Ramification theory of reciprocity sheaves, II Higher local symbols

We construct a theory of higher local symbols along Paršin chains for reciprocity sheaves. Applying this formalism to differential forms, gives a new construction of the Paršin–Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato’s higher local class field theory in the geometric case. The higher local symbols satisfy various reciprocity laws. The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.


Introduction
In this note we apply the results from [RS23b] to obtain a theory of higher local symbols for reciprocity sheaves.These symbols are higher dimensional generalizations of the local symbols defined by Rosenlicht-Serre [Ser84] in the 1-dimensional case for commutative algebraic groups.Higher local symbols are defined along Paršin chains and satisfy various reciprocity laws.Applying this formalism to differential forms, gives a new construction of the Paršin-Lomadze residue maps, and applying it to the torsion characters of the fundamental group gives back the reciprocity map from Kato's higher local class field theory in the geometric case.The main result of the paper is a characterization of the modulus attached to a section of a reciprocity sheaf in terms of the higher local symbols.This result will be an essential ingredient in [RS] and [Sai23].
1.1.We fix a perfect field k.Reciprocity sheaves were introduced by Kahn, Saito, and Yamazaki in [KSY22].A reciprocity sheaf F is a Nisnevich sheaf with transfers which has the additional property, that any section a ∈ F (U) over a smooth kscheme U has a modulus, i.e., there is a proper k-scheme X and an effective Cartier K.R. was supported by the DFG Heisenberg Grant RU 1412/2-2.S.S. is supported by the JSPS KAKENHI Grant (20H01791).
divisor D on X, such that U = X \ D and the pair (X, D) measures the defect of a being regular outside of U. Though one should think of the modulus as a measure for the pole order or the depth of ramification of a along D, the interest comes from the fact that it is defined in a motivic way, namely by requiring that the action of certain finite correspondences is zero on a, see 2.1 and the references there for details.The subgroup of sections of F (U) with modulus (X, D) is denoted by F (X, D).If X is projective of dimension d over k, then in [RS23b, Proposition 6.7] we construct a pairing (1.1.1)(−, −) (X,D) K /K : For particular reciprocity sheaves this gives back several pairings which were constructed in the literature by different methods, e.g., if F = Hom cont (π ab (−), Q/Z) and K is a finite field, this pairing (or at least the pro-system over larger and larger D) was constructed in [KS86] to obtain higher dimensional geometric class field theory, or if k has characteristic zero and F (U) denotes the absolute rank one connections on U, this pairing was constructed by Bloch-Esnault in the case U has dimension 1, see [BE01,(4.8)].A disadvantage of the motivic definition of F (X, D) is that it is hard to decide which sections of F (U) have modulus (X, D).To study the pairing in other interesting examples, e.g., F (U) = H 1 (U fppf , G) for G a finite k-group scheme, or F (U) = H 0 (U, R n+1 ε * Q/Z(n)) with Q/Z(n) the étale motivic complex of weight n with Q/Z-coefficient and ε : X ét → X Nis the change of sites map, it is desirable to get a better hold on F (X, D).Easier-to-handle global descriptions of F (X, D) are given in [RS23b] and [RS].In the present article we give a purely local description, at least under certain extra assumptions on (X, D).
1.2.Let K be a function field over k.Recall that a Paršin chain (or maximal chain) on an integral finite-type K-scheme X of dimension d is a sequence x = (x 0 , . . ., x d ) of points of X with x i < x i+1 , i.e., x i is a strict specialization of x i+1 , for all i = 0, . . ., d − 1.Let F be a reciprocity sheaf.For any maximal chain x on X, we define in section 5 the higher local symbol (1.2.1) (−, −) X/K,x : F (K h X,x ) ⊗ Z K M d (K h X,x ) → F (K), where K h X,x is the henselization of O X,x 0 along the chain x, see 3.2 for details.The definition of this pairing relies on the map c x : ) already considered in [KS86] and the pushforward H d (X Nis , j !F d U ) → F (K) constructed in [RS23b] (and relying on the pushforward constructed in [BRS22]).Using the natural map K(X) ֒→ K h X,x (1.2.1) also induces a semi-local pairing (−, −) X/K,x : F (K(X)) ⊗ Z K M d (K(X)) → F (K).The family of these symbols (for all X and all x) is uniquely determined by the properties (HS1)-(HS4) which resemble the properties used by Serre to characterize and construct his local symbols on curves for commutative algebraic groups in [Ser84,III].This uniqueness property can be used to check that the higher local symbols coincide for F = Ω q with those defined by Paršin and Lomadze ([Par76], [Lom81]), for details on this and further examples see 5.6.The property (HS3) roughly says that the symbol (−, −) X/K,x vanishes on ) at the chain (x 0 , . . ., x d−1 ), see (3.2.2).The property (HS4) is the reciprocity theorem , and i ∈ {0, . . ., d}, where in the case i = 0, we have to assume X projective.Interestingly, in [Lom81] (and many similar constructions) property (HS3) follows easily from the definition of the local symbol and the reciprocity law (HS4) is a theorem, whose proof requires a more involved argument, whereas in our case (HS4) is a formal consequence of the construction and (HS3) follows from the pairing (1.1.1),which is one of the main results from [RS23b].
The main result of the present paper is the following theorem (the statement in the body of the text is a bit stronger).
Theorem 1.3 (see Theorem 6.1, Proposition 7.3).Let X be a smooth k-scheme of pure dimension d and D an effective Cartier divisor on X whose support has simple normal crossings.Let U = X \ |D| and a ∈ F (U). Assume that there exists an open dense immersion X ֒→ X into a smooth and projective k-scheme, such that (X \ U) red has simple normal crossings.Let V ⊂ X be an open neighborhood of the generic points of |D|.Then the following conditions are equivalent: If furthermore D is a reduced divisor with simple normal crossings, then the same is true without assuming the existence of the smooth projective compactification X with SNCD boundary.
If F has level ≤ 3 (see 6.5) one can also get around the assumption on the existence of the smooth compactification with SNCD boundary, see Corollary 6.6.
The proof of Theorem 1.3 uses the main results from [RS23b].The stronger statement for D reduced relies on [Sai23, Corollary 2.5], [Sai20], and an additional diagonal argument explained in section 7. Theorem 1.3 and the properties (HS1)-(HS5) of the higher local symbols play an important role in the proofs of the main result of [RS] and in the proof of [Sai23, Theorem 4.2].
Acknowledgement.The authors thank the referee for his comments which helped to clarify the exposition.
(1) In this paper k denotes a field and Sm the category of separated schemes which are smooth and of finite type over k.For k-schemes X and Y we write X × Y := X × k Y .For n ≥ 0 we write P n := P n k , A n := A n k .
(2) Let F be a Nisnevich sheaf on a scheme X and x ∈ X a point.Then we denote by F x its Zariski stalk and by F h x = lim − →x∈U/X F (U) the Nisnevich stalk, where the limit is over all Nisnevich neighborhoods U → X of x.
(3) For a reduced ring R, Frac(R) denotes its total ring of fractions.(4) For a scheme X we denote by X (i) (resp.X (i) ) the set of i-dimensional (resp. i-codimensional) points of X.

Preliminaries on reciprocity sheaves and pairings
This paper builds on [RS23b].In this section we recall some notations and results, see loc.cit.and the references there for more details.
In this section k is a perfect field.

2.1.
A modulus pair (X, D) in the sense of [KMSY21a], [KMSY21b] consists of a separated scheme of finite type over k and an effective (possibly empty) Cartier divisor D on X, such that the complement U = X \ D is smooth over k.The modulus pair (X, D) is called proper, if X is proper over k.Let F be a presheaf with transfers and U ∈ Sm.A modulus for an element a ∈ F (U) is a proper modulus pair (X, D) with U = X \ D, such that for all S ∈ Sm and all integral closed subschemes Z ⊂ A 1 × S × U, which are finite and surjective over a connected component of A 1 S and such that the normalization Z of the closure of A reciprocity sheaf in the sense of [KSY22] is a presheaf with transfers F which is a Nisnevich sheaf on Sm and for which any section a ∈ F (U) has a modulus (X, D).We denote by RSC Nis the category of reciprocity sheaves.For a proper modulus pair (X, D) with X \ D = U we set where the limit is over the cofiltered ordered set of compactifications (Y, E) of (X, D), see [KMSY21a,1.8].We also regularly work with pairs (X, D), which are equal to a projective limit lim ← −i∈I (X i , D i ) with (X i , D i ) modulus pairs and I some filtered set (e.g., X is of finite type over a function field K/k , D is an effective Cartier divisor on X and U = X \ D is regular).In this case we set 2.2.Let F ∈ RSC Nis .Let K be a function field over k and U a regular quasiprojective K-scheme of dimension d.Choose a factorization of the structure map U → Spec K with X reduced, j open dense, and f projective.Building on the results from [BRS22], we define in [RS23b,4.]for such a factorization, a pushforward map where "j !" denotes the extension-by-zero functor.Here F d ∈ RSC Nis is the dth twist of F introduced in [RSY22, 5.5] and F d U denotes its restriction to U Nis .
There is a natural map of Nisnevich sheaves denotes the Nisnevich sheafification of the improved Milnor K-theory from [Ker10], which induces a morphism on X Nis (2.2.1) It is a factorization of the usual pairing induced by finite correspondences from Spec K to U in the following sense: ), induced by the Gersten resolution (see [Ker10, Proposition 10, (8)]).Composing with the natural map ) and taking the sum over all closed points x yields the map (2.2.4) By [RS23b, Lemma 6.6] we have for all a ∈ F (U) and where [ζ] on the left denotes the image of ζ under (2.2.4) and on the right we view ζ as a finite correspondence from Spec K to U.

2.3.
Let X be a reduced noetherian excellent separated scheme of dimension d < ∞ over a field, such that X (d) = X (0) .Let D ⊂ X be a nowhere dense closed subscheme.We define for r ≥ 1 This sheaf is very close to the relative Milnor K-sheaf K M r (O X , I D ) defined in [KS86, (1.3)], where I D denotes the ideal sheaf of D. In fact the two sheaves agree for r = 1 and they have the same stalks at all points with infinite residue field.Since by Grothendieck-Nisnevich vanishing the cohomological dimension of the Nisnevich cohomology on a noetherian scheme is bounded by its dimension, see [KS86, (1.2.5)] or [Nis89, 1.32 Theorem], we find that the natural inclusion ) is therefore surjective by [KS86, Theorem 2.5]. 1 2.4.We recall the main result from [RS23b].Let F ∈ RSC Nis .Let K be a function field over k.Let X be an integral projective K-scheme of dimension d and j : U ֒→ X a regular dense open subscheme.Let D ⊂ X be a closed subscheme (not necessarily a divisor) such that D red = X \ U. Let ν : Y → X be the normalization of X.We define (2.4.1) where Y h y = Spec O h Y,y and D h y = D× X Y h y .We define R(X|D) by the exact sequence 0 ) → 0. Theorem 2.5 ([RS23b, Proposition 6.7, Theorem 6.8]).Assumptions as in 2.4.
(1) The pairing (2.2.2) induces a pairing (2) Assume X ∈ Sm is projective over k and D is an effective Cartier divisor with simple normal crossing support.Then for a ∈ F (U) with U = X \ D we have where in the set on the right, K runs over all function fields over k, X K = X ⊗ k K, and a K is the pullback of a to F (U K ).
In this paper we give a purely local description of the right hand side in (2), using Paršin chains and higher local rings.
A chain on X is a sequence The chain x is a maximal Paršin chain (or maximal chain) if n = d and x i ∈ X (i) .
Note that the assumptions on X imply x i ∈ {x i+1 } (1) .We denote c(X) = {chains on X} and mc(X) = {maximal chains on X}.
A maximal chain with break at r ∈ {0, . . ., d} is a chain (3.1.1)with n = d − 1 and x i ∈ X (i) , for i < r, and x i ∈ X (i+1) , for i ≥ r.We denote mc r (X) = {maximal chain with break at r on X}.

Let S ⊂ X be a finite subset contained in an affine open neighborhood of X.
A strict Nisnevich neighborhood of S is an étale map u : U → X such that U is affine, the base change u −1 (S) → S of u is an isomorphism, and every connected component of U intersects u −1 (S).Let x = (x 0 , . . ., x n ) be a chain on X.A strict Nisnevich neighborhood of x is a sequence of maps for all i = 0, . . ., n.There is an obvious notion of morphism between two strict Nisnevich neighborhoods and picking a representative in each isomorphism class yields a filtered set Assume x ∈ mc r (X) and y ∈ b(x).If U is a strict Nisnevich neighborhood as above, then repeating U r−1 in the rth spot yields a map of filtered sets The Nisnevich stalk of a presheaf F on Note that for x ∈ mc r (X) and y ∈ b(x) the map (3.2.1) induces a natural map where we recursively define: where is the finite set of prime ideals in R i−1 lying over the prime ideal in O X,x 0 corresponding to x i ; this is also the definition used in [KS86].
and it follows from [Gro67, Proposition (18.6.8)], that Nisnevich neighborhoods of the form U × X Y are cofinal in N(y).

3.7.
Let F be a presheaf of abelian groups on X Zar and x = (x 0 , . . ., x n ) ∈ c(X).We can define the Zariski stalk F x of F at x as above, but in fact F x = F xn .If x ∈ mc(X), we can define also the map analogous to (3.4.3): In [KS86] Proposition 3.5 is deduced by induction on the dimension from the coniveau spectral sequence for Nisnevich cohomology and the Grothendieck-Nisnevich vanishing.Since the Zariski analogue of both statements hold, we also have a Zariski analogue of Proposition 3.5.In particular, for w ∈ X (0) , the map is surjective.This follows from the Zariski analogue of Proposition 3.5 applied to the (d − 1)-dimensional scheme Spec(O X,w )\{w}.

Some auxiliary results for relative Milnor K-theory
In this section k denotes any field and X is a reduced noetherian excellent separated k-scheme of dimension d < ∞, such that X (d) = X (0) .

4.1.
Let T be a noetherian reduced purely 1-dimensional and excellent semilocal scheme with total ring of fractions κ(T ) and denote by ν : T → T the normalization.Writing T as a union of irreducible components T = ∪ i T i we obtain κ(T ) = i κ(T i ) and T = i Ti with the obvious notation.Let S be the set of closed points of T and set κ(S) = s∈S κ(s).Then we define where v s ′ denotes the discrete valuation on Frac(O T ,s ′ ) defined by s ′ , ∂ v s ′ is the classical tame symbol, and Nm κ(s ′ )/κ(s) is the norm map.
Here we use the following notation: S d−1 is the set of closed points of the reduced 1-dimensional and excellent semi-local ring O h X,x ′ , where we view x ′ as a chain on X x ′ , by Lemma 3.3.For x ∈ mc(X) as above and i ∈ {0, . . ., d} set x i = (x 0 , . . ., x i ) ∈ mc({x i }).We denote by ∂ X,x the following composition ).For d = 0 this is the identity (by convention).
Lemma 4.2.Assume x 0 is contained in X reg the regular locus of X.Then the following diagram commutes 8 8 q q q q q q q q q q q q where c x,0 is the map (3.4.3), ∂ X,x is the map (4.1.2),and θ x 0 the map (2.2.3).
Proof.We may assume X = X reg .By [Ker10, Proposition 10(8)] the Gersten complex viewed as complex on X Nis is a resolution of K M d,X ; since its terms are furthermore acyclic for the global section functor, we may use it to compute the local cohomology.This in particular yields the identifications in the diagram below for 0 where Y := {x d−j }, Z := {x d−j−1 }, and S d−j−1 denotes the set of closed points in O h Y,(x 0 ,...,x d−j−1 ) and the other notation is taken from 3.4 and 4.1.Composing the diagrams for j = 0, . . ., d − 1 yields the statement.Lemma 4.3 ([KS86, Proposition 2.9]).Let i : Y → X be a closed immersion with Y integral of dimension e and assume Y ∩ X reg = ∅.Let D ⊂ X be a closed subscheme which does not contain Y .Then there exists a proper closed subscheme E ⊂ Y and a map (see 2.4 for notation) which is uniquely determined by the requirement that for any regular open U ⊂ X \D and regular open where the horizontal maps are the maps (2.3.2).Moreover, for any y ∈ mc(Y ) and any x = (y, x e+1 , . . ., x d ) ∈ mc(X), the following diagram is commutative: where we set (using the notation from 4.1) with x j = (y, x e+1 , . . ., x j ), for j ≥ e + 1.
Proof.This is essentially [KS86, Proposition 2.9] and the same proof works.Since in loc.cit. the assumptions and the formulation are slightly different, we repeat the argument for the convenience of the reader.
is zero.We fix this E in the following.For y = (y 0 , . . ., y e ) ∈ mc(Y ) we consider the composition ), where the first isomorphism is induced by the Gersten resolution together with the definition of the local cohomology group in 3.4 and we set δ i = δ (y 0 ,...,y i ) with the notation from (3.4.1).The family {χ y } y∈mc(Y ) satisfies the condition (3.5.1): for r ∈ {0, . . ., e − 1} this follows from the definition of the δ i , for r = e it follows from our choice of E above and (2.3.1).Thus by Proposition 3.5 there is a map i * as in the statement, such that ), all y ∈ mc(Y ).The commutativity of (4.3.3)follows immediately from this equality together with the commutativity of (4.2.1).

4.4.
We recall some constructions and results from [KS86, §4].Let D be a closed subscheme of X which is nowhere dense and is defined by the ideal sheaf I ⊂ O X .We define the Nisnevich sheaf V r,X|D on X by where U runs over the étale X-schemes and D U = D × X U. Note that this sheaf agrees with the sheaf K M r (O X , I) defined in loc.cit.for r = 1 and, if d ≥ 2, for all r ≥ 1: For r = 1, this is immediate.
x since the residue fields k(x), for x ∈ U (1) , have infinitely many elements (see 2.4).Note that we have a natural map V r,X|D → V r,X|D .The cokernel of this map is supported in codimension 2 and the kernel in codimension 1. Hence Grothendieck-Nisnevich vanishing yields an isomorphism (4.4.1) We will need the following statement from loc. cit.: Proposition 4.5 ([KS86, Proposition 4.2]).Let f : Y → X be a finite morphism and assume Y is reduced and f (Y (0) ) ⊂ X (0) .Assume r = 1 or d ≥ 2. Then the norm map on Milnor K-theory where i η : η ֒→ Y and i ξ : ξ ֒→ X are the natural inclusions, induces a morphism for some large enough nowhere dense closed subscheme E ⊂ Y containing D × X Y .
Corollary 4.6.Let f : Y → X be a separated morphism of finite-type with f (Y (0) ) ⊂ X (0) and dim Y = dim X = d.Assume r = 1 or d ≥ 2 and r ≥ 1.Let D ⊂ X be a closed subscheme and set and β 1 , . . ., β n ∈ η∈Y (0) K M r (k(η)).Assume that there exists an open affine subscheme in X containing all the points x 1 , . . ., x n .Then there exits an element γ ∈ η∈Y (0) K M r (k(η)) such that for all i = 1, . . ., n γ − where Nm : is the norm map.Proof.We may replace f by a compactification and hence assume that f is proper.Furthermore, we may replace X by its semi-localization at the points x 1 , . . .
By the Approximation Lemma, we find an element γ ∈ η∈Y (0 The statement follows from this and (4.6.1).

Higher local symbols
We introduce higher local symbols along maximal chains for reciprocity sheaves.These generalize local symbols for curves, see [KSY16, Proposition 5.2.1] and [Ser84] for the classical case of commutative k-groups.Furthermore, we obtain a unified construction for several higher local symbols defined in the literature, e.g., by Paršin, Lomadze, Kato and many more.The results will be used in section 6 to give a characterization of the modulus in terms of local symbols.The content of this section will also play a crucial role in [RS].
In this section k is a perfect field, K is function field over k, and X is an integral scheme of finite type over K and dimension d.We fix F ∈ RSC Nis .
Definition 5.1.Let x = (x 0 , . . ., x d ) ∈ mc(X).We define the pairing Choose an open subscheme V ⊂ X which is quasi-projective and contains x 0 .Choose a dense open regular subscheme U ⊂ V .Choose a dense open immersions j V : V ֒→ Y into an integral projective K-scheme (a projective compactification) with structure map f : Y → Spec K and denote by j : U ֒→ Y the induced immersion.Note that x ∈ mc(Y ).We define (5.1.1)as the composition where the first map is the stalk at x of (2.2.1), c x is the map (3.4.2), and (f, j) * is the pushforward recalled in 2.2.It follows from Lemma 5.2(1), (2) below that this definition is independent of the choice of V , U, and Y .Take r, s ∈ {0, . . ., d}.Precomposing (5.1.1)with the natural map 3), we obtain pairings (denoted by the same symbol) (5.1.2) (−, −) X/K,x : F (K h X,(xr,...,x d ) ) ⊗ Z K M d (K h X,(xs,...,x d ) ) → F (K), in particular, for r = d and s = d − 1, we get the pairing )) → F (K).We call (5.1.1),(5.1.2),and (5.1.3)the higher local symbol of F at x. Lemma 5.2.Let the situation be as above.
(1) The definition of the higher local symbol (5.1.1)is independent of the choice of the quasi-projective open V ⊂ X containing x 0 , the regular dense open subset U ⊂ V , and the choice of the projective compactification V ֒→ Y .(2) Let X be quasi-projective, U ⊂ X a regular dense open, and j : U ֒→ X ֒→ X a projective compactification.Let a ∈ F (U) and (1).We start by showing the independence of the choice of the projective compactification of V .Thus assume we have two projective compactifications j : U ֒→ V ֒→ Y and j ′ : U ֒→ V ֒→ Y ′ and denote by f : Y → Spec K and f ′ : Y ′ → Spec K the projective structure maps.It suffices to consider the situation, where we have a projective morphism g : Y ′ → Y such that g • j ′ = j and f • g = f ′ .In this case the independence follows from the following commutative diagram where the vertical map in the middle is induced by the natural map j !→ Rg * j ′ !, see [RS23b,(4. Here the commutativity of the right triangle holds by [RS23b, Lemma 4.7(2)] and the one of the left triangle is obvious.It remains to check the independence of the choice of V .To this end, let V, V ′ ⊂ X be two open quasi-projective subschemes containing x 0 , let V ֒→ Y → Spec K and V ′ ֒→ Y ′ → Spec K be projective compactifications, and let U ⊂ V and U ′ ⊂ V ′ be two open regular subschemes.We obtain the open immersions U ֒→ Y and U ′ ֒→ Y ′ .Let V ′′ ⊂ V ∩ V ′ be an affine open neighborhood of x 0 and let U ′′ ⊂ U ∩ U ′ ∩ V ′′ be a dense open regular subscheme.We have two induced open immersions U ′′ ֒→ Y and U ′ ֒→ Y ′ .Denote by (5.1.1)(U,V,Y ) the pairing (5.1.1)constructed using U ֒→ V ֒→ Y → Spec K.
where the first equality holds by the independence of the choice of U proven above, the second equality holds by definition of the pairing (it only depends on the maps U ֒→ Y → Spec K), and the third equality holds by the independence of the choice of the compactification of V ′′ proven above.This together with the analog reasoning for (2).This follows form the compatibility of the boundary maps from the localization sequence with cup-products, see [RS23b, (6.7.5)]Proposition 5.3.The pairings (5.1.1)satisfies the following properties for all a ∈ F (K(X)): (HS1) Let X ֒→ X ′ be an open immersion where X ′ is an integral K-scheme of dimension d.Then for all where a(x d−1 ) ∈ F (K(X d−1 )) is the restriction of a and ∂ x is defined in (4.1.1),and Tr K(X)/K : F (K(X)) → F (K) is the trace for the finite map Spec K(X) → Spec K. (HS3) Let D ⊂ X be a closed subscheme such that X \ D is nonempty and regular.
Assume a ∈ F gen (X, D).Then, for x = (x 0 , . . ., x d ) ∈ mc(X) and x ′ = (x 0 , . . ., x d−1 ) ∈ mc d (X), we have (See 4.4 for the definition of If either r ≥ 1 or r = 0, X is quasi-projective, and the closure of x 1 in X is projective over K, where x ′ = (x 1 , . . ., x d ), then , where X is running through all integral schemes of finite type of dimension ≤ d, is uniquely determined by (HS1)-(HS4).
Proof.(HS1) follows from Lemma 5.2(1).(HS2).For d = 0 this follows from the fact that in this case the pushforward (f, j) * appearing in Definition 5.1 is the pushforward along the finite map f : Spec K(X) → Spec K constructed in [BRS22], which is equal to the trace by [BRS22, Proposition 8.10(3)].Now assume d ≥ 1 and take a . By (HS1) we may assume that X → Spec K is projective.We find a closed subscheme D ⊂ X with x d−1 ∈ D, such that U = X \ D is regular and a ∈ F gen (X, D).Denote by i : ; we can view ζ as a finite correspondence from Spec K to U ′ .We denote by This yields (HS2).Property (HS3) follows from Lemma 5.2(2), Theorem 2.5(1), (4.4.1), and the vanishing of (3.6.2).For (HS4) in the case r ≥ 1 choose a quasiprojective open V ⊂ X which contains the closed point x 0 of x ′ , and take a projective compactification V ֒→ Y .Note that x ′ also defines a chain on Y and the set b(x ′ ) does not change when we consider x ′ as a chain on X or Y .Hence in this case the statement follows directly from Definition 5.1 and (3.6.1)applied to F = j !K M d,U , where j : U ֒→ Y is the inclusion of a dense open regular subscheme.In the case r = 0, we can take a projective compactification X ֒→ X and view x ′ as a chain on X.By the assumption that the closure of x 1 in X is projective, the set b(x ′ ) does not change when we consider x ′ as a chain on X or on X. Hence the statement follows from (3.6.1) also in this case.
It remains to prove the uniqueness part.Let { −, − X/K,x | x ∈ mc(X)} dim X≤d be another family of symbols satisfying (HS1)-(HS4).By (HS1) it suffices to show −, − X/K,x = (−, −) X/K,x for all affine K-schemes X; applying (HS1) again we may assume X is projective.We proceed by induction.If dim X = 0 the symbol is uniquely determined by (HS2).Now we assume dim X = d ≥ 1 and the symbols coincide on all closed subschemes of X of dimension strictly smaller than d.Set L := K(X).Let a ∈ F (L), β ∈ K M d (L), and x = (x 0 , . . ., x d ) ∈ mc(X).Let D ⊂ X be a strict closed subscheme such that X \ D is regular, x d−1 ∈ D, and a ∈ F gen (X, D).By Corollary 4.6 (with f = id) we find an element The same computation with −, − X/K,x and induction yields the desired equality.
The formulation of the above proposition was inspired by the treatment of local symbols in [Ser84, III, §1].But note that the construction is completely different.The next proposition is however a formal consequence of (HS1)-(HS4) (and properties of Milnor K-theory) in the same way [Ser84, III, Proposition 4] is a consequence of the properties written in Definition 2 of loc.cit.
Proposition 5.4.Let f : Y → X be a projective and surjective K-morphism between two integral K-schemes of the same dimension d.Then we have for all a ∈ F (K(X)) and where Proof.First note that the sum in (HS5) is finite.Indeed given points x i ∈ X (i) and y i ∈ Y (i) with f (y i ) = x i , then f induces a projective and surjective K-morphism f i : {y i } → {x i }.As source and target of f i have the same dimension it follows that for any x i−1 ∈ {x i } (1) the preimage f −1 i (x i−1 ) consists of 1-codimensional points in {y i }, in particular it is a discrete noetherian topological space and hence is finite.Thus there are only finitely many maximal chains in Y lying over x.
To prove the equality in (HS5) we proceed by induction on the dimension d.Set E := K(X) and L := K(Y ).For d = 0, (HS5) translates by (HS2) into . Now assume d ≥ 1 and the formula holds in dimension ≤ d − 1. Write x = (x 0 , . . ., x d ).We consider two cases.
1st case: a ∈ F (O X,x d−1 ).Set u := x d−1 ∈ X (1) and denote by X ′ = {u} ⊂ X the closure.Set x ′ = (x 0 , . . ., x d−1 ) ∈ mc(X ′ ).We first collect some standard commutative diagrams for Milnor K-theory, also to clarify the notation used later; (5.4.1) where ι's are the natural maps and ∂'s are induced by the tame symbol, see 4.1; (5.4.2) (5.4.3) where z ∈ Y d−1 with f (z) = u.The commutativity of the above diagrams follows from standard relations in Milnor K-theory, e.g., in [Ros96, 1.] see R3a for (5.4.1),R1c for (5.4.2), and R3b for (5.4.3).With this notation we want to show assuming a ∈ F (O X,u ) and that the equality holds in smaller dimensions.We compute where ξ is the generic point of Y .This completes the proof of the first case.2nd case: a ∈ F (E).By (HS1) we may assume X to be projective.Let D ⊂ X be a closed subscheme such that X \ D is regular and a ∈ F gen (X, D).Enlarging D we may assume by (iii), (iv), (HS3) Note that we can apply (HS4) also in the case d = 1, since X and Y are projective.This completes the proof of the proposition.
The following corollary will be used in the proof of Proposition 7.3 and in [RS].
Proof.Set E := K(X) and L := K(Y ).Note that y is an isolated point in ) can be represented by a zero-cycle in Y K(X) \ E K(X) , which spreads out to a finite correspondence α from U to Y \ E for some smooth open U ⊂ X containing the closed point x 0 of x, then it follows from (HS2), that we have

Characterization of the modulus via higher local symbols
In this section k is a perfect field and F ∈ RSC Nis .The main result of this section is the following.Theorem 6.1.Let (X, D) be a modulus pair (see 2.1) with X of pure dimension d.Let U = X \ |D| and a ∈ F (U).For a function field K/k denote by X K = X ⊗ k K the base change and by a K ∈ F (U K ) the pullback of a.Let W ⊂ |D| be a set of closed points which contains at least one point of every irreducible component of |D|.Consider the following conditions (see Definition 5.1): (i) a ∈ F (X, D). (ii) For any function field K and x = (x 0 , . . ., x d ) ∈ mc(X K ) we have (a K , β) X K /K,x = 0, for all β ∈ (V d,X K |D K ) h (x 0 ,...,x d−1 ) .(iii) For any function field K and x = (x 0 , x 1 , . . ., x d ) ∈ mc(X K ) with x 0 ∈ W K and x d−1 ∈ D K , we have Then, we have the implication (i) =⇒ (ii) =⇒ (iii).Assume furthermore that there exists an open dense immersion X ֒→ X into a smooth and projective k-scheme, such that X \ U is an SNCD, then all the above statements are equivalent.
We stress the fact that (V d,X|D ) h (x 0 ,...,x d−1 ) in (ii) is the limit over all Nisnevich neighborhoods of (x 0 , . . ., x d−1 ) (see 3.2), whereas (V d,X|D ) x d−1 in (iii) is the Zariski stalk at the one codimensional point x d−1 ∈ X (1) .In section 7 we will see that in case D is reduced, the assumption on the existence of a smooth compactification is superfluous (but still X has to be smooth and D is a SNCD).
6.2.Before we prove Theorem 6.1 we recall the following result: Let X be a separated scheme of finite type over a field K of dimension d.Assume no irreducible component of dimension d of X is proper.Then for any coherent sheaf F on X, we have This theorem was conjectured by Lichtenbaum and proven by Grothendieck, see [Har67, Theorem 6.9] for Grothendieck's proof in the quasi-projective case relying on duality theory, see [Kle67] for a more elementary proof in the stated generality.We will use the following consequence (cf. the proof of [Har67, Theorem 6.9]): Let Y be a proper K-scheme of dimension . Assume X has a smooth compactification X such that X \ U is an SNCD.It remains to show that in this case also (iii) ⇒ (i) holds.Let D ⊂ X K be the closure of D K .By Theorem 2.5(2) and Lemma 5.2(2) it suffices to prove the following: Claim 6.2.1.Let B ⊂ X K be an effective Cartier divisor supported on X K \ X K and n ≥ 1. Set Y = |D + B| which we will view as a reduced effective Cartier divisor.Then the sequence Here, the second sum is over all x w = (w, x 1 , . . ., x d−1 , x d ) ∈ mc(X K ) such that x d−1 ∈ D (0) and the first map is the sum of the maps x w → H d (X K,Nis , V d,X K |D+nY +B ) where x ′ w = (w, x 1 , . . ., x d−1 ) for x w = (w, x 1 , . . ., x d ) and the last map is induced by c Considering the same claim with D + B on the right replaced by D + mY + B and D + nY + B in the middle replaced by D + (m + 1)Y + B, for m = 0, . . ., n − 1, we are reduced to the case n It is supported on Y , and by [RS23a, Corollary 2.12] we have an exact sequence (6.2.2) Then, we obtain surjections where the last map is surjective due to (6.2.1) and c Zar x ′ w ,0 is the map (3.7.1) and it is surjective by Remark 3.7.This proves Claim 6.2.1 and hence the theorem.Remark 6.3.If F = W n is the sheaf of p-typical Witt vectors of length n, where p = char(k), the equivalence (i) ⇔ (iii) in Theorem 6.1 is reminiscent of [KR10, Proposition 7.5] (the case ♭ fil F m ).Though in loc.cit.k is assumed to be algebraically closed and it is not necessary to consider all function fields K/k.Definition 6.4.For (X, D) ∈ MCor with U = X − |D|, We define By Theorem 6.1 we always have an inclusion F (X, D) ⊂ F LS (X, D) and this is an equlaity if X has a smooth projective compactification X such that X \ U is SNCD.
6.5.We say that a reciprocity sheaf F has level n ≥ 0, if for any smooth k-scheme X and any a ∈ F (A 1 × X) the following implication holds: denotes the set of points in X whose closure has dimension ≤ n − 1, and for a smooth scheme S we identify F (S) with its image in F (A 1 × S) via pullback along the projection map.This is equivalent to the motivic conductor of F having level n in the language of [RS23c].The A 1 -invariant sheaves with transfers are precisely the reciprocity sheaves of level 0. By [RS23c, Part 2], the presheaves X → G(X), with G a commutative algebraic group, X → Hom(π ab 1 (X), Q/Z), X → Lisse 1 (X), the lisse Q ℓ -sheaves of rank 1, and X → Conn 1 int (X), the integrable rank 1 connections on X (char(k) = 0), are reciprocity sheaves of level 1; and the presheaves X → Ω 1 (X), X → ZΩ 2 (X) (both in char(k) = 0), and X → H 1 (X fppf , G), with G a finite flat k-group scheme, are reciprocity sheaves of level 2.
We say that resolutions of singularities hold over k in dimension ≤ n, if for any integral projective k-scheme Z of dimension ≤ n and any effective Cartier divisor E on Z, there exists a proper birational morphism h : Z ′ → Z such that Z ′ is regular and |h −1 (E)| has simple normal crossings.This is known to hold if char(k) = 0 by Hironaka or if n ≤ 3 by [CP09].Corollary 6.6.Assume F has level n ≥ 0 and resolutions of singularities hold over k in dimension ≤ n.Let (X, D) ba a modulus pair.Assume X is quasi-projective and set U = X \ |D|.Let a ∈ F (U).The following statements are equivalent Proof.By (HS3) we only have to show the implication (ii) ⇒ (i).Let h : Z → X be as in (ii).By resolution of singularity in dimension ≤ n, Theorem 6.1 together with Theorem 2.5(2) imply where z 1 ∈ Z 1 is the generic point of Spec O X L ,x 0 /(θ 1 , . . ., θ d−1 ) and Z 1 is its closure in X L .By the choice of the θ i , we have Z 1 = Spec L[t] ⊂ P 1 L and hence we may identify x 0 with 0 L ∈ P 1 L .Applying (HS1) and (HS4) one more time we find (a L , β 0 ) X L /L,x = ± y∈P L \0 L a L (z 1 ), t − s t − 1 P 1 L /L,(y,z 1 ) .
Shrinking around the generic point of D, we may assume further D = Div(t) for some t ∈ O(X) defining an étale morphism v : X → A 1 D , which satisfies v −1 (0 D ) = D.By [Sai20, Lemmas 4.2, 4.4] the morphism To this end we first observe that by a similar argument as was used around (7.3.3), the condition in (2) also holds with (V d,X K |D K ) x d−1 replaced by its Nisnevich stalk (V d,X K |D K ) h x d−1 and we may as well consider the Nisnevich stalk of V d,A 1 . Now let γ and x ∈ mc(A 1 D K ) be as in (7.3.4).By the construction we can lift x uniquely to y ∈ mc(X K ).Take β ∈ (V d,X K |D K ) h x d−1 with Nm(β) = γ.Then 0 = (a, β) X K /K,y = (v * b, β) X K /K,y = (b, Nm(β)) A 1 D K /K,x = (b, γ) A 1 where the first equality holds by the condition in (2) and the third equality holds by (HS5 ′ ) in Corollary 5.5.This yields (7.3.4) and completes the proof.
x n and f by the base change.The semi-localization exists since x 1 , . . ., x n are contained in an affine open in X.Thus X is affine, integral, excellent, and 1-dimensional and f : Y → X is a proper, dominant, and quasi-finite morphism, whence it is finite and surjective.Let ν : Y → Y be the normalization.Thus Y is a finite disjoint union of Dedekind schemes.By Proposition 4.5 we find a closed subscheme E ⊂ Y containing D Y such that (4.6.1) 3.3)].The right triangle commutes by [RS23b, Lemma 4.7(3)].The left triangle commutes since both maps labeled c x factor over H d x 0 (V, F d V ).Next we show the independence of the choice of U. By the above, it suffices to consider a dense open immersion ν : U ′ ֒→ U with projective compactifications j : U ֒→ V ֒→ Y and j ′ = j • ν : U ′ ֒→ U ֒→ V ֒→ Y .In this case the independence follows from the commutative diagram by defn of corr.action, = (a(x d−1 ), [ζ]) U ′ ⊂Y /K , by (2.2.5), = (a(x d−1 ), ∂ x (β)) Y /K,x ′ , by Lem.5.2(2).

Corollary 5. 5 .
Let f : Y → X be a dominant and quasi-projective K-morphism between integral K-schemes of the same dimension d.Let x = (x 0 , . . ., x d ) ∈ mc(X) and u := x d−1 .Let y ∈ Y (1) with f (y) = u.We assume that f induces a projective morphism between the closures of the points y and u.Then K h Y,y is finite over K h X,u (see 3.2 for notation) and for all a ∈ F (K(X)) and β ∈ K M d (K h Y,y ), we have(HS5 ′ ) z∈mc d−1 (Y ), z<y f (z(y))=x (f * a, β) Y /K,z(y) = a, Nm y/u (β) X/K,x ,where z < y means z d−2 < y with z = (z 0 , . . ., z d−2 , z d ) and z(y) = (z 0 , . . ., z d−2 , y, z d ), and Nm y/u : K y is finite and injective.Let Y ֒→ Y f − → X be a projective compactification of f .Take a closed subscheme D ⊂ X such that X \ D and Y \ D Y are regular, where D Y = Y × X D, and a ∈ F gen (X, D), f * a ∈ F gen (Y , D Y ).Set X ′ = Spec O h X,u and denote by f ′ : Y ′ → X ′ the base change of f .Note that the total fraction ring of Y ′ is equal to z∈ f −1 (u) K h Y ,z .By Corollary 4.6 applied to f ′ and (β z ) ∈ z∈ f −1 (u) K M d (K h Y ,z) with β y = β and β z = 0 for z = y, we find an element γ
1 S = S[z].Let L = k(X) be the function field and set X L = X⊗ k L and S L = S⊗ k L. Let ι : S L = V (t) ֒→ X L be the closed immersion defined by t = 0, where t = z ⊗ 1. Denote by s 0 ∈ S L the image of the generic point of the diagonal in S × S under the map S × k S → S × k X → S L where the first map is the base change of the closed immersion S ֒→ X defined by z = 0. Denote by η ∈ X L the generic point of the irreducible component containing ι(s 0 ).Let a ∈ F (P 1 S \ 0 S , ∞ S ) ⊂ F (X[z −1 ]).(1) Assume for all y = (y 0 , . . ., y d−1 ) ∈ mc(S L ) with y 0 = s 0 we have (a L , β) X L /L,(ι(y),η) = 0, for all β ∈ K M d (O X L ,ι(y d−1 ) ).Then a ∈ Im(F (S) → F (P 1 S \ 0 S , ∞ S )).

F
(X, D) becomes an isomorphism when we shrink D to its generic point.Thus we may assume a = v * b for some b ∈ F (A 1 D \ 0 D ).By [Sai20, Lemma 5.9], we haveF (A 1 D \ 0 D ) F (A 1 D , 0 D ) ∼ = F (P 1 D \ 0 D , ∞ D ) F (P 1 D , 0 D + ∞ D ) , so we may assume b ∈ F (P 1 D \ 0 D , ∞ D ).It remains to show b ∈ F (P 1 D , 0 D + ∞ D ).By Corollary 7.2(2) it therefore suffices to show, that for all K/k and all x = (x 0 , . . .,x d ) ∈ mc(A 1 D K ) with x d−1 ∈ D K (0) (where D K is embedded in A 1 D K along the zero-section) we have (7.3.4) (b, γ) A 1 D K /K,x = 0, for all γ ∈ (V d,A 1 D K |0 D K ) x d−1 .
First note that if a map i * as in the statement exists such that (4.3.3)commutes, then (4.3.2) commutes as well, by Lemma 4.2.This later commutativity uniquely characterizes the map i * , as the horizontal maps in (4.3.2) are surjective, see (2.3.2).Thus it remains to construct a map i * (for an appropriate E) which makes (4.3.3)commutative.By [KS86, Proposition 2.7] there exists a proper closed subscheme E ⊂ Y with ideal sheaf I E , such that for any étale map X ′ → X and any y ′ ∈ X ′ over the generic point of Y with closure Y ′ = {y ′ } and any w ∈ Y ′ (1) the composition d. Let W ⊂ Y be a set of closed points which contains at least one closed point of each irreducible component of dimension d of Y .Then the natural map Zar , F ) → H d (Y Zar , F ) is surjective for all coherent O Y -modules F .Indeed, this holds as H d ((Y \ W ) Zar , F ) vanishes by the above result.