2-2. Part of the work was done while he was a visiting professor at the TU München. He thanks Eva Viehmann for the invitation and the support. The second author is supported by JSPS KAKENHI Grant (15H03606) and the DFG SFB/CRC 1085 "Higher Invariants". Fix a perfect field k and let Sm be the category of separated smooth k-schemes. Let Cor be the category of finite correspondences: Cor has the same objects as Sm and morphisms in Cor are finite correspondences (see 2.1 for a precise definition). Let PST be the category of additive presheaves of abelian groups on Cor, called presheaves with transfers. In this note we give a construction which associates to each F ∈ PST a collection of functions

Conductors for presheaves with transfers
where N is the set of non-negative integers, Φ is the collection of henselian discrete valuation fields which are the fraction fields of the henselization O h X,x of X ∈ Sm at points x of codimension one in X, and where V → X ranges overétale neighborhoods of x and D x is the closure of x in V . We call c F the motivic conductor for F . Our main aim is to convince the reader that our construction deserves such a pretentious terminology. Indeed, it gives a unified way to understand different conductors such as the Artin conductor of a character of the abelian fundamental group π ab 1 (X) with X ∈ Sm along a boundary of X, the Rosenlicht-Serre conductor of a morphism from a curve to a commutative algebraic k-group, and the irregularity of a line bundle with connections on X ∈ Sm along a boundary of X . It also gives rise to a new conductor for G-torsors with G a finite flat k-group scheme. The latter conductor specializes to the classical Artin conductor in case G is constant.
Our construction of the motivic conductors is rather simple once we have the new categorical framework introduced in [KMSY21a], [KMSY21b] at our disposal (see (1.0.1) below). The main aim of loc. cit. is to develop a theory of motives with modulus generalizing Voevodsky's theory of motives in order to capture non-A 1 -invariant phenomena and objects. The basic principle is that the category Cor should be replaced by the larger category of modulus pairs, MCor: Objects are pairs X = (X, X ∞ ) consisting of a separated k-scheme of finite type X and an effective (possibly empty) Cartier divisor X ∞ on it such that the complement X \ X ∞ is smooth. Morphisms are given by finite correspondences between the smooth complements satisfying certain admissibility conditions (see §3 for the precise definition). Let MCor ⊂ MCor be the full subcategory consisting of objects (X, X ∞ ) with X proper over k. We then define MPST (resp. MPST) as the category of additive presheaves of abelian groups on MCor (resp. MCor). We have a functor ω : MCor → Cor, (X, X ∞ ) → X − |X ∞ |, and two pairs of adjunctions where τ * is induced by the inclusion τ : MCor → MCor and τ ! is its left Kan extension, and ω * is induced by ω and ω ! is its left Kan extension (see 3.3 for more concrete descriptions of these functors). A basic notion is the -invariance, where = (P 1 , ∞) ∈ MCor: F ∈ MPST is called -invariant if F (X ) ≃ F (X ⊗ ) for all X ∈ MCor (see 3.1 for the tensor product ⊗ in MCor). It is an analogue of the A 1 -invariance 1 exploited by Voevodsky in his theory of motives. We write CI for the full subcategory of MPST consisting of -invariant objects. We know ([KSY, Lem 2.1.7]) that the inclusion CI → MPST admits a right adjoint h 0 which associates to F ∈ MPST the maximal -invariant subobject of F . We define the functor and writeF = τ ! ω CI F ∈ MPST, for F ∈ PST. Then the motivic conductor c F for F ∈ PST is defined by (1.0.1) c F L (a) = min{n| a ∈F (O L , m −n L )}, for a ∈ F (L). Here, for G ∈ MPST, L = Frac(O h X,x ) ∈ Φ, and n ∈ Z ≥1 , we put where V → X ranges overétale neighborhoods of x and D x is the closure of x in V and nD x is its n-th thickening in V . By convention, For G =F there are natural inclusionsF (O L , m −n L ) ֒→ F (L), which are used to define (1.0.1). It turns out that {F (O L , m −n L )} n∈Z ≥0 induces an increasing filtration on F (L) which is exhaustive if F ∈ RSC. Here RSC is the full subcategory of PST consisting of the objects belonging to the essential image of CI under ω ! . Objects of RSC are called reciprocity presheaves and play a key role in this note. We know (see [KSY,Cor 2.3.4]) that RSC contains all A 1 -invariant objects in PST. Moreover it contains many interesting objects F which are not A 1invariant. In this note we consider in particular the following examples (X runs over objects of Sm): (i) F (X) = Hom Sm (X, Γ), where Γ is a smooth commutative algebraic k-group which may have non-trivial unipotent part (for example Γ = G a ). (ii) F (X) = H 1 et (X, Q/Z) = Hom cont (π 1 (X) ab , Q/Z). (iii) F (X) = Conn 1 (X) (resp. Conn 1 int (X)) the group of isomorphism classes of (resp. integrable) rank 1 connections on X.

Theorem 1.
(1) In case (i), c F L agrees with the conductor of Rosenlicht-Serre ([Ser84]) if L has perfect residue field. If ch(k) = p is positive and F = W n is the group scheme of p-typical Witt vectors of length n, then c F L agrees with a conductor defined by Kato-Russell in [KR10] for any L.
(2) In case (ii), c F L agrees with the Artin conductor Art L of Kato-Matsuda (see §7.1) 2 .
(3) In case (iii), c F agrees with the irregularity of connections.
As far as we know, the motivic conductor c F in the case (iv) is new and we give an explicit description only in case the infinitesimal unipotent part of G is α p , where p = ch(k) (see Theorem 9.12).
An amusing application of the motivic conductor c F is to give an explicit description of the maximal A 1 -invariant part of F : Let HI ⊂ PST be the full subcategory of A 1 -invariant objects. The inclusion HI → PST admits a right adjoint h 0 A 1 which associates to F ∈ PST the maximal A 1 -invariant subobject of F (see 4.30 for an explicit description of h 0 A 1 ). Let NST ⊂ PST be the full subcategory of Nisnevich sheaves, i.e., those objects F ∈ PST whose restrictions to Sm ⊂ Cor are sheaves with respect to the Nisnevich topology.
Theorem 2. For F ∈ RSC ∩ NST and X ∈ Sm, we have where ρ ranges over all morphisms Spec L → X with L ∈ Φ.
In case F = H 1 et (−, Q/Z) from (ii) (resp. F = Conn 1 int from (iii)), Theorem 2 asserts that the maximal A 1 -invariant part of F is precisely the subsheaf of tame characters (resp. regular singular connections).
In what follows we fix F ∈ RSC ∩ NST and introduce a class of collections of functions c = {c L : F (L) → N} L∈Φ which may be called conductors for F . Let Func Φ (F, N) be the partially ordered set consisting of collections of functions with partial order given by c ≤ c ′ , if c L (a) ≤ c ′ L (a) for all L ∈ Φ and a ∈ F (L). Let CI(F ) be the partially ordered set consisting of subobjects G of ω CI F such that the induced maps ω ! G → ω ! ω CI F are isomorphisms and with partial order given by inclusion. Then every G ∈ CI(F ) gives rise to an exhaustive increasing filtration {τ ! G(O L , m −n L )} n≥0 on F (L) and we define c G ∈ Func Φ (F, N) by c G L (a) = min{n | a ∈ τ ! G(O L , m −n L )}, for a ∈ F (L). By definition the motivic conductor c F of F is c ω CI F and c F ≤ c G , for all G ∈ CI(F ). Now a question is whether there is a simple characterization of the poset {c G | G ∈ CI(F )} in Func Φ (F, N). We answer it in the following refined form. Let n be a positive integer or ∞. Let Φ ≤n ⊂ Φ be the collection of such L that trdeg k (L) ≤ n.  (1) c G ∈ Cond(F ) sc for every G ∈ CI(F ).
(2) There exists an order reversing map Cond(F ) sc ≤n → CI(F ), c →F c such that c = (cF c ) ≤n . For X = (X, X ∞ ) ∈ MCor with X = X − |X ∞ | we havê where c X (a) ≤ X ∞ means that for any L ∈ Φ ≤n and any morphism ρ : Spec O L → X such that ρ(Spec L) ∈ X, c L (ρ * a) is not more than the multiplicity of the pullback of X ∞ along ρ.
As a consequence, we obtain the following (see Theorem 4.25(4)).
We call c ∞ the canonical extension of c. For example, the Kato-Matsuda Artin conductor is the canonical extension of the classical Artin conductor. We say F has level n, if (c F ) ≤n ∈ Cond(F ) sc ≤n ; in this case c F is the canonical extension of (c F ) ≤n , by Theorem 4.25(5). We show that F = H 1 et (−, Q/Z) in (ii) is of level 1 (see Theorem 8.8), F = Conn 1 (resp. F = Conn 1 int ) from (iii) is of level 2 (resp. 1) (see Theorem 6.11), and F = H 1 fppf (−, Γ) from (iv) is of level 1 if the infinitesimal unipotent part of Γ is trivial and else is of level 2 (see Theorem 9.12).
We give a description of the content of each section: In section 2 we explain how to extend a presheaf with transfers to the category of regular schemes over k which are pro-smooth; this is well-known and we include it only for lack of reference. In section 3 we recall the necessary constructions and results from the theory of motives with modulus as developed in [KMSY21a], [KMSY21b], [KSY16], [KSY], and [Sai20]. Then we introduce in section 4 the notion of (semi-continuous) conductors and prove Theorems 3 and 2. We close the section with a discussion of the relation between the motivic conductor of a reciprocity sheaf with certain vanishing properties of its associated symbol. This is needed in order to prove in the later sections that a certain conductor is equal to the motivic one; the main point being Corollary 4.40. In the second part we consider various conductors which are mostly classical and show that they are motivic in our sense. Kähler differentials and rank one connections are considered in section 6, where ch(k) = 0. In the following sections we assume ch(k) = p > 0. In section 7 it is shown that one of the conductors defined by Kato-Russell for W n is motivic. We use this in section 8 to show that the Kato-Matsuda conductor for characters is motivic, which yields also a description of the motivic conductor for lisseQ ℓ -sheaves of rank 1. Finally, in section 9 we define and investigate a conductor for torsors under finite flat k-groups, which we believe to be new. The general pattern of these computations is always the same: First we show that the collection c = {c L } defined in the various situations defines a semi-continuous conductor (of a certain level) in the sense of Definitions 4.3 and 4.14, then we do a symbol computation to show that this conductor is actually motivic. Note however, that the actual computations in the various cases differ quite a bit.
Part 1. The general theory

Presheaves with transfers on pro-smooth schemes
The material in this section is well-known, we give some details for lack of reference.
2.1. Denote by Cor the category of finite correspondences of Suslin-Voevodsky. Recall that the objects are the smooth k-schemes and morphisms are given by correspondences, i.e., Cor(X, Y ) is the free abelian group generated by prime correspondences, i.e., integral closed subschemes V ⊂ X ×Y which are finite and surjective over a connected component of X. Given two prime correspondences V ∈ Cor(X, Y ) and W ∈ Cor(Y, Z) their composition is given by the intersection product (see e.g. [Ser65, V, C]) Denote by ProCor the pro-category of Cor, i.e., objects are functors I o → Sm, i → X i , where I is a filtered essentially small category, and the morphisms between two pro-objects (X i ) i∈I and (Y j ) j∈J are given by Definition 2.2. We define the category Cor pro as follows: The objects are the noetherian regular schemes over k of the form where (X i ) i∈I is a projective system of smooth k-schemes indexed by a partially ordered set and with affine transition maps X i → X j , i ≥ j. If X and Y are two objects in Cor pro , then Cor pro (X, Y ) = Cor(X, Y ) is the free abelian group generated by prime correspondences in the sense of 2. (2) Note, that for X, Y ∈ Cor pro the cartesian product X × Y does not need to be noetherian; but if Y ∈ Sm and X ∈ Cor pro , then X × Y ∈ Cor pro .
Lemma 2.4. Let A be a k-algebra which is noetherian, regular, and is a directed limit A = lim − →i∈I A i , where the A i are smooth and of finite type over k and the transition maps A i → A j , j ≥ i are flat. Let X be a regular quasi-projective A-scheme. Then X ∈ Cor pro .
Choose an Sembedding X ⊂ P n S . We find an i 0 and a subscheme X i 0 ⊂ P n Then the transition maps X j → X i , j ≥ i ≥ i 0 , are affine and flat, hence so is the projection τ i : Lemma 2.5. There is a (up to isomorphism) canonical and faithful functor Proof. For any X ∈ Cor pro we choose once and for all a projective system (X i ) i∈I as in (2.2.1). In particular, Cor pro and let V ⊂ X ×Y be a prime correspondence. For any scheme S over k we denote by the projection and transition maps of (X i ×S) and (Y j ×S), respectively. By assumption all these maps are affine. For all j, the morphism V → X × Y j induced by σ j is a morphism of finite type X-schemes.
is proper and affine, hence finite. Since X is noetherian σ j (V ) is finite over X, hence we obtain a well defined correspondence σ j * V ∈ Cor(X, Y j ) with the property Furthermore, since X × Y j is noetherian, we find an index i (depending on j) and a correspondence V i,j ∈ Cor(X i , Y j ) such that Cor(X i , Y j ). Therefore we obtain a well-defined element V j are defined; the base change formula (2.5.6) will be only applied in cases where one of the maps f or h is flat, hence the tor-independence condition will be automatic. (But note that h might not be flat so there might appear higher Tor's in the computation of h * and h ′ * , respectively.) This finishes the proof.
2.6. A presheaf with transfers in the sense of Suslin-Voevodsky is a functor F : Cor o → Ab; they form the category PST. We extend it to a functor F : ProCor o → Ab by the formula Precomposing F with the functor from Lemma 2.5 we obtain presheaves on Cor pro , which we again denote by F , between k-schemes which are objects in Cor pro , then we set
3.1. A modulus pair X = (X, X ∞ ) consists of a separated and finite type k-scheme X and an effective Cartier divisor X ∞ ≥ 0 such that the open complement X := X \ |X ∞ | is smooth. We say X is a proper modulus pair if X is proper over k. A basic example is the cube An admissible prime correspondence from X to Y is a prime correspondence V ∈ Cor(X, Y ) satisfying the following condition We denote by Cor adm (X , Y) ⊂ Cor(X, Y ) the subgroup generated by admissible correspondences. Assume X is a proper modulus pair. Recall from [KSY, Lem 2.2.2], that the presheaf with transfers h 0 (X ) ∈ PST is defined by where we write S instead of (S, ∅) and i ε : S ֒→ A 1 S is the ε-section, ε ∈ {0, 1}. We have a natural quotient map Z tr (X) → h 0 (X ), where Z tr (X) is the presheaf with transfers representing X, i.e., Z tr (X)(S) = Cor(S, X).
Definition 3.2 ([KSY, Def 2.2.4]). Let F ∈ PST, X ∈ Sm and a ∈ F (X). We say a has SC-modulus (or just modulus) X , if X = (X, X ∞ ) is a proper modulus pair with X = X \ |X ∞ | and the Yoneda map a : Z tr (X) → F , factors via i.e., for any S ∈ Sm and any correspondence γ ∈ Cor adm ( × S, X ) ⊂ Cor(A 1 × S, X) we have i * 0 γ * a = i * 1 γ * a. We say F has SC-reciprocity, if for all X ∈ Sm any a ∈ F (X) has a modulus. We denote by RSC ⊂ PST the full subcategory consisting of presheaves with transfers which have SC-reciprocity. Further we set where NST ⊂ PST is the full subcategory of Nisnevich sheaves with transfers. [KSY] that the presheaves in RSC are in fact induced by presheaves on modulus pairs in the following way: Let X = (X, X ∞ ) and Y = (Y , Y ∞ ) be modulus pairs with corresponding opens X and Y , respectively. An admissible correspondence from X to Y (see 3.1.1) is called left proper, if the closure in X × Y of all its irreducible components is proper over X. We denote by MCor(X , Y) ⊂ Cor(X, Y ) the subgroup of all left proper admissible correspondences. This subgroup is stable under composition of correspondences (see [

It is shown in
where MSm(X) is the subcategory of MCor with objects the proper modulus pairs with corresponding opens X and only those morphism which map to the identity in Cor(X, X), and Comp(U) is the category of compactifications of U = (U , U ∞ ), i.e., objects are proper modulus pairs X = (X, U ∞ + Σ), where U ∞ and Σ are effective Cartier divisors such that X \ |Σ| = U and U ∞|U = U ∞ , and the morphisms are those which map to the identity in MCor(U, U), see [KMSY21a, Lem 2.4.2]. The functors ω ! , ω ! , τ ! are exact and we have ω ! = ω ! τ ! .
We denote by CI the full subcategory of MPST of cube invariant objects, i.e., those F ∈ MPST, which satisfy that for any proper modulus pair X the pullback along X ⊗ → X induces an isomorphism By [KSY,Prop 2.3.7] we have ω ! (CI) = RSC and there is a fully faithful left exact functor ω CI : RSC → CI given by 3.4. We recall some more definitions and results from [KMSY21a], [KMSY21b], and [Sai20] related to Nisnevich sheaves. For F ∈ MPST and X = (X, X ∞ ) ∈ MCor we denote by F X the presheaf on Xé t defined by We denote by MNST the full subcategory of MPST consisting of those F such that F X is a Nisnevich sheaf on X, for any X = (X, X ∞ ) ∈ , where X = (X, X ∞ ) ∈ MCor, F X ,Nis denotes the Nisnevich sheafification of the presheaf F X on Xé t , and the limit is over all proper birational morphisms (2) τ ! restricts to an exact functor τ ! : MNST → MNST and satisfies It follows that a Nis = τ * a Nis τ ! and a Nis ω * = ω * a V Nis , a Nis ω * = ω * a V Nis , where a V Nis : PST → NST is Voevodsky's Nisnevich sheafification functor (see [Voe00b, Lem 3.1.6]), and we obtain induced functors ω * : NST → MNST, ω * : NST → MNST .
Lemma 3.5. For F ∈ RSC Nis we have ω CI F ⊂ a Nis ω CI F ⊂ ω * F in MPST (see Definition 3.2 and (3.3.4) for notation). Here the first inclusion is given by the unit of adjunction.
Proof. By definition ω CI F ⊂ ω * F . We obtain the following commutative diagram in which the vertical maps are induced by adjunction. The vertical map on the right is an isomorphism since ω * F ∈ MNST, the top horizontal map is an inclusion since a Nis is exact. This gives the statement.
Remark 3.6. It follows from Corollary 4.16 below that the first inclusion in Lemma 3.5 is actually an equality.

3.7.
We define the category MCor pro as follows: The objects are pairs X = (X, X ∞ ), where (1) X is a separated noetherian scheme over k of the form X = lim ← −i∈I X i , with (X i ) i∈I a projective system of separated finite type k-schemes indexed by a partially ordered set with affine transition maps τ i,j : Lemma 3.8. There is a (up to isomorphism) canonical and faithful functor We have to show that the injection (2.5.2) restricts to To this end let V ∈ MCor pro (X , Y) be a left proper admissible correspondence. For j ∈ J denote by σ j (V ) the image of V under the projection X × Y → X × Y j . Then σ j (V ) is a finite prime correspondence as was observed in the proof of Lemma 2.5. Let V ⊂ X × Y be the closure of V . By assumption V is proper over X. Since X × Y j is separated and of finite type over X the image of V in X × Y j is closed and proper over X; hence it is equal to the closure σ j (V ) of σ j (V ).
with the notation from (3.1.2). As in the proof of Lemma 2.5 we find an index i 0 ∈ I and a finite correspondence V i 0 ,j ⊂ X i 0 × Y j which pulls back to σ j (V ). We can also assume (after possibly enlarging i 0 ) that the closure V i 0 ,j ⊂ X i 0 × Y j of V i 0 ,j pulls back to σ j (V ). We obtain the cartesian diagram Since the upper horizontal arrow is proper, the lower horizontal arrow becomes proper after possibly enlarging i 0 , see [EGA IV 3 , Thm (8.10.5), (xii)]. Hence by our construction and (3.8.2), the scheme is a left proper admissible correspondence from X i 0 to Y j and gives a well-defined element This shows that (2.5.2) restricts to (3.8.1).
3.9. Let F ∈ MPST. Using Lemma 3.8 we can extend F to a presheaf on MCor pro by the formula

Conductors for presheaves with transfers
Definition 4.1.
(1) We say that L is a henselian discrete valuation field of geometric type (over k) (or short that L is a henselian dvf) if L is a discrete valuation field and its ring of integers is equal to the henselization of the local ring of a smooth kscheme U in a 1-codimensional point x ∈ U (1) , i.e., Note that in positive characteristic Φ ≤1 consists precisely of the henselian dvf's with perfect residue field.
(2) Let X be a smooth k-scheme. A henselian dvf point of X is a k-morphism Spec L → X, with L ∈ Φ. (3) Let X = (X, X ∞ ) be a modulus pair with X = X \ |X ∞ |. A henselian dvf point of X is a henselian dvf point ρ : Spec L → X extending to Spec O L → X. Note, if it exits, such an extension is unique, and if X is proper, then there always exists an extension. We will denote this extension also by ρ. We will also write ρ : Spec L → X for the henselian dvf point of X defined by ρ.
(1) Let F ∈ PST and X ∈ Sm. A henselian dvf point ρ : η = Spec L → X is a morphism in Cor pro (see 2.2). Hence we get a morphism (see 2.6) Also η = Spec L → Spec O L = η is in Cor pro and we get an (2) Let X = (X, X ∞ ) be a modulus pair with X = X \ |X ∞ | and ρ : Spec L → X a henselian dvf point. Then we denote by the multiplicity of X ∞ pulled back along ρ.
satisfying the following properties for all L ∈ Φ ≤n and all X ∈ Sm: e(L ′ /L) ⌉, for any finite morphism f : Spec L ′ → Spec L and any a ∈ F (L ′ ). Here e(L ′ /L) denotes the ramification index of L ′ /L and ⌈−⌉ is the round up.
x ,∞ ) and ρ x : Spec k(x)(t) ∞ → A 1 X is the natural map. Then a ∈ π * F (X), with π : A 1 X → X the projection. (c5) For any a ∈ F (X) there exists a proper modulus pair X = (X, X ∞ ) with X = X \ |X ∞ |, such that for all ρ : Spec L → X we have c L (ρ * a) ≤ v L (X ∞ ). A conductor of level ∞ will be simply called conductor.
(1) If F is homotopy invariant, then setting c L (a) = 0, if a ∈ Im(F (O L ) → F (L)), and c L (a) = 1 else, defines a conductor (of any level).
(2) If c = {c L } is a conductor for F . Then for any L we have Definition 4.5. Let F ∈ PST and let c = {c L } be a conductor of level n for F . Let X = (X, X ∞ ) be a modulus pair with X = X \ |X ∞ |. For a ∈ F (X), we write c X (a) ≤ X ∞ to mean c L (ρ * a) ≤ v L (X ∞ ), for all henselian dvf points ρ : Spec L → X with trdeg(L/k) ≤ n (see Definition 4.1).
Lemma 4.6. Let c be a conductor of some level for F ∈ PST, X ∈ Sm, and a ∈ F (X). Let X = (X, X ∞ ) be any proper modulus pair with X = X \ X ∞ . Then there exists a natural number n ≥ 1 such that c X (a) ≤ n · X ∞ .
Proof. By 4.3(c5), there exists a proper modulus pair X 1 = (X 1 , X 1,∞ ) with corresponding open X and such that c L (ρ * a) ≤ v L (X 1,∞ ), for all ρ. We find a proper normal k-scheme X 2 with k-morphisms f : . Hence the statement.
Proposition 4.7. Let F ∈ PST and let c be a conductor of level n for F . Then defines an object in MPST. Furthermore (see 3.3 for notations): (1) For any X ∈ MCor the pullback along the projection map Proof. We start by showing F c ∈ MPST. Let X = (X, X ∞ ) and Y = (Y , Y ∞ ) be two modulus pairs with corresponding opens X and Y , respectively. We have to show that a left proper admissible prime . Take a ∈ F c (Y) and a henselian dvf point ρ : η = Spec L → X with trdeg(L/k) ≤ n. We have to show with some multiplicities m i ∈ N. For each i we get a commutative diagram ) (see 2.6 for the notation). Thus Since the closure V of V in X × Y is proper over X and ρ extends to ρ, we see that ρ i extends to ρ i as in the diagram Since V satisfies the modulus condition (3.1.2) we get where the last equality follows from v L j (X ∞ ) = e(L j /L)v L (X ∞ ). This proves (4.7.1) and hence that F c is in MPST. Next, we prove (1). Let X = (X, X ∞ ) be a modulus pair with X = X \ |X ∞ |. Denote by π : X × A 1 k → X the projection and by i 0 : X ֒→ X × A 1 k the zero section. These define morphisms π ∈ MCor(X ⊗ , X ) and i 0 ∈ MCor(X , X ⊗ ). We have to show that π * : F c (X ) → F c (X ⊗ ) is an isomorphism. Since i * 0 π * = id Fc(X ) , it suffices to show that π * is surjective. Take a ∈ F c (X ⊗ ). For any henselian dvf point ρ : Spec L → (P 1 X , {∞} X ), with trdeg(L/k) ≤ n, we have Hence by 4.3(c4), there exists an element b ∈ F (X) with π * (b) = a. We have to check that b ∈ F c (X ). Take ρ : Spec L → X a henselian dvf point with trdeg(L/k) ≤ n. Then i 0 • ρ : Spec L → X ⊗ is a henselian dvf point and thus Hence b ∈ F c (X ). Statement (2) follows directly from (3.3.2) and 4.3(c5). Finally (3). For X = (X, X ∞ ), the presheaf F c,X on Xé t (see (3.4.1)) is given by We have to show that this is a Nisnevich sheaf. Since F is a Nisnevich sheaf it suffices to show the following: Let u : U → X be anétale map, a ∈ F (U \|u * X ∞ |) and assume there is a Nisnevich cover To this end, observe that if ρ : Spec L → (U, u * X ∞ ) is a henselian dvf point with trdeg(L/k) ≤ n and x ∈ U is the image point of the closed point of Spec O L , then by the functoriality of henselization ρ factors via . This completes the proof.
4.8. Let F ∈ PST and let c be a conductor of some level for F . Let F c ∈ MPST be as in Proposition 4.7. We set (see 3.3 for notation) By adjunction we have a natural map By Proposition 4.7 and [KMSY21b, Lem 4.2.5] (or a similar argument as in the proof of 4.7(3)) we have 4.9. Let F ∈ RSC. Denote by CI(F ) the partially ordered set consisting of those subobjects G ⊂ ω CI F in MPST, such that the induced map ω ! G → ω ! ω CI F = F is an isomorphism, and where the partial order is given by inclusion G 1 ⊂ G 2 We set Lemma 4.10. Let F ∈ RSC and G ∈ CI(F ). Then G 1 = τ ! G ∈ MPST has the following properties: Proof. Note that (2) follows directly from τ * τ ! = id. We show (1) and (3). The inclusion G ֒→ ω CI F yields a commutative diagram Here the top horizontal row is injective by the exactness of τ ! , the vertical maps are induced by adjunction, the vertical map on the right is injective by (3.3.4). It follows that the vertical map on the left is injective; furthermore the injectivity of the top horizontal map and [Sai20, Lem 1.15, 1.16] imply that G 1 is -invariant.
Lemma 4.12. Let F ∈ PST and let c be a conductor of some level for Proof. By Proposition 4.7(2), it suffices to show that there is an inclusion τ * F c ֒→ ω CI F inside ω * F . For X a proper modulus pair set Z tr (X ) := MCor(−, X ), and Since F c is cube invariant, by Proposition 4.7, the Yoneda map a : Z tr (X ) → τ * F c factors via the quotient map Z tr (X ) → h 0 (X ). Applying ω ! = ω ! τ ! we see that the Yoneda map a : Z tr (X) → F in PST defined by a ∈ F (X) factors via Z tr (X) → h 0 (X ), i.e., a ∈ ω CI F (X ). This proves the lemma.
Notation 4.13. Let L ∈ Φ. Denote by s ∈ S := Spec O L the closed point. For all n ≥ 1 we have (S, n · s) ∈ MCor pro (see 3.7). Let G ∈ MPST; we extend it to a presheaf on MCor pro . For n ≥ 0 we introduce the following notation: Definition 4.14. Let F ∈ RSC Nis and G ∈ CI(F ) (see 4.9). We denote by This is well-defined since In case G = ω CI F we write (4.14.1) and call c F the motivic conductor of F .
Theorem 4.15. Let F be a presheaf with transfers.
(1) If F has a conductor c of some level, then F ∈ RSC.
(2) If F ∈ RSC Nis and G ∈ CI(F ) (see 4.9), then the family c G = {c G L } (see Definition 4.14) is a conductor for F in the sense of Definition 4.3. In particular, c F is a conductor for F .
and for all L ∈ Φ and n ≥ 0, we have (4) Let F ∈ RSC Nis and let c be a conductor for F (of some level). ThenF where c F is the motivic conductor, see (4.14.1). In particular, F ∈ RSC Nis ⇐⇒ F ∈ NST and F has a conductor (of some level).
Proof. (1). We have F = ω ! τ * F c ∈ ω ! (CI) ⊂ RSC, by Proposition 4.7 and [KSY, Prop 2.3.7]. Next (2). We check the properties from Definition 4.3. Set where s L (resp. s L ′ ) are the closed points. This yields the commutative diagram Hence, we obtain the following inequality which implies (c3): The following claim clearly implies (c4): Thm 6] and [KSY, Cor 3.2.3]; thus it suffices to show a K ∈ F (K). Set G 1,Nis := a Nis (G 1 ) (see 3.4). Consider the Nisnevich localization exact sequence . Hence our assumption implies a K comes from G 1,Nis (P 1 K , ∞) and the desired assertion follows from the cube invariance of G 1,Nis , see [Sai20, Thm 10.1] (and Remark 4.11), . Next we prove (c5). Let X ∈ Sm and a ∈ F (X). We can assume that X is not proper over k. Take any X = (X, X ∞ ) ∈ MCor such that X = X − |X ∞ |. We have and hence a ∈ G(X, n · X ∞ ), for some n. Then, for any henselian dvf . This completes the proof of (2).
(3). It follows directly from the definition of F c G in Proposition 4.7, Furthermore, the equality in the second part of the statement comes from the inclusions where the first inclusion comes from the above and the second holds by definition. Finally (4). The inclusionF c ⊂ τ ! ω CI F follows from Lemma 4.12. The equalityF c F = τ ! ω CI F , now follows from this and (3). This completes the proof.
Notation 4.17. Let F ∈ RSC Nis . In the following we will simply writeF : By Corollary 4.16 we have τ * F ∈ CI(F ) Nis (see 4.9).
Corollary 4.18. Let F ∈ RSC Nis . Denote by (c F ) ≤n the restriction of the motivic conductor to trdeg ≤ n. Assume (c F ) ≤n is a conductor of level n. ThenF Proof. ClearlyF c F ⊂F (c F ) ≤n , and '⊃' holds by Theorem 4.15(4).
Proposition 4.19. Let F 1 ⊂ F 2 be an inclusion in RSC Nis . Then the restriction of the motivic conductor of F 2 to F 1 is equal to the motivic conductor on F 1 , i.e., Proof. Let a ∈ F 1 (X). By the definition of the motivic conductor it suffices to show: a has modulus (X, X ∞ ) as an element in F 2 (X), if and only if it has the same modulus as an element in F 1 (X). This is obvious, see Definition 3.2.
Direct from Definition 4.14.
Proposition 4.21. Let k 1 /k be an algebraic (hence separable) field extension and let F ∈ RSC Nis,k 1 (i.e. F is a contravariant functor Cor k 1 → Ab which is a Nisnevich sheaf on Sm k 1 and has SCreciprocity). Denote by R k 1 /k F : Sm = Sm k → Ab the functor given by where X k 1 = X × Spec k Spec k 1 . Then R k 1 /k F ∈ RSC Nis and its motivic conductor is given by Proof. The first statement follows from the definition of RSC Nis ; for the second observe that for L ∈ Φ the k 1 -algebra L ⊗ k k 1 = i L i is unramified over L, hence (see 4.17 for notation) This yields the statement.
Definition 4.22. Let F ∈ PST and let c be a conductor of level n ∈ [1, ∞] for F . We say c is semi-continuous if it satisfies the following condition: (c6) Let X ∈ Sm with dim(X) ≤ n and Z ⊂ X a smooth prime divisor with generic point z and where a U (resp. a K ) denotes the restriction of a to U (resp. K).
Lemma 4.23. Let F ∈ PST and let c be a conductor of level n for F . The following statements are equivalent: (1) c is semi-continuous; ( From this description we see that this '⊂' inclusion in (2) always holds, while this '⊃' inclusion is equivalent to the semi-continuity of c.
Corollary 4.24. Let F ∈ RSC Nis and let c be a semi-continuous conductor of level n for F . Then (c F ) ≤n ≤ c, i.e., for all L ∈ Φ ≤n and all a ∈ F (L) we have c F L (a) ≤ c L (a). Proof. Follows from Theorem 4.15(4) and Lemma 4.23.
We can summarize part of the above as follows: (1) Any G ∈ CI(F ) (see 4.9) defines a semi-continuous conductor c G (see 4.14). For for all proper modulus pairs X = (X, (4) Let c be a semi-continuous conductor of level n for F (possibly only defined on trdeg ≤ n). Then there exists a unique semicontinuous conductor c ∞ for F with the following properties: We call c ∞ the canonical extension of c. We finish this section with some lemmas which are needed later on.
Definition 4.26. Let F ∈ RSC Nis . We say F is proper if the following equivalent conditions are satisfied: (1) For all X ∈ Sm and any dense open U ⊂ X the restriction map F (X) (For this (2) ⇒ (1) implication use that (c4) implies that F ∈ HI Nis and then the statement follows from Voevodsky's Gersten resolution, cf. [KY13, Lem 10.3].) be an exact sequence in NST and with F 1 , F 2 ∈ RSC Nis and assume F 1 is proper.
Then F ∈ RSC Nis . Any (semi-continuous) conductor c of level n on F 2 , induces a (semi-continuous) conductor cψ = {c L • ψ} L of level n on F . Furthermore, the motivic conductor of F is given by c F = c F 2 ψ Proof. Let c be a conductor of level n on F 2 . Then cψ clearly satisfies (c2), (c3), (c5) (and (c6) if c does). By the properness of F 1 we have This shows that cψ satisfies (c4). Therefore, cψ is a conductor of level n. Thus Theorem If c satisfies (c1) (resp. (c2), (c3), (c6) ), then so doesc.
Furthermore, if ϕ has the following property: For all X ∈ Sm there exists a proper modulus pair (X, X ∞ ) with X = X \ X ∞ , such that for all x ∈ X the map ϕ induces a surjection (c3). Let f : Spec L ′ → Spec L be a finite extension with ramification index e and let a ∈ G(L ′ ). Take a liftã ∈ F (L ′ ) withc L ′ (a) = c L ′ (ã). Then by (c3) for c (c6). Let X, z ∈ Z, K be as in (c6) and a ∈ G( we find a Nisnevich neighborhood U → X of z and an elementã ∈ F (U \ Z) which restricts toã K . After possibly shrinking U around z, we may assume that ϕ(ã) = a |U \Z U . By (c6) for F , we may shrink U further around z to obtain a compactification (c5)(assuming (4.28.1)). Let X ∈ Sm and a ∈ G(X). Let X = (X, X ∞ ) be a proper modulus pair with X = X \ |X ∞ | as in (4.28.1). This condition implies that we find a finite Nisnevich cover and inducing a morphism of proper modulus pairs Y i → X . By (c5) for c and (the proof of) Lemma 4.6 we find an integer

Homotopy invariant subsheaves.
Corollary 4.29. Let F ∈ NST be A 1 -invariant (in particular F ∈ RSC Nis ). Then the motivic conductor of F is given by Proof. The right hand side defines a conductor, as already remarked in 4.4; it is clearly semi-continuous. By Corollary 4.24 we get '≤' in the statement and (c1) forces it to be an equality.

4.30.
We denote by HI the category of A 1 -invariant presheaves with transfers and set HI Nis := HI ∩ NST. It follows immediately from Definition 3.2 that we have HI ⊂ RSC and HI Nis ⊂ RSC Nis .
Let F ∈ PST. For X ∈ Sm, we denote by We immediately see that furthermore, it has the following universal property: Let F ∈ PST and let c be a conductor of level n for F . Then defines a homotopy invariant sub-presheaf with transfers of F . If F ∈ NST, then F c≤1 ∈ HI Nis .
Proof. To show F c≤1 ∈ PST is equivalent to the following: let V ∈ Cor(X, Y ) be a finite prime correspondence and a ∈ F c≤1 (Y ); then for all henselian dvf points ρ : Spec L → X with trdeg(L/k) ≤ n, we have This follows from the calculation in (4.7.4). The A 1 -invariance of F c≤1 follows directly from (c4). The last statement is proven similarly as in Proposition 4.7(3).
Corollary 4.32. Let F ∈ RSC Nis with motivic conductor c F . Then . Let X ∈ Sm be proper over k and U ⊂ X dense open. Then In particular, if F satisfies (4.33.1), then X → F (X) is a birational invariant on smooth proper schemes.
Proof. By Corollary 4.32 All together yields the statement. Let F ∈ RSC Nis . If L/K is a finite field extension of finitely generated fields over k, we denote by Tr L/K : F (L) → F (K) the map induced by the transfer structure on F . For X ∈ Cor pro , x ∈ X, and a ∈ F (X) we denote a(x) ∈ F (x) the pullback of a along x ֒→ X.
Let K be a function field over k and C a regular projective K-curve. Note that C ∈ Cor pro by Lemma 2.4. For x ∈ C (0) a closed point we write v x for the corresponding normalized discrete valuation on K(C) × , x be an effective Cartier divisor on C and a ∈F (C, D) (see 4.17 for the notationF ). Then there exists a family of maps (0) which is uniquely determined by the following properties: It follows from the uniqueness that the family {(a, −) C/K,x } does not depend on the chosen modulus D. Furthermore, from the uniqueness one can deduce the following properties: Let K ′ /K be a finite field extension, C ′ /K ′ ∈ Cor pro a projective curve, and π : C ′ → C a finite morphism over Spec K ′ → Spec K, then: where in both cases the sum is over all y ∈ C ′ mapping to x.
Lemma 4.35. Let F ∈ RSC Nis , C be a regular projective and geometrically connected K-curve. Let K ′ /K be a finitely generated field extension, denote by τ : Spec K ′ → Spec K the induced map, and by τ C : for all a ∈ F (K(C)), f ∈ K(C) × , and x ∈ C (0) .
Proof. Let U ⊂ C be open with a ∈ F (U). Using the Approximation Lemma, (LS1), and (LS3) we can assume that for a given m ≥ 1 we . The formula in the statement now follows by applying τ * to this equality, using the base change formula τ * and using (LS1) -(LS4) backwards.
Lemma 4.36. Let L ∈ Φ. Let C be a regular curve over a k-function field K. Assume there exists a closed point x ∈ C and a k-morphism u : Then there is an isomorphisms induced via pullback along u If O C,x has a coefficient field then we have an isomorphism where for a local ring A ∈ Cor pro with maximal ideal m we set Proof. We prove the first isomorphism. The natural map in the statement is compatible with pullbacks and pushforwards on both sides. Thus we can apply the standard trick replacing k by its maximal proℓ extensions for various primes ℓ, to assume k is infinite. By Gabber's Presentation Theorem (see, e.g., [CTHK97, 3.1.2]) we find an open U ⊂ C containing x, a k-function field E and anétale morphism ϕ : . We obtain the first isomorphism of the statement by taking the limit over all Nisnevich neighborhoods v. For the second isomorphism observe that if a coefficient field σ : κ ֒→ O C,x exists, then σ * induces a splitting of the restriction to the closed point where the last map is given by  Proof.
Let v : C ′ → C be a K-morphism between regular projective K-curves, let x ∈ C and x ′ ∈ C ′ be closed points such that v isétale in a neighborhood of x ′ and induces an isomorphism x ′ ≃ − → x. Assume that O C,x has a coefficient field. Let E = K(C) and E ′ = K(C ′ ) be the function fields. Then it suffices to show, that for all a ∈ F (E) and f ∈ E × we have etc. Then the composition is induced by v * and is an isomorphism with inverse induced by the norm. Thus we can use the Approximation Lemma, (LS3), and the continuity of the norm map to choose g ∈ E ′ × close to v * f at x ′ and close to 1 at all which yields the statement.
Proof. We have a L 1 ∈F (O L 1 , m −er L 1 ) and hence the statement follows from the construction of the symbol in 4.37 and (LS3) .
Lemma 4.41. Let F ∈ RSC Nis . Let K/k be a function field, X a normal affine integral finite type K-scheme with function field E. Let x i ∈ X (1) , i = 1, . . . , r, be distinct one codimensional points. Then for all integers n i ≥ 0 the natural map Let A be the semi-localization of X at the points x i and denote by D = i n i x i the divisor on U := Spec A. (Note that we allow |D| {x 1 , . . . , x r }.) We claim The natural map in the statement is compatible with pullbacks and pushforwards on both sides. Thus we can apply the standard trick replacing k by its maximal pro-ℓ extensions for various primes ℓ, to assume k is infinite. By Gabber's Presentation Theorem (see, e.g., [CTHK97, 3.1.2]) we find a function field K 1 /k and an essentiallyétale morphism ϕ : U → A 1 K 1 such that {x 1 , . . . , x r } = ϕ −1 ϕ({x 1 , . . . , x r }) ∼ =   ( We can spread out the situation as follows: There exists a finite and surjective morphism π : C ′ → C between regular and projective K-curves, with function fields We prove (1): By Lemma 4.41 we find an element Since π * f ∈ O × C ′ ,y , for all y/x, we obtain (4.42.1) ; this together with (4.42.1) and (LS1) implies formula (1). Now (2): By the Approximation Lemma we find g 1 ∈ E ′ × such that (π * ã , g 1 ) C ′ /K,y = 0, y ∈ π −1 (x) \ {x ′ }, and (π * ã , g 1 ) C ′ /K,x ′ = (π * a, g) L ′ ,σ ′ .
Furthermore we have the following equality in L × If g 1 is close enough to 1 at the points for N >> 0. Thus we can choose g 1 with the additional property (ã, Nm E ′ /E (g 1 )) C/K,x = (a, Nm L ′ /L (g)) L,σ .
The formula (2) now follows from (LS8) and the above.

Algebraic groups and the local symbol
In this section k is a perfect field and G is a commutative algebraic k-group. Note that as sheaves on Sm we have G = G red and hence we can always identify G with the smooth commutative k-group G red . We fix an algebraic closurek of k; note Speck ∈ Cor pro . 5.1. Let G be a commutative algebraic k-group. Then G ∈ RSC Nis , by [KSY, Cor 3.2.5]. Let L ∈ Φ ≤1 have residue field κ. Let ι : κ ֒→k be a k-embedding. We denote by L sh ι the strict henselization of L with respect to ι. Note that L sh ι is a henselian dvf of geometric type overk. We write for the symbol (−, −) L sh ι ,σ from 4.37 with σ :k ֒→ O sh L,ι the unique coefficient field; in this case this is the symbol defined by Rosenlicht-Serre, see [Ser84, III, §1]. If we choose a different k-embedding ι ′ : κ ֒→ k, then we find an automorphism τ :k →k with τ , τ (f )) L sh ι ′ . We will usually drop the ι from the notation and write L sh = L sh ι . We define the Rosenlicht-Serre conductor of a ∈ G(L) by Note that it is independent of the choice of ι : κ ֒→k.
Theorem 5.2. Let G be a commutative algebraic k-group. (2) Let c G be the motivic conductor of G (see Definition 4.14) and denote by (c G ) ≤1 its restriction to Φ ≤1 . Then RoSe = (c G ) ≤1 . In particular, the motivic conductor extends the Rosenlicht-Serre conductor to henselian dvf 's over k with non-perfect residue field and we have G = G RoSe (see 4.8 and 4.17 for notation).
Proof. The last statement follows from Corollary 4.18. For (1) we check that RoSe satisfies the properties from Definition 4.3. (c1) and (c2) are obvious. Let L ′ /L be a finite extension of henselian dvf's with trdeg(L/k) = 1 and a ∈ G(L ′ ). Let κ ֒→ κ ′ be the induced map on the residue fields and fix an embedding κ ′ ⊂k. Then L ′ sh is finite over L sh and e(L ′ sh /L sh ) = e(L ′ /L). Thus (c3) follows directly from Lemma 4.42(1). To check (c4) first observe, if a ∈ G(A 1 X ) is not in G(X) (via pullback), then we find a closed point x ∈ X such that a A 1 x is not in G(x). (Since G is a finite type k-scheme and X is Jacobson.) Thus it suffices to show the following: Claim. Let κ/k be a finite field extension and set κ(t) ∞ = Frac(O h P 1 κ ,∞ ). Assume a ∈ G(A 1 κ ) has RoSe κ(t)∞ (a) ≤ 1. Then a ∈ G(κ). Else a ∈ G(κ). Then its pullback ak ∈ G(A 1 k ) is not in G(k) and we can thus find two points x, y ∈ A 1 (k) =k such that ak(x) = ak(y). Take f = (t − x)/(t − y) ∈k(t). Then f ∈ U where the first equality follows from RoSe κ(t)∞ (a) ≤ 1 and the second from (LS4) and (LS2). This yields a contradiction and thereby proves the claim. (c5) follows from the fact that G is a reciprocity sheaf and Corollary 4.40. Finally (c6) (semi-continuity for n = 1). Assume C is a smooth k-curve, x ∈ C a closed point and ) and a x ∈ G(L x ) denotes the pullback of a. Let C be the smooth compactification of C and let C ∞ = (C\C) red . Choose N such that RoSe Ly (a y ) ≤ N, for all y ∈ |C ∞ |. Then (C, n · {x} + N · C ∞ ) is a compactification of (C, n · {x}) and we claim L sh , where n x = n and n y = N, for y = x. By Lemma 4.42(2) we have (a L , u) L sh = (a Ly , Nm L sh /L sh y (u)) L sh y , which vanishes by Nm L sh /L sh y (u) ∈ U (ny ) L sh y and RoSe Ly (a y ) ≤ n y . This proves the claim (5.2.1), hence (c6), and finishes the proof of (1).
By Corollary 4.24 we have c G,1 ≤ RoSe. Thus for (2) it suffices to show: If a ∈G(O L , m −r L ), for some L ∈ Φ ≤1 and r ≥ 1, then RoSe L (a) ≤ r. This follows from Corollary 4.40.
Remark 5.3. An extension of RoSe to dvf's of higher transcendence degree over k was also constructed in [KR12] (char 0) and [KR10] (char p > 0). The construction essentially coincides with the extension from Theorem 5.2, but in loc. cit. the log version is considered, whereas here non-log one, c.f. Theorem 7.20 below.

Differential forms and irregularity of rank 1 connections
In this section we assume that the base field k has characteristic 0. We fix a ring homomorphism R → k which induces the structure of an R-scheme on any k-scheme. (Of main interest are R = k or Z.) 6.1. Kähler differentials.
Lemma 6.2. The differential d : Ω q /R → Ω q+1 /R is a map in RSC Nis . Proof. We have to show, that if α ∈ Cor(X, Y ) is a finite correspondence, X, Y ∈ Sm, then α * d = dα * as maps Ω q /R (Y ) → Ω q /R (X). Since the restriction Ω q /R (X) → Ω q /R (U) is injective for any dense open U ⊂ X (by [KSY16, Thm 6]), it suffices to verify the equality after shrinking X arbitrarily around its generic points. In particular we can assume, that X is connected and α = Z ⊂ X × Y is a prime correspondence which is finiteétale over X (here we use char(k) = 0). Denote by f : Z → X and g : Z → Y the maps induced by projection. Then Z * = f * g * . The compatibility of d with g * is clear. Hence it remains to show f * d = df * for a finiteétale map f : Z → X between smooth schemes. In this case, Theorem 6.4. For all q ≥ 0, the collection c dR = {c dR L } defined in 6.3 coincides with the motivic conductor, i.e., (see Definition 4.14) Furthermore, the restriction (c dR ) ≤q+1 is a semi-continuous conductor.
Proof. We start by showing that c dR is a semi-continuous conductor of level q + 1. Properties (c1) and (c2) of Definition 4.3 are obvious.
(c3). Let L ′ /L be a finite extension of henselian dvf with ramification index e = e(L ′ /L), and denote by f : Spec L ′ → Spec L the induced map. Let a ∈ Ω q L ′ /R . We have to show: We know that f * restricts to Ω q O L ′ /R → Ω q O L /R and by the well-known formula f * dlog = dlog • Nm L ′ /L also to . Moreover, f * is continuous with respect to the m L -adic topology (which on Ω q L ′ /R is the same as the m L ′ -adic topology). We may therefore replace Ω q L ′ /R and Ω q L/R by the corresponding completed modules. Furthermore, it suffices to treat the two cases separately in which L ′ /L is either totally ramified or unramified.
1st case: e = 1. In this case a local parameter t ∈ O L is also a local parameter of O L ′ and hence (6.4.1) follows directly from (6.4.2) and the L-linearity of f * .
2nd case: e > 1, L, L ′ complete and O L /m L = O L ′ /m L ′ . Let K ֒→ O L be a coefficient field; it also defines a coefficient field of O L ′ . Let τ ∈ O L ′ and t ∈ O L be local parameters. Then we can identify L ′ = K((τ )) and 1 τ n−1 · Ω q O L ′ /R (log) with the τ -adic completion of Since f * commutes with the differential (by 6.2) we are reduced to show: · a ∈ O L , for any a ∈ L = K((t)). (c4) for c dR,q+1 follows directly from the following facts, where A is a finite type smooth k-algebra: /k ); (ii) for any non-zero α ∈ Ω q A/R there exists a prime ideal p ⊂ A with trdeg(k(p)/k) = q, where k(p) = A p /p, such that the image of α in Ω q k(p)/R is non-zero; (iii) H 0 (P 1 k , Ω 1 P 1 /k (log ∞)) = 0, H 0 (P 1 , O P 1 ) = k. (Note, (ii) is easy for R = k and follows in general from the natural map Ω q /R → Ω q /k .) For (c5) it suffices to observe that if a ∈ H 0 (X, Ω q X/R ⊗ O X O X (D)), for some proper modulus pair (X, D), then c dR X (a) ≤ D. Finally, (c6). Let U = Spec A be smooth affine and Z ⊂ U a smooth divisor which we can assume to be principal Z = Div(t). Let a = 1 t r−1 a 1 + 1 t r−1 a 2 dlog t, a 1 ∈ Ω q A/R , a 2 ∈ Ω q−1 A/R , r ≥ 1. Let (Y , Z + Σ) be a compactification of (U, Z) with Z |U = Z and Y normal. Let Y = ∪V i be an open covering such that V i = Spec B i , Σ |V i = Div(f i ), and Z |V i = Div(τ i ), with τ i , f i ∈ B i . Note that Let E i be the Cartier divisor on V i defined by e i . We have |E i | ⊂ |Σ |V i |. By Lemma 6.5 below, there exists Hence c dR Y (a) ≤ (r · Z + N · Σ), which proves (c6). Thus c dR is a semi-continuous conductor on Ω q /R and Theorem 4.15(3) yields for n ≥ 1 where the local symbol on the left hand side is the one from 4.37 for Ω q /R . Since the local symbol for Ω q /R is uniquely determined by (LS1) -(LS4), we see that it is given by (a, 1 − xt r ) Lx,σ = Res t (a dlog(1 − xt r )), where we use the isomorphism L x = K(x)((t)) defined by σ to compute the residue symbol on the right. To prove the implication (6.4.4) it suffices to consider a modulo fil r ; we have a ≡ 1 t r α + β dt t r+1 mod fil r , for α ∈ Ω q K/R , β ∈ Ω q−1 K/R . We compute in Ω q K(x)/R Res t (a dlog(1 − xt r )) = −rxα + βdx.
This shows (6.4.4) and completes the proof.
Lemma 6.5. Let X be a noetherian integral normal scheme, E, F two Cartier divisors on X and assume F is effective. If |E| ⊂ |F |, then there exists N ≥ 1, such that for all maps Spec O → X whose Proof. The question is local on X; hence we can assume E and F are given by functions e, f ∈ k(X) × . Let Div(e), Div(f ) be the associated Weil divisors. Since |E| ⊂ |F | and F is effective we find N ≥ 1, such that Div(e) ≤ N · Div(f ), which by the normality of X implies f N /e ∈ Γ(X, O X ). This yields the statement.
Remark 6.6. The proof of Theorem 6.4 also shows that t n−1 · Ω q O L /R }, else, defines a semi-continuous conductor on Ω q , but it coincides with the motivic one, only for q = 0.
Proof. The formula for c ZΩ q /R follows from Proposition 4.19. It remains to show that it has level q. Let a ∈ ZΩ q /R (A 1 X ) with c dR k(x)(t)∞ (a) ≤ 1, for all points x ∈ X with trdeg(k(x)/k) ≤ q − 1. This implies a ∈ H 0 (X, k[t] ⊗ k Ω q X/R ) ∩ ZΩ q /R (A 1 X ), cf. the proof of (c4) in Theorem 6.4. Hence a ∈ ZΩ q /R (X). This shows that (c ZΩ q /R ) ≤q satisfies (c4).
(1) Let X = (X, D) ∈ MCor be a proper modulus pair. Then Ω q /R (X ) = H 0 (X 1 , Ω q X 1 /R (log D 1 )(D 1 − D 1,red )), where π : X 1 → X is any resolution of singularities which is an isomorphism over X \ D and such that D 1 := π * D is supported on a simple normal crossings divisor. (See 4.17, for the notation Ω q /R .) (2) Let h 0 , where X is any smooth compactification of X with simple normal crossing divisor D at infinity.
Let X = (X, D) be a proper modulus pair with D red a simple normal crossings divisor. Write D = i r i · η i , with η i ∈ X (1) and set L η i := Frac(O h X,η i ). Then it is direct to check that we have c dR L (ρ * a) ≤ v L (D), for all henselian dvf points ρ : Spec L → X if and only if c dR Lη i (a) ≤ r i , for all i. Thus the corollary follows from Theorem 6.4, Theorem 4.15(4), and Corollary 4.32.
Lemma 6.9. The homomorphism dlog : O × X → Ω 1 X/R , X ∈ Sm, induces a morphism dlog : The proof is similar to the one of Lemma 6.2, except that we have to replace the formula f * d = df * by f * dlog = dlog Nm Z/X , where f : Z → X is a finiteétale map between smooth schemes.
6.10. Denote by Conn 1 (X) the group of isomorphism classes of rank 1 connections on X ∈ Sm, and by Conn 1 int (X) the subgroup of integrable connections. We have canonical group isomorphisms . Indeed, the first isomorphism is well-known (use that the first Zariski cohomology can be computed asČech cohomology); we show the second as follows: Letk X be the algebraic closure of k in k(X); we consider it as a constant sheaf on X. We obtain the isomorphism , in the derived category of abelian sheaves on X Zar ; similar with ZΩ 1 /k . Observe that Ω 1 /k and O × are already Nisnevich sheaves, hence /k . This yields the second isomorphisms. By Lemma 6.9 and [Sai20, Thm 0.1] we obtain Conn 1 , Conn 1 int ∈ RSC Nis .

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS 45
For E ∈ Conn 1 (X) we denote by ω E ∈ H 0 (X, (Ω 1 /k / dlog O × ) Nis ), the element corresponding to E under the above isomorphism.
Let L ∈ Φ and let t ∈ O L be a local parameter. Recall (e.g. from [Kat94, Def. 1.12]) that the irregularity of E ∈ Conn 1 (Spec L) ∼ = Ω 1 L/k / dlog L × is defined by irr L (E) = min n ≥ 0 | ω E ∈ Im 1 t n · Ω 1 O L /k (log) → Ω 1 L/k / dlog L × . Theorem 6.11. Notations are as in 6.10. The motivic conductor of E ∈ Conn 1 (L) is given by Moreover, on Conn 1 the motivic conductor restricts to a level 2 conductor and on Conn 1 int it restricts to a level 1 conductor.
L/k lift of a}, see 6.3 for the definition of c dR . It suffices to prove the following identity for the motivic conductor of H 1 (6.11.1) c H 1 = c irr , and that (c irr ) ≤2 and (c irr ) ≤1 |H 1 int satisfy (c4). It follows directly form Theorem 6.4 and Lemma 4.28, that c irr satisfies (c1)-(c6) except maybe (c4) and (c5). For (c5), note that given X ∈ Sm we find by resolution of singularities a compactification X = (X, X ∞ ) with X ∈ Sm. In particular, for all x ∈ X the local ring O h X,x is factorial and hence so is any of its localizations. Therefore, it follows from the exact sequence , for any integral scheme Y over k, that the condition (4.28.1) from Lemma 4.28 is satisfied; hence c irr satisfies (c5). Next (c4). Take a ∈ H 1 (A 1 X ) with (6.11.2) c irr k(x)(t)∞ (a x ) ≤ 1, for all x ∈ X with trdeg(k(x)/k) ≤ 1, where a x is the restriction of a to k(x)(t) ∞ . Consider the exact sequence (using the A 1 -invariance of X → H i (X, O × X )) Let π : A 1 X → X be the projection and i : X ֒→ A 1 X a section. By (6.11.3) there exists anã ∈ H 0 (A 1 X , Ω 1 A 1 X /k ) mapping to a − π * i * a and any such lift satisfies (6.11.2) with c irr replaced by c Ω 1 /k . Thus a ∈ H 0 (X, Ω 1 X/k ), by (c4) for (c Ω 1 /k ) ≤2 ; hence (c irr ) ≤2 satisfies (c4). Similarly, one proves (c4) and (c5) for (c irr ) ≤1 Hence c irr is a semi-continuous conductor and we obtain c H 1 ≤ c irr . We show the other inequality. Let L ∈ Φ and let σ : K ֒→ O L be a coefficient field. Denote by fil n ⊂ H 1 (L) the image of fil n = 1 t n−1 Ω 1 O L /k (log). Take a ∈ fil r+1 . Similar as in the proof of Theorem 6.4 (around (6.4.4), and with the notation from there) it suffices to show the implication Letã ∈ fil r+1 be a lift of a; writẽ a = 1 t r α + β dt t r+1 mod fil r with α ∈ Ω 1 K/k and β ∈ K. Then the left hand side of (6.11.4) is equivalent to Computing the residue symbol yields (6.11.5) − rxα + βdx = dlog f in Ω 1 K(x)/k . We claim this can only happen if α = β = 0. Indeed, first observe that if h ∈ K((x)) × is a formal Laurent series such that there exists a form γ ∈ Ω 1 K/k with dlog(h) = x · γ in Ω 1 K((x))/k , then γ = 0 = dlog(h). Thus (6.11.5) implies that dlog(f · exp(−βx)) = 0 in Ω 1 K((x))/k . Hence there exists an element λ ∈ k 1 the algebraic closure of k in K such that which is only possible if β = 0; it follows α = 0. Thus a ∈ fil r , which proves (6.11.4) and completes the proof.
Corollary 6.12. Let X ∈ Sm. Then h 0 A 1 (Conn 1 int )(X) is the group of isomorphism classes of regular singular rank 1 connections on X (see 4.30 for notation).

Witt vectors of finite length
In this section we assume that k is a perfect field of characteristic p > 0. Denote by W n the ring scheme of p-typical Witt vectors of length n. We will denote by W n O X the (Zariski-, Nisnevich-,étale-) sheaf on X defined by W n . The restriction of W n to k-schemes, which -by abuse of notation -we will again denote by W n , is in particular a smooth commutative group over k. Hence W n ∈ RSC Nis (see 5.1).
7.1. Let A be a ring. Recall, that there is an isomorphism of groups

Assume A is normal and we have an inclusion of rings A ֒→ B making B a finite A-module. Then B[[T ]] is finite over the normal ring A[[T ]] and hence the norm map, Nm : B[[T ]] × → A[[T ]] × induces a trace
Tr : W n (B) → W n (A), see e.g. [Rül07b,Prop A.9]. Now assume f : Y → X is a finite and surjective k-morphism, where X is a normal k-scheme. Then the local traces above glue to give Lemma 7.2. In the situation above, Tr f equals f * : W n (Y ) → W n (X), the map used to define the transfer structure on the group scheme W n .
Proof. Let a ∈ W n (Y ) and d = deg(f ). Recall the element f * (a) is defined by the composition It suffices to check that Tr f (a) and f * (a) coincide on a dense open subset. Thus we can assume that X is affine integral and f : Y → X is finite free. Furthermore W n is a direct factor of the scheme of big Witt vectors W p n and Tr and f * extend to the big Witt vectors. Thus it suffices to show the equality on the big Witt vectors W r , for r ≥ 1. Let S r = Spec k[t]/(t r+1 ) and denote by ε : S = Spec k ֒→ S r the S-section.
We have the following isomorphism of S-group schemes (cf. (7.1.1)) where Res Sr/S (G m ) denotes the Weil restriction. Denote by f r : Y r → X r the base change of f along S r → S. Let b ∈ W r (Y ) which we can view as an element in Res Sr/S (G m )(Y ). Then the image of f * (b) in W r (X) ⊂ Res Sr/S (G m )(X) is equal to the S r -morphism Now the statement follows from the fact that f * = Nm on G m , see [SGA 4 3 , Exp. XVII, Ex 6.3.18 ].
7.3. Let L ∈ Φ. Denote by fil log j W n (L), j ≥ 0, the Brylinski-Kato filtration (see [Bry83], [Kat89]), i.e., where [x] denotes the Teichmüller lift of x and F : W n (L) → W n (L) is the Frobenius, which by contravariant functoriality is induced by the Frobenius of L (or by covariant functoriality by the base change over Spec k of the Frobenius on the Spec(F p )-ring scheme W n ). We observe that for s ≥ 0 we have fil j W n (L) = fil log j−1 W n (L) + V n−r (fil log j W r (L)), j ≥ 1, where r = min{n, ord p (j)}. (This is equal to Matsuda's fil ′ j−1 W n (L).) Assume r = ord p (j) < n, then (a 0 , . . . , a n−1 ) ∈ fil j W n (L) This is the description given in [KR10,4.7]. (They denote by ♭ fil j W n (L) what we call fil j W n (L).) One directly checks that For a ∈ W n (L), we define the Brylinski-Kato-Matsuda-Russell conductor γ n,L (a) (cf. [KR10,Thm 8.7]) by Note that fil where Tr = Tr L ′ /L , see Lemma 7.2. This is immediate if r = 0. Thus we can assume r ≥ 2 and write a = j≥0 F j (a j ), with a j ∈ fil r W n (L ′ ).
We have Tr(a j ) ∈ fil log s W n (L). Indeed, this follows from By the injectivity ofθ s in (7.4.3) it suffices to show (7.5.2) m s L · F n−1 d Tr(a j ) ∈ Ω 1 O L , all j ≥ 0. By [Rül07b, Thm 2.6] the trace Tr extends to a trace between the de Rham-Witt complexes Tr : W n Ω · L ′ → W n Ω · L which is compatible with the differential and Frobenius, is W n Ω · L -linear, and equals the classical trace on Kähler differentials for n = 1. We obtain This completes the proof of (c3).
Next we show that the restriction of γ to Φ ≤1 satisfies (c4). Let X ∈ Sm and a ∈ W n (A 1 X ) with (7.5.3) x ,∞ ). We have to show a ∈ W n (X). We may assume X = Spec A, and thus a ∈ W n (A[t]). If a is not constant, then we find a closed point x ∈ X such that the image of a in W n (k(x)[t]) is not constant. Hence a k(x)(t)∞ ∈ W n (O k(x)(t)∞ ), i.e., γ(a k(x)(t)∞ ) ≥ 2, contradicting our assumption (7.5.3).
(c5). Let X ∈ Sm and a ∈ W n (X) = H 0 (X, W n O X ). Let X = (X, X ∞ ) be a proper modulus pair with X = X \|X ∞ |. For an effective Cartier divisor E on X denote by W n O X (E) the invertible subsheaf of

under the map induced by the Teichmüller lift. If e is an equation for
. There exists an integer N such that a ∈ H 0 (X, W n O X (N · X ∞ )). i.e., γ L (ρ * a) ≤ rm = v L (r · X ∞ ), proving Claim 7.5.1. Finally, (c6). Let X ∈ Sm and Z ⊂ X a smooth prime divisor with generic point z.
Then there exists an affine Nisnevich neighborhood U = Spec A → X of z such that Z U = div(t) on U and a U = s≥0 F s (a s + V n−r (b s )), where r = min{ord p (j), n} and Let (Y , Z + Σ) be a compactification of (U, Z) with Z |U = Z and Y normal. Let Y = ∪V i be an open covering such that Let E i be the Cartier divisor on V i defined by e i . We have |E i | ⊂ |Σ |V i |. By Lemma 6.5, there exists N 1 ≥ 0, such that f N 1 i /e i ∈ B i , for all i. By (7.5.4), there exists an N 2 ≥ 0 such that for all i and all s Choose N ≥ j · N 1 + N 2 , such that p n | N. We obtain for all i Let ρ : Spec L → U, L ∈ Φ. Assume the closed point of Spec O L maps into |Z + Σ|. Then it follows from the above formula that ρ * a s ∈ fil log v L ((j−1)·Z+(N −1)·Σ) W n (K) ⊂ fil v L (j·Z+N ·Σ) W n (K) and ρ * b s ∈ fil log v L (j·Z+N ·Σ) W r (K). By the choice of N we have This proves (c6) and completes the proof of the proposition.
The above proposition gives c Wn ≤ γ n by Corollary 4.24. We show in Theorem 7.20 below, that equality holds using symbol computations. If we restrict to trdeg(L/k) = 1 and k is infinite, this follows, e.g., from [KR10, Prop 6.4, (3)]. To handle the case of higher transcendence degree we need some preparations. We start by identifying the local symbol for W n on regular projective curves over function fields. 7.6. Let X ∈ Sm. We denote by W n Ω • X the de Rham-Witt complex of length n on X (see [Ill79]). By [KSY,Cor 3.2.5] we have W n Ω q ∈ RSC Nis . See also [Gro85] and [CR12] for details on how to define the transfers structure. If f : X → Y is a morphism in Sm, then the morphism Γ * f = f * : W n Ω q (Y ) → W n Ω q (X) induced by its graph Γ f ∈ Cor(X, Y ), is the natural pullback morphism induced by the functoriality of the de Rham-Witt complex. If f is finite and surjective, then the transpose of the graph defines an element Γ t f ∈ Cor(Y, X) and Γ t * f = f * , where f * is the pushforward defined using duality theory.
(1) The restriction, Verschiebung, Frobenius, and the differential (which are part of the structure of the de Rham-Witt complex) define morphisms in RSC Nis (2) Let W n be the algebraic group of Witt vectors of length n considered as a presheaf on Sm. Then there is a unique structure of presheaf with transfers on W n , for all n, which is unique with the following properties (a) the restriction R : W n+1 → W n is compatible with the transfer structure, for all n; is the pullback from the presheaf structure. In particular, the Nisnevich sheaf with transfers W n Ω 0 = W n O from 7.6 coincides with the Nisnevich sheaf with transfers defined by the algebraic group W n (see [KSY, Cor 3.2.5]).

Proof.
(1). We have to show, that if α ∈ Cor(X, Y ) is a finite correspondence, then the following morphisms are equal on This follows from [CR12, Proof of Prop 3.5.4].
(2). The existence of such a transfer structure follows, e.g., from 7.6. The last part of the statement follows since the two transfer structures satisfy (2)a, (2)b.
It remains to prove the uniqueness. Assume we have two transfer actions on W n with (2)a, (2)b. For α ∈ Cor(X, Y ) a finite correspondence denote by α * , α ⋆ : W n (Y ) → W n (X) the two actions. We have to show they are equal. Let f : X → Y be a morphism. By assumption we have Γ * f = Γ ⋆ t =: f * ; if f is finite and and surjective we set f * := (Γ t f ) * and f ⋆ := (Γ t f ) ⋆ . In general for α as above we want to show α * = α ⋆ . It suffices to check this after shrinking X around its generic points. Hence we can assume, that X is connected and α = Z ⊂ X ×Y with Z smooth, integral, and finite free over X. Denote by f : Z → X and g : Z → Y the maps induced by the projections. Then α ⋆ = f ⋆ g * and α * = f * g * . It remains to show f ⋆ = f * . We may shrink X further and hence assume that f : Z = Spec L → X = Spec K is induced by a finite field extension L/K of function fields over k. By transitivity it suffices to consider the two cases where L/K is either separable or purely inseparable of degree p.
1st case: L/K separable. Let K ′ /K be a Galois hull of L/K and set X ′ = Spec K ′ . We obtain the cartesian diagram where the vertical map on the left is induced by the universal property of the coproduct from the identity on X ′ , u is induced by the inclusion K ֒→ K ′ , and the σ i : X ′ → Z, i = 1, . . . , n, are induced by be all the K-embeddings L ֒→ K ′ . For a ∈ W n (L) we obtain and similar with u * f ⋆ . Thus u * f * = u * f ⋆ and since u * : W n (K) ֒→ W n (K ′ ) is injective we have proven the claim in this case. 2nd case: L/K purely inseparable of degree p. In this case we have Let p : W n → W n+1 be the map lift-and-multiply-by-p; thus it sends a Witt vector (a 0 , . . . , a n−1 ) in W n (A), where A is some F p -algebra, to (0, a p 0 , . . . , a p n−1 ). Let b ∈ W n (L). Clearly we find an element a ∈ W n+1 (K) such that f * a = p(b). We obtain The same computation works for f ⋆ b. Thus p(f * b) = p(f ⋆ b), and the claim follows the injectivity of p.
Lemma 7.8. Let f : Y → X be a finite and surjective morphism in Sm. Then for all u ∈ H 0 (Y, O × Y ) and all n ≥ 1 we have Note that f is flat by [Mat89,Thm 23.1], hence also finite locally free, so that Nm Y /X is defined. It suffices to prove the equality after shrinking X around its generic points. Thus we can assume that f corresponds to a finite field extension L/K. By transitivity it suffices to consider the cases where L/K is separable or purely inseparable of degree p.
1st case: L/K finite separable. We have W n Ω q L = W n (L) ⊗ Wn(K) W n Ω q K (see [Ill79, I, Prop 1.14]). By the projection formula and Lemma 7.7(2), we have f * = Tr L/K ⊗id. Let K sep be a separable closure of K. Note that W n (K) → W n (K sep ) is faithfully flat (since it is indetale and Spec W n (K) is one point). Hence byétale base change and fppf descent the natural map W n Ω 1 K → W n Ω 1 K sep is injective. Thus it suffices to check the equality in W n Ω 1 K sep . Let σ 1 , . . . , σ r : L ֒→ K sep be all K-embeddings, then by the above we have in W n Ω 1 2nd case: L/K is purely inseparable of degree p. We have Nm L/K (u) = u p ∈ K. Since the map lift-and-multiply-by-p, p : W n Ω 1 K → W n+1 Ω 1 K is injective by [Ill79, I, Prop 3.4] and commutes with f * the statement follows from the following equality in W n+1 Ω 1 K : This completes the proof of the lemma. Let K be a function field over k and C a regular projective connected curve over K with function field E = K(C). Recall from [Rül07a, Def-Prop 1] that the residue map at a closed point x ∈ C is defined as follows: by a result of Hübel-Kunz we find an integer m 0 ≥ 0 such that for all m ≥ m 0 the curve C m := Spec(O C ∩K(E p m )) is smooth over K and, if x m denotes the image of x under the finite homeomorphism C → C m , then the residue field K m := K(x m ) is separable over K. Hence O h Cm,xm has a unique coefficient field containing K, which we identify with K m . Set E m := K(C m ) = K(E p m ). The choice of a local parameter t ∈ O Cm,xm yields a canonical inclusion E m ֒→ K m ((t)). We define Res C/K,x as the composition Here we should observe that if π : Spec L → Spec K is a finite extension, then the trace Tr L/K : W n Ω q L → W n Ω q K from [Rül07b, Thm 2.6] is equal to the pushforward π * from 7.6. Indeed in the case q = 0 this follows from Lemma 7.7(2) and Lemma 7.2; by transitivity, the general case is reduced to a simple extension L = K[a] in which case it follows from the fact that both maps commute with V , F , d, satisfy a projection formula, and the equality Remark 7.10. In [Rül07b, 2.] and [Rül07a], where the trace and the residue symbol mentioned above are constructed it is always assumed that the characteristic is not 2. The reason for this that the structure theorem by Hesselholt and Madsen which in loc. cit. is cited as Theorem 2.1 was only known for Z (p) -algebras, with p odd at that time. This theorem is used in Proposition 2.4 and Lemma 2.9 of loc. cit. which are needed to define the trace and the formal residue symbol, respectively. However, the Theorem 2.1 of loc. cit. is also available for Z (2) -algebras by [Cos08, 4.2] hence all the results from loc. cit. extend to the case p = 2.
Lemma 7.11. Let C/K and x ∈ C be as in 7.9. Then the corresponding local symbol of W n Ω q (see 4.34) is given by In particular, if L ∈ Φ with coefficient field σ : K ֒→ O L and local parameter t ∈ O L , then the local symbol (−, −) L,σ : W n Ω q L × L × → W n Ω q K (see 4.37) is given by the composition Rest − −→ W n Ω q K , where we denote byσ : L ֒→ K((t)) the canonical inclusion.
To this end choose m as in 7.9 above. Then K(x)/K(x m ) is purely inseparable of degree, say, p s and we can write where p e is the ramification index of x/x m . Denote by p s : W n Ω q → W n+s Ω q the map lifting-and-multiplying by p s ; it is injective, by [Ill79, I, Prop 3.4]. Denote by σ : x the inclusion of the coefficient field. By [Rül07b, Thm 2.6(iii)] there exists a β ∈ W n+s Ω q Km mapping to p s α(x) ∈ W n+s Ω q K(x) and we have (7.11.1) Tr K(x)/Km (α(x)) = R s (β).
By the choice of β, we have Since the kernel is the differential graded ideal generated by W n+s (m x ) we obtain in W n+s Ω q = v x (f ) · p s Tr Km/K (R s (β)) = v x (f ) · p s Tr K(x)/K (α(x)), (7.11.1).
Here the first equality follows from the fact that Res C/K,x commutes with the restriction R. (This follows from the definition and the fact that Res t from (7.9.1) and Tr commute with R, for the latter see, e.g., Lemma 7.7(1).) The statement follows from the injectivity of p s .
7.12. Let A be a Z (p) -algebra. For an A-algebra B we denote by W n Ω • B/A the relative de Rham-Witt complex of Langer-Zink (see [LZ04]). It is equipped with R, F, V, d as usual. If B[x] is the polynomial ring with coefficients in B, we denote by I r ⊂ W n Ω • B[x]/A the differential graded ideal generated by W n (x r B[x]). We define the x-adic comple- Lemma 7.13. The following equalities hold in W n Ω 1 Proof. We prove this by induction over n. The case n = 1 is clear. Assume n ≥ 2. By [LZ04, Cor 2.13] we find unique elements a i ∈ W n (Z (p) ) and b s,j ∈ W n−s (Z (p) ) such that Applying F n−1 we obtain in Ω 1 By induction hypothesis we have for all i, j, and for s = 1, . . . , n − 2 , with e i , f s,j ∈ Z (p) . Comparing coefficients we obtain in Z (p) 1 = F n−1 (a i ) = 1 + p n−1 e i , and for s = 1, . . . , n − 2 1 j = F n−s−1 (b s,j ) = 1 j + p n−s−1 f s,j , hence e i = f s,j = 0; further we find b n−1,j = 1/j ∈ W 1 (Z (p) ).
(1). By the Lemmas 7.11 and 7.13 we have Now the claim follows from 7.14. The proof of (2) is similar.
Lemma 7.16. Let L ∈ Φ and let t ∈ O L be a local parameter. Let K ֒→ O L be a coefficient field. Then, for r ≥ 1, any element a ∈ fil log r W n (L)/W n (O L ) can be written uniquely in the following way where a i ∈ W n (K) and b s,j ∈ W n−s (K).
Proof. We can assume L is complete and hence have L = K((t)). By [HM04, Lem 4.1.1] (see also [Rül07b,Lem 2.9]) we can write any element a in W n (K((t)))/W n (K[[t]]) uniquely in the form with a i ∈ W n (K) and b s,j ∈ W n−s (K). Now, a ∈ fil log r W n (L)/W n (O L ) is equivalent to the following equality in W n (K((t)))/ This yields the statement. (1) Assume e = 0. Write r − 1 = p e 1 r 1 with e 1 ≥ 0 and (r 1 , p) = 1.
Then a ∈ fil F r−1 W n (L). Proof. Since k is perfect, a p-basis over k is the same as a separating transcendence basis over k,  4.40). Thus in the following we may replace a by a + b with b ∈ fil F r−1 W n (L). We will use σ 0 to identifyL = K 0 ((t)). Write r = p e r 0 with e ≥ 0 and (r 0 , p) = 1. We distinguish four cases. 1st case: e = 0. Write r − 1 = p e 1 r 1 with (r 1 , p) = 1 and e 1 ≥ 0. By Corollary 7.18(1) we have gr r W n (L) = 0, if e 1 ≥ n, and there is Hence b h = 0, for all h ≥ 0, which completes the proof of the first case. 2nd case: r = p = 2. By Corollary 7.18(2), (3) we have Hence by (1) We obtain b 0 = 0 and c h = b 2 h+1 , all h ≥ 0. Thus reshuffling the sum defining a we obtain 3rd case: 1 ≤ e ≤ n − 1 and r > 2. By Corollary 7.18(2) we have where b h ∈ K 0 and c h ∈ W e (K 0 ). By a similar computation as in the first case, the vanishing (a, 1 − xt r−1 ) Lx,σ 0 = 0 together with r − 1 > r 0 and Lemma 7.15, (1) and (2), imply b h = 0, for all h ≥ 0. Thus It suffices to show , which lies in F fil log r/p W n (L) ⊂ F fil log r−2 W n (L) (use r ≥ 3 for the last inclusion).
If m = trdeg(κ/k) = 0, then K 0 is perfect and (7.19.1) holds. This completes the proof of the implication: (1) ⇒ a ∈ fil r−1 W n O L . Now assume m ≥ 1. We prove (7.19.1) by contradiction using (a, 1 − xt r−1 ) Lx,σ 1 = 0 with σ 1 : ) and V n−e : W e (K j (x)) → W n (K j (x)), j = 0, 1, are injective, the element a ′ also satisfies (a ′ , 1 − xt r−1 ) Lx,σ j = 0, j = 0, 1. Thus we can assume n = e and h 0 = 0, i.e., c 0 ∈ F W e (K 0 ) and we want to find a contradiction. Since the elements z 1 , . . . , z m ∈ K 0 from the statement form a p-basis we can write c 0 as follows: Since we want to compute the local symbol with respect to the coefficient field σ 1 : K 1 (x) ֒→ O Lx , we have to rewrite c 0 as an element in W n (K 1 [[t]]). Set Then are not constant. The composition F e−1 d : W e (−) → Ω 1 is a morphism of reciprocity sheaves (see Lemma 7.7). Hence F e−1 d commutes with the local symbol, which on Ω 1 is given by (α, f ) Lx,σ 1 = Res K 1 ((t)) (α ∧ dlog f ) (see Lemma 7.11). Using F e−1 dF = 0 on W e , we obtain the following equalities in Ω 1 K 1 (x) : Denote byσ We have . Hence the element in the brackets has to be a p-th power, i.e., by (7.19.6) Since y 1 , . . . , y m ∈ K 1 form a p-basis over k, we obtain, similar as above, a I,e−2 = 0, for all I = 0. We may proceed in this way and obtain a I,j = 0, for all I = 0, j ≥ 0.
where c h ∈ W n (K 0 ) and b h ∈ K 0 . As before it follows from (a, 1 − xt r−1 ) Lx,σ 0 = 0 and Lemma 7.15 that b h = 0, for all h ≥ 0. Thus Applying V e−n+1 we obtain where c ′ h = V e−n (c h ) ∈ W e (K 0 ). Since V e−n+1 (a) ∈ fil r W e+1 (L) we can apply the third case, in particular (7.19.1) to conclude c h ∈ F W n (K 0 ), and then also a ∈ fil F r−1 W n (L). This completes the proof of the proposition.
Theorem 7.20. Let L ∈ Φ and r ≥ 0. Then ) and σ is running through all coefficient fields σ : K ֒→ O L . Furthermore, we know for any a ∈ W n (O L , m −r ) there exists some m ≥ r such that a ∈ fil F m W n (L). Hence the statement follows from Proposition 7.19.

Lisse sheaves of rank 1 and the Artin conductor
In this section k is a perfect field of characteristic p > 0. X → H 1 (X) := H 1 et (X, Q/Z) = Hom cont (π 1 (X) ab , Q/Z) is a presheaf with transfers, which we denote by H 1 in the following.
Note that H 1 ∈ NST as follows from the following Lemma.
is an exact sequence ofétale sheaves with transfers on Sm, where F : W n → W n is the base change over Spec k of the Frobenius on the F pgroup scheme W n .
Proof. The exactness of the sequence (8.3.1) on Xé t is classical. The map F − 1 : W n → W n is a morphism of k-group schemes hence is compatible with transfers; for the inclusion Z/p n Z ֒→ W n this follows directly from Lemma 7.2.
Proposition 8.5. The collection is a semi-continuous conductor on H 1 , as is its restriction Art ≤1 .
Proof. By Proposition 7.5 and Lemma 4.28, Art satisfies (c1)-(c6) except possibly for (c4). (For (c5) note, that W n (Y ) → H 1 p n (Y ) is surjective for any affine scheme over k.) It remains, to show that Art ≤1 satisfies (c4). Let X be a smooth k-scheme and a ∈ H 1 (A 1 X ) with (8.5.1) X is the natural map. We want to show : a ∈ H 1 (X). Since with H 1 {p ′ } the A 1 -invariant subsheaf of prime-to-p-torsion, we can assume a ∈ H 1 p n (A 1 X ). Furthermore, the question is local on X, hence Lemma 8.6. Let K be a field of positive characteristic, x an indeterminate, and g ∈ W n (K(x)). Assume F (g) − g = V n−1 (bx) for some b ∈ K. Then g ∈ Z/p n Z, i.e., F (g) − g = 0.
Proof. If n = 1, then g p − g = bx forces g to be constant and hence g p − g = 0, i.e., g ∈ F p . If n ≥ 2, then F (g) − g is zero when restricted to W n−1 (K(x)). Hence g = m · [1] + V n−1 (f ) with f ∈ K(x), m ∈ Z.
Thus F (f ) − f = bx, and we conclude with the case n = 1.
Proof. The last statement follows from the first and Proposition 8.5. By Corollary 4.29 it suffices to show the corresponding statement on the subsheaf of p n -torsion, for all n ≥ 1. Here the proof is the same as in Theorem 7.20 if we replace everywhere W n by H 1 p n , fil F by fil, the reference to Proposition 7.5 by a reference to Proposition 8.5, and the reference to Proposition 7.19 by a reference to Proposition 8.7. 8.2. Lisse sheaves of rank 1. In this subsection we fix a prime number ℓ = p, an algebraic closure Q ℓ of Q ℓ , and a compatible system of primitive roots of unity {ζ n } ⊂ Q × ℓ . 8.9. We denote by Lisse 1 (X) the group of isomorphism classes of lissē Q ℓ -sheaves on X of rank 1, with group structure given by ⊗. Note that where E runs over sub-extensions of Q ℓ /Q ℓ which are finite over Q ℓ , and O E and m E denote the ring of integers and the maximal ideal, respectively. Indeed, a sheaf M ∈ Lisse 1 (X) corresponds uniquely to a continuous morphism π ab 1 (X) → Q × ℓ , which in particular implies that it factors as a continuous morphism π ab 1 (X) → E × , with some E as above (e.g., [Del80, 1.1]). Since any representation of a profinite group in a finite dimensional E-vector space has an O E -lattice, we see that such a morphism factors via a continuous map The isomorphism classes of such maps correspond uniquely to elements in lim ← −n H 1 et (X, (O E /m n E ) × ). By 8.1 and Lemma 8.2 the isomorphism (8.9.1) induces the structure of a Nisnevich sheaf with transfers on X → Lisse 1 (X), i.e., Lisse 1 ∈ NST . Write Then µ ℓ r E −1 (Q ℓ ) ⊂ O × E and the roots of unity fixed at the beginning of this subsection induce a canonical isomorphism E is a pro-ℓ group this yields the following decomposition where fil j H 1 p ∞ (L) = ∪ n fil j H 1 p n (L) is defined in 8.4. Corollary 8.10. Let the notation be as in 8.9 above. Then (1) Lisse 1 ∈ RSC Nis ; (2) the motivic conductor is given by furthermore it restricts to a level 1 conductor.

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS 71
(3) let X ∈ Sm be proper over k and U ⊂ X dense open, then , see 4.30 for notation.
Proof. Note Lisse 1,p ′ ∈ HI Nis . Hence (1) and (2) follow directly from Theorem 8.8 together with the Corollaries 4.29 and Lemma 4.20. For (3) observe that by Theorem 8.8 and the definition of the Artin conductor, we have H 1 ; hence the statement follows from Corollary 4.33.
Remark 8.11. Let U ∈ Sm and denote by π ab,t 1 (U/k) the abelian tame fundamental group in the sense of [KS10,7]; it is a quotient of π ab 1 (U). Denote by Tame 1 (U) the subgroup of Lisse 1 (U) consisting of those lisse sheaves of rank one whose corresponding representation factors via π ab,t 1 (U/k). Then h 0 A 1 (Lisse 1 )(U) = Tame 1 (U). Indeed, we classically have Tame 1 (C) = Lisse 1,p ′ (C) ⊕ H 1 p ∞ (C), in case C ∈ Sm is a curve over k with smooth compactification C. Hence this ⊂ inclusion follows from Corollary 8.10(3) and the description of π ab,t 1 (U/k) via curve-tameness, see [KS10]. The other inclusion follows from the A 1 -invariance of Tame 1 .

Torsors under finite group schemes over a perfect field
In this section k is a perfect field of positive characteristic p. We fix an algebraic closurek of k. The term k-group is short for commutative group scheme of finite type over k.
Lemma 9.1. Let G be a finite k-group. Then there exists an exact sequence of sheaves on (Sch/k) fppf , the fppf-site on k-schemes, with H i , i = 1, 2, smooth k-groups. Furthermore, if we denote by u : (Sch/k) fppf → (Sch/k)é t the morphism from the fppf-site to thé etale site, then the above sequence induces a canonical isomorphism in the derived category of abelian sheaves on (Sch/k)é t . In particular, for all n ≥ 0 the presheaf on Sm where L⊗ kk = i L i and c H 1 (Gk) is computed in Theorem 8.8 (note that Gk = ⊕ j Z/p n j ). In particular, (c H 1 (G) ) ≤1 is a conductor. Moreover, if X is smooth proper and U ⊂ X is dense open, then h 0 A 1 (H 1 (G))(U) = H 1 (G)(X) (see 4.30 for notation).
Proof. Since G is infinitesimal, we have G(Y ) = 0 for all reduced schemes Y over k. There is also a Hochschild-Serre spectral sequence for the fppf-cohomology (e.g., [Mil80, III, Rem 2.21]); by the above remark the fppf-version of the exact sequence E(X) from Lemma 9.4 yields the first isomorphism. By Lemma 9.1 this isomorphism is compatible with the transfer structure. It remains to show that H 1 (G) is a Nisnevich sheaf. By the remark from the beginning of this proof any sequence (9.1.1) yields an injection H 1 ֒→ H 2 when restricted to Sm. Thus the isomorphism (9.1.2) implies in the derived category ofétale sheaves on Sm, where (H 2 /H 1 )é t denotes theétale sheafification of the presheaf X → H 2 (X)/H 1 (X). Hence H 1 (G)(X) = H 0 (X, (H 2 /H 1 )é t ). It follows that H 1 (G) is even anétale sheaf.
Lemma 9.8. Assume G is an infinitesimal finite k-group of multiplicative type. Then H 1 (G) ∈ HI Nis .
Proof. By Lemma 9.7 we may assume k =k. In this case G is diagonalizable and we find an exact sequence (9.1.1) with H i = G n i m , some n i ≥ 1, see [DG70a, IV, §1, 1.5 Cor]. The statement follows from the A 1 -invariance of X → H i (X Zar , G m ), i = 0, 1, and Hilbert 90.
9.9. We denote α p := Ker(F : G a → G a ), where F is the absolute Frobenius on the additive group. Then α p is a unipotent infinitesimal finite k-group. Let L ∈ Φ and let t ∈ O L be a local parameter. Recall from 7.3 that fil j G a (L) := fil j W 1 (L) is given by (9.9.1) fil j G a (L) = We denote by (9.9.2) fil j H 1 (α p )(L) the image of fil j G a (L) under the connecting homomorphism δ : G a (L) → H 1 (α p )(L) = H 1 (Spec L fppf , α p ).
Note that fil j H 1 (α p )(L) is also equal to the image of the Frobenius saturated filtration fil F j W 1 (L). Proposition 9.10. We have H 1 (α p ) ∈ RSC Nis and the motivic conductor c H 1 (αp) on H 1 (α p ) is given by Proof. Denote the collection of maps H 1 (α p )(L) → N 0 defined by the right hand side of (9.10.1) by c. By Proposition 7.5 and Lemma 4.28, c satisfies (c1)-(c6) except possibly for (c4). (For (c5) note, that G a (Y ) → H 1 (α p )(Y ) is surjective for any affine scheme Y over k.) We claim that c ≤2 satisfies (c4). Let X be a smooth k-scheme and b ∈ H 1 (α p )(A 1 X ) with (9.10.2) c k(x)∞ (ρ * x b) ≤ 1, for all x ∈ X with trdeg(k(x)/k) ≤ 1, where ρ x : Spec k(x)(t) ∞ = Spec Frac(O h P 1 x ,∞ ) → A 1 X is the natural map. We want to show : b ∈ H 1 (α p )(X). This is equivalent to b = π * i * b in H 1 (α p )(A 1 X ); by the definition of c and Lemma 9.7, we can therefore assume k is algebraically closed. Furthermore, the question is local on X, hence we can assume X = Spec A affine. Note, for a general β ∈ H 1 (α p )(L) \ H 1 (α p )(O L ) we have c L (β) ≥ 2, as follows directly from (9.9.1). Hence condition (9.10.2) implies x ,∞ ) → H 1 (α p )(k(x)(t) ∞ )). Denote by b(x) the restriction of b to A 1 x . Since H 1 (α p ) is a Nisnevich sheaf we conclude b(x) ∈ H 1 (α p )(P 1 x ) = H 1 (α p )(x). Thus we find a polynomialb = b 0 + b 1 t + . . . + b n t n ∈ A[t] mapping to b such that for all points x ∈ X with trdeg(k(x)/k) ≤ 1 there exist c x ∈ k(x) and g x ∈ k(x)[t] with (9.10.3)b(x) = c x + g p x , in k(x)[t]. It follows immediately thatb ∈ A[t p ] and it remains to show b i ∈ A p , for all i ≥ 1, since then b = b 0 in H 1 (α p )(A 1 X ). Thus we are reduced to show the following: Let X = Spec A → A d = Spec k[x 1 , . . . , x d ] be anétale map and a ∈ A \ A p . Then there exists a smooth connected curve i : C ֒→ X such that i * a ∈ O(C) \ O(C) p . If a ∈ A p we find a variable -say x 1 -such that a = a 0 + a 1 x 1 + . . . + a n x n 1 , where a i ∈ A p [x 2 , . . . , x d ] := B and a ∈ B[x p 1 ]. A tuple λ = (λ 2 , . . . , λ d ) ∈ k d−1 induces a closed immersion i λ : A 1 → A d given by x 1 → x 1 , x i → λ i , i = 2, . . . , d. Denote by C λ the pullback of X along i λ . Since k is algebraically closed we find a tuple λ such that a |C λ ∈ O(C λ ) p . This proves the above claim; hence c ≤2 satisfies (c4).
Since this hold for all σ, Proposition 7.19 (in the case n = 1) yieldsb ∈ fil F r−1 G a (L), hence b ∈ fil r−1 H 1 (α p )(L). This completes the proof. Proposition 9.11. Let G be a finite unipotent infinitesimal k-group.
Indeed, by Lemma 9.7 this sequence is in NST; hence it suffices to check its exactness on any smooth affine k-scheme X, in which case it follows from H 0 (X fppf , α p ) = 0 = H 2 (X fppf , G r ). By Proposition 9.10 the motivic conductor of H 1 (α p ) restricts to a level 2 conductor and by induction we may assume that so does the motivic conductor of H 1 (G r−1 ). We deduce that the motivic conductor of H 1 (G r ) restricts to a level 2 conductor from (9.11.2) and a similar argument as at the end of the proof of Proposition 8.5.
(3). We claim (9.11.3) H 1 (G)(O L , m −1 L ) = H 1 (G)(O L ). The claim is true for G = α p , by the explicit formula of the motivic conductor in Proposition 9.10. Consider the sequence (9.11.1) and assume the claim is proven for G r . Let b ∈ H 1 (G r−1 )(O L , m −1 ). By the exact sequence (9.11.2) and the claim for α p we find a c ∈ H 1 (G r−1 )(O L ) such that b − c is in the image of H 1 (G r )(L). By Proposition 4.19 we find b − c ∈ H 1 (G r )(O L , m −1 ) = H 1 (G r )(O L ), which proves (9.11.3). Hence (3) follows from Corollary 4.33.
In summary: Theorem 9.12. Let G be a finite k-group. Then: (1) H 1 (G) ∈ RSC Nis ; (2) the motivic conductor of H 1 (G) restricts to conductor of level 2, and if G has no infinitesimal unipotent factor, to a conductor of level 1; (3) write G = G ′ × G unip with G unip unipotent and G ′ without any unipotent subgroup, and let X be smooth proper over k and U ⊂ X dense open. Then h 0 A 1 (H 1 (G))(U) = H 1 (G ′ )(U) ⊕ H 1 (G unip )(X). Proof. By [DG70a, IV, §3, 5.9] we can decompose G uniquely into a product where G em isétale multiplicative, i.e., it is anétale k-group without p-torsion, G eu isétale unipotent, i.e., it is anétale k-group with pprimary torsion, G im is infinitesimal and of multiplicative type, and G iu is an infinitesimal unipotent k-group. Hence the statement follows from Lemma 9.5, Lemma 9.6, Lemma 9.8, and Proposition 9.11.
Remark 9.13. Let G be a finite unipotent k-group. Note that by Theorem 9.12(3) above, the functor X → H 1 (X fppf , G) is a birational invariant for smooth proper k-schemes. This gives a new proof of this (probably) well-known result (it follows, e.g., also from [CR11]).