Ramified Class Field Theory and Duality Over Finite Fields 3

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The corresponding duality for the ℓ-adic étale cohomology is the classical Poincaréduality for étale cohomology for a prime ℓ ≠ p. If U is a smooth quasi-projective variety over k which is not complete, then there is a perfect duality (see [43]) for the ℓ-adic étale cohomology in the form i m 2d+1−i m m (1.2) Het´ (U, Z~ℓ (r)) × Hc (U, Z~ℓ (d − r)) → Z~ℓ , i m where Hc(U, Z~ℓ (j)) is the ℓ-adic étale cohomology of U with compact support. However, there is no known p-adic analogue of the étale cohomology of U with compact support such that a p-adic analogue of (1.2) could hold. Construction of this duality is yet an open problem. Recall that one of the applications (which is the interest of this paper) of such a duality theorem is to the study of the mod-p étale fundamental group of U, which is in general a very complicated object. A duality theorem such as above would allow one to study this group in terms of a more tractable étale cohomology of U with compact support. In [22], Jannsen-Saito-Zhao proposed an approach in a special case when U is the complement of a divisor D on a smooth projective variety X over k such that Dred is a simple normal crossing divisor. They constructed a relative version of the logarithmic r Hodge-Witt sheaves on X, denoted by WmΩXSD,log. Using these sheaves, they showed that there is a semi-perfect pairing (see Definition 8.1) i r d+1−i d−r m (1.3) H (U, WmΩ )× lim H (X,WmΩ ) → Z~p et´ U,log ←Ðn et´ XSnD,log of topological abelian groups, where the first group has discrete topology and the second has profinite topology. This pairing is perfect when i = 1 and r = 0. In this paper, we propose a different approach to the p-adic duality for U. This new approach has the advantage that it allows D to be an arbitrary divisor on X. This is possible by virtue of the choice of the relative version of the logarithmic Hodge-Witt sheaves. Instead of using the Hyodo-Kato de Rham-Witt sheaves with respect to a suitable log structure considered in [22], we use a more ingenuous version of the relative r logarithmic Hodge-Witt sheaf, which we denote by WmΩ(X,D),log. The latter is defined r r to be the kernel of the canonical surjection WmΩX,log ↠ WmΩD,log. Our main result on the p-adic duality for U is roughly the following. We refer to Theorem 8.6 for the precise statement. Theorem 1.1. Let X be a smooth projective variety of dimension d over a finite field k of characteristic p. Let D ⊂ X be an effective Cartier divisor with complement U. Let r, i ≥ 0 and m ≥ 1 be integers. Then there is a semi-perfect pairing of topological abelian groups i r d+1−i d−r m H (U, WmΩ )× lim H (X,WmΩ ) → Z~p . et´ U,log ←Ðn et´ (X,nD),log This pairing is perfect if Dred is a simple normal crossing divisor, i = 1, r = 0 and one of the conditions {d ≠ 2, k ≠ F2} holds.

We show that Theorem 1.1 recovers the duality theorem of Jannsen-Saito-Zhao if Dred is a simple normal crossing divisor.

et´ 1 As a consequence of Theorem 1.1, we obtain a filtration {filnDH (K)}n, where K is 1 1 the function field of X and H (K) is a shorthand for Het´ (K, Q~Z) (see § 8.2). We let 1 1 filDH (K) be the subgroup of H (K) introduced in [15, Definition 7.12]. This coincides 1 with the filtration defined in [29, Definition 2.9] by [16, Theorem 1.2]. filDH (K) can be described as the subgroup of continuous characters of the absolute Galois group of K whose Artin conductors (see [37, Definition 3.2.5]) at the generic points of D are 1 bounded by the multiplicities of D in Dred. This is a more intricate subgroup of H (K) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 3

et´ 1 than filDH (K) because the latter can be described in terms of the simpler objects such as the cohomology of the relative logarithmic Hodge-Witt sheaves. On the other 1 hand, filDH (K) determines the ramification theory of finite étale coverings of U. It is therefore desirable to know if and when these two filtrations agree. Our next result is the following.

Theorem 1.2. Let X be a smooth projective variety of dimension d over a finite field k of characteristic p. Let D ⊂ X be an effective Cartier divisor such that Dred is regular. Assume that either d ≠ 2 or k ≠ F2. Then

1 et´ 1 ﬁlDH (K) = ﬁlDH (K).

1.2. The reciprocity theorem. The purpose of reciprocity theorems in class field theory over a perfect field is to be able to represent the abelianized étale fundamental groups of varieties over the field in terms of idele class groups which are often described in terms of explicit sets of generators and relations. Let us assume that the base field k is finite of characteristic p. In this case, such a reciprocity theorem for smooth projective varieties over k is due to Kato-Saito [25] which describes the abelianized étale fundamental group in terms of the Chow group of 0-cycles. For the more intricate case of a smooth quasi-projective variety U which is not complete, an approach was introduced by Kato-Saito [26] whose underlying idea is to study the so called ‘étale fundamental group of X with modulus D’, where D ⊂ X is a fixed divisor which is supported on the complement of U in a normal compactification U ⊂ X. This group characterizes the finite étale coverings of U whose ramification along X ∖U is bounded by D in a certain sense. There are various ways to make sense of this bound on the ramification, and they give rise to several definitions of the étale fundamental group with modulus. It turns out that depending on what one wants to do, each of these has certain advantage over the others. ab Kato and Saito were able to describe π1 (U) in terms of the limit (over D) of the idele d M class groups with modulus Hnis(X, Kd,(X,D)), where d = dim(X). In [16, Theorems 1.1], it d M adiv adiv was shown that Hnis(X, Kd,(X,D)) describes π1 (X, D) for every D if we let π1 (X, D) 1 be the Pontryagin dual to the Matsuda filtration filDH (K). It was also shown in loc. adiv ab cit. that π1 (X, D) coincides with the fundamental group with modulus π1 (X, D), introduced earlier by Deligne and Laumon [34] if X is smooth. The latter was shown to coincide (along the degree zero parts) with the Chow group of 0-cycles with modulus CH0(XSD) by Kerz-Saito [29] (when p ≠ 2) and Binda-Krishna-Saito [3]. If we use Kato’s Swan conductor instead of Matsuda’s Artin conductor to bound the ramification in terms of a divisor supported away from U, we are led to a different notion of the abelianized étale fundamental group with modulus which we denote by abk bk 1 π1 (X, D). This is defined as the Pontryagin dual to the subgroup filD H (K). The latter is the subgroup of continuous characters of the absolute Galois group of K whose Swan conductors (defined by Kato [24]) at the generic points of D are bounded by the multiplicities of D in Dred. abk One can now ask if π1 (X, D) could be described by an idele class group with modulus, similar to the K-theoretic idele class group of Kato-Saito and the cycle-theoretic idele class group of Kerz-Saito. Our next result solves this problem. ̂M ̂M Let Kr,X be the improved Milnor K-theory sheaf of Gabber and Kerz [27]. Let Kr,XSD M be the relative Milnor K-theory sheaf, defined locally as the image of the map K1,(X,D)⊗Z × × ̂M j∗OU ⊗Z ⋯⊗Z j∗OU → Kr,X . We refer to Lemma 3.1 for the proof that this map is defined. This sheaf was considered by Rülling-Saito [42] when X is smooth and Dred is a simple 4 RAHULGUPTA,AMALENDUKRISHNA

d ̂M normal crossing divisor. There are degree maps deg∶ Hnis(X, Kd,XSD) → Z (see § 3.1) and ′ abk ̂ d ̂M abk ′ deg ∶ π1 (X, D) → Z. We let Hnis(X, Kd,XSD)0 = Ker(deg) and π1 (X, D)0 = Ker(deg ). Theorem 1.3. Let X be a normal projective variety of dimension d over a finite field. Let D ⊂ X be an effective Cartier divisor whose complement is regular. Then there is a continuous reciprocity homomorphism ′ d ̂M abk ρXSD∶ Hnis(X, Kd,XSD) → π1 (X, D) with dense image such that the induced map ′ d ̂M ≅ abk ρXSD∶ Hnis(X, Kd,XSD)0 Ð→ π1 (X, D)0 is an isomorphism of finite groups.

i Let HM(XSD, Z(j)) be the motivic cohomology with modulus. This is defined as the Nisnevich hypercohomology of the sheafified cycle complex with modulus zj(XSD, 2j−●), introduced in [4]. Using Theorem 1.3 and [42, Theorem 1], we obtain the following.

Corollary 1.4. Assume in Theorem 1.3 that X is regular and Dred is a simple normal crossing divisor. Then there is an isomorphism of ﬁnite groups

′ 2d ≅ abk cycXSD∶ HM(XSD, Z(d))0 Ð→ π1 (X, D)0.

abk This result can be viewed as the cycle-theoretic description of the π1 (X, D), analo- adiv gous a similar result for π1 (X, D), proven in [29], [3] and [16]. 1.3. Failure of Nisnevich descent for Chow group with modulus. Recall that the classical higher Chow groups of smooth varieties satisfy the Nisnevich descent in i 2i−j the sense that the canonical map CH (X, j) → HM (X, Z(i)) is an isomorphism for i, j ≥ 0, where the latter is the Nisnevich hypercohomology of the sheafified (in Nisnevich topology) Bloch’s cycle complex. However, this question for the higher Chow groups with modulus is yet an open problem. Some cases of this were verified by Rülling-Saito [42, Theorem 3]. As an application of Corollary 1.4, we prove the following result which provides a counterexample to the Nisnevich descent for the Chow groups with modulus. abk This was one of our motivations for studying the reciprocity for π1 (X, D). Theorem 1.5. Let X be a smooth projective surface over a finite field. Let D ⊂ X be an effective Cartier divisor such that Dred is a simple normal crossing divisor. Then the canonical map 4 CH0(XSD) → HM(X, Z(2)) is not always an isomorphism. Using Theorem 1.5, one can show that the Nisnevich descent for the Chow groups with modulus fails over infinite fields too. Recall that the map in Theorem 1.5 is known to be an isomorphism if X is a curve. This is related to the fact that for a Henselian discrete valuation field K with perfect residue field, the Swan conductor and the Artin conductor agree for the characters of the absolute Galois group of K. In this context, we also remark that one could attempt to ab define an analogue of the Deligne-Laumon fundamental group with modulus π1 (X, D) by using the Brylinski-Kato filtration along all integral curves in X not contained in D. adiv However, the resulting fundamental group will coincide with π1 (X, D) and not yield anything new for the same reason as above. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 5

1.4. Overview of proofs. We give a brief overview of our proofs. The main idea behind the proof of our duality theorem is the observation that the naive relative logarithmic Hodge-Witt sheaves are isomorphic to the naive relative mod-p Milnor K-theory sheaves (in the sense of Kato-Saito) in the pro-setting. The hope that the étale cohomology of relative mod-p Milnor K-theory sheaves are the correct objects to use for duality origi- nated from our results in previous papers which showed that the Nisnevich cohomology of the Kato-Saito relative Milnor K-theory yields a reciprocity isomorphism without any condition on the divisor. To implement the above ideas, we need to prove many results about the relative logarithmic Hodge-Witt sheaves and their relation with various versions of relative Milnor K-theory. One of these is a pro-isomorphism between the relative logarithmic Hodge- Witt sheaves and the twisted de Rham-Witt sheaves. The latter are easier objects to work with because duality for them follows from the Grothendieck-Serre coherent duality. The next step is to construct higher Cartier operators on the twisted de Rham-Witt sheaves. This allows us to construct the relative version of the two-term complexes considered by Milne [39]. The proof of the duality theorem is then reduced to coherent duality using an induction procedure. To prove Theorem 1.2, we use our duality theorem and the reciprocity theorem of [16] to reduce it to proving an independent statement that the top Nisnevich cohomology of the relative Milnor K-theory sheaf coincides with the corresponding étale cohomology if the reduced divisor is regular (see Theorem 4.9). To prove Theorem 1.3, we follow the strategy we used in the proof of a similar result for adiv π1 (X, D) in [16]. But there are some new ingredients to be used. The first key ingre- d ̂M dient is a result of Kerz-Zhao [30] which gives an idelic presentation of Hnis(X, Kd,XSD). The second key result is a theorem of Kato [24] which gives a criterion for the characters of the absolute Galois group of a Henselian discrete valuation field to annihilate various subgroups of the Milnor K-theory of the field under a suitable pairing (see Theorem 9.1). What remains after this is to prove Proposition 9.3. We prove various results about relative Milnor K-theory in sections 2, 3 and 4. We study the relative logarithmic Hodge-Witt sheaves and prove some results about their cohomology in § 5. We introduce the Cartier operators on the twisted de Rham-Witt sheaves in § 6. We construct the pairing for the duality theorem and prove its perfectness in sections 7 and 8. The reciprocity theorem is proven in § 9 and the failure of Nisnevich descent for the Chow groups with modulus is shown in § 10. 1.5. Notation. We shall work over a field k of characteristic p > 0 throughout this paper. We let Schk denote the category of separated and essentially of finite type schemes over (q) k. The product X ×Spec (k) Y in Schk will be written as X × Y . We let X (resp. X(q)) denote the set of points on X having codimension (resp. dimension) q. We let Schk~zar (resp. Schk~nis, resp. Schk~et´ ) denote the Zariski (resp. Nisnevich, resp. étale) site of Schk. We let ǫ∶ Schk~ét → Schk~nis denote the canonical morphism of sites. If F is a sheaf ∗ on Schk~nis, we shall denote ǫ F also by F as long as the usage of the étale topology is clear in a context. For X ∈ Schk, we shall let ψ∶ X → X denote the absolute Frobenius morphism. 1 For an abelian group A, we shall write TorZ(A, Z~n) as nA and A~nA as A~n. The ′ tensor product A ⊗Z B will be written as A ⊗ B. We shall let A{p } denote the subgroup of elements of A which are torsion of order prime to p. We let A{p} denote the subgroup of elements of A which are torsion of order some power of p. 6 RAHULGUPTA,AMALENDUKRISHNA

2. Relative Milnor K-theory In this section, we recall several versions of relative Milnor K-theory sheaves and establish some relations among their cohomology. M For a commutative ring A, the Milnor K-group Kr (A) (as defined by Kato [23]) is M the r-th graded piece of the graded ring K∗ (A). The latter is the quotient of the tensor × algebra T∗(A ) by the two-sided graded ideal generated by the homogeneous elements a1 ⊗⋯⊗ ar such that r ≥ 2, and ai + aj = 1 for some 1 ≤ i ≠ j ≤ r. The residue class of × M a1 ⊗⋯⊗ ar ∈ Tr(A ) in Kr (A) is denoted by the Milnor symbol a = {a1, . . . , ar}. Given M an ideal I ⊂ A, the relative Milnor K-theory K∗ (A, I) is defined as the kernel of the M M restriction map K∗ (A) → K∗ (A~I). ̃ M ̃ M Let Kr (A) denote the r-th graded piece of the graded ring K∗ (A), where the latter × is the quotient of the tensor algebra T∗(A ) by the two-sided graded ideal generated by ̃ M the homogeneous elements a1 ⊗ a2 such that a1 + a2 = 1. We let K∗ (A, I) be the kernel ̃M ̃ M of the restriction map K∗ (A) → K∗ (A~I). It is clear that there is a natural surjection ̃ M M K∗ (A) ↠ K∗ (A). This is an isomorphism if A is a local ring with infinite residue field (see [27, Proposition 2]). The following says that a similar thing holds also for the relative K-theory. Lemma 2.1. Let A be a local ring and let I ⊂ A be an ideal. Then the following hold. M ̃M (1) Kr (A, I) and Kr (A, I) are generated by the Milnor symbols {a1, . . . , ar} such M that ai ∈ K1 (A, I) for some 1 ≤ i ≤ r. ̃ M M (2) The canonical map Kr (A, I) → Kr (A, I) is surjective. This is an isomorphism if A has infinite residue field. M Proof. The assertion (1) for Kr (A, I) is [26, Lemma 1.3.1] and the proof of (1) for ̃ M M Kr (A, I) is completely identical to that of Kr (A, I). The second part of (2) follows from the corresponding result in the non-relative case mentioned above. To prove the ̃ M first part of (2), we fix an integer r ≥ 1. Let a = {a1, . . . , ar} ∈ Kr (A) be such that ̃ M M ̃ M ai ∈ K1 (A, I) = K1 (A, I) for some 1 ≤ i ≤ r. It is then clear that a ∈ Kr (A, I). Using M this observation, our assertion follows directly from item (1) for Kr (A, I). ̂ M For a ring A as above, we let K∗ (A) denote the improved Milnor K-theory defined ̂ M independently by Gabber and Kerz [27]. For an ideal I ⊂ A, we define K∗ (A, I) to be ̂ M ̂ M the kernel of the map K∗ (A) → K∗ (A~I). Let K∗(A) denote the Quillen K-theory of A. We state the following facts as a lemma and refer to [27] for their source. Lemma 2.2. There are natural maps M αA ̃ M βA ̂ M γA (2.1) K∗ (A) ←Ð K∗ (A) Ð→ K∗ (A) Ð→ K∗(A), where αA is always surjective and βA is surjective if A is local. These two maps are isomorphisms if A is either a field or a local ring with infinite residue field. Let X be a Noetherian scheme and ι∶ D ↪ X a closed immersion. We shall say that (X, D) is a modulus pair if D is an effective Cartier divisor on X. This Cartier divisor may be empty. If P is a property of schemes, we shall say that (X, D) satisfies P if X does so. We shall say that (X, D) has dimension d if X has Krull dimension d. M Given a Noetherian scheme X and an integer r ≥ 1, let Kr,X denote the sheaf on Xnis M h M whose stalk at a point x ∈ X is Kr (OX,x). We let Kr,(X,D) be the kernel of the restriction M M M M map Kr,X ↠ Kr,D ∶= ι∗Kr,D. We shall usually refer to Kr,(X,D) as the Kato-Saito relative ̃M ̂M Milnor K-sheaves. We define the sheaves Kr,(X,D) and Kr,(X,D) in an analogous way. We let Kr,X be the Quillen K-theory sheaf on Xnis. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 7

Lemma 2.3. Let X be a reduced Noetherian scheme of Krull dimension d and let D ⊂ X be a nowhere dense closed subscheme. Then the canonical map d ̃M d M Hnis(X, Kr,(X,D)~n) → Hnis(X, Kr,(X,D)~n) is an isomorphism for every integer n ≥ 0. Proof. We look at the commutative diagram

̃M / ̃M (2.2) Kr,(X,D) Kr,X

M / M Kr,(X,D) Kr,X .

Lemma 2.1 says that the left vertical arrow is surjective. We have seen above that the kernel of the right vertical arrow is supported on a nowhere dense closed subscheme of X (this uses reducedness of X). Hence, the same holds for the left vertical arrow. But this easily implies the lemma when n = 0. For n ≥ 1, we use the commutative diagram

d ̃M n / d ̃M / d ̃M / Hnis(X, Kr,(X,D)) Hnis(X, Kr,(X,D)) Hnis(X, Kr,(X,D)~n) 0

d M n / d M / d M / Hnis(X, Kr,(X,D)) Hnis(X, Kr,(X,D)) Hnis(X, Kr,(X,D)~n) 0.

It is easily seen that the two rows are exact. The n = 0 case shows that the left and the middle vertical arrows are isomorphisms. It follows that the right vertical arrow is also an isomorphism.

For D ⊂ X as above and n ≥ 1, let

̃M ̃M ̃M ̃M ̃M ̃M (2.3) Kr,(X,D)~n ∶= Kr,(X,D)~(Kr,(X,D) ∩ nKr,X ) = Image(Kr,(X,D) → Kr,X ~n). ̃M ̃M Since the map nKr,X → nKr,D is surjective, it easily follows that there are exact sequences

̃M ̃M ̃M (2.4) 0 → Kr,(X,D)~n → Kr,X ~n → Kr,D~n → 0;

̃M ̃M ̃M nKr,D → Kr,(X,D)~n → Kr,(X,D)~n → 0. ̂M ̂M ̂M ̂M We let Kr,(X,D)~n ∶= Kr,(X,D)~(Kr,(X,D) ∩ nKr,X). It is then clear that the two exact sequences of (2.4) hold for the improved Milnor K-sheaves as well. The following is immediate from (2.4). Lemma 2.4. Let X be a Noetherian scheme of Krull dimension d and let D ⊂ X be a nowhere dense closed subscheme. Then the canonical map

d ̃M d ̃M Hnis(X, Kr,(X,D)~n) → Hnis(X, Kr,(X,D)~n) is an isomorphism for all n ≥ 1. To proceed further, we need the following result about the sheaf cohomology. Given a ﬁeld k and a topology τ on Spec (k), let cdτ (k) denote the cohomological dimension of k for torsion τ-sheaves. 8 RAHULGUPTA,AMALENDUKRISHNA

Lemma 2.5. Let k be a ﬁeld and X ∈ Schk. Let F be a torsion sheaf on Xnis such that q Fx = 0 for every x ∈ X(q) with q > 0. Then Hτ (X, F) = 0 for q > cdτ (k) and τ ∈ {nis, et´ }. In particular, given an exact sequence of Nisnevich sheaves

0 →F→F ′ → F ′′ → 0,

q ′ q ′′ the induced map Hτ (X, F ) → Hτ (X, F ) is an isomorphism for q > cdτ (k) and τ ∈ {nis, et´ }.

Proof. We fix τ ∈ {nis, et´ }. Let J be the set of finite subsets of X(0). Given any α ∈ J, let Sα ⊂ X be the 0-dimensional reduced closed subscheme defined by α and let Uα = X ∖Sα. Let ια∶ Sα ↪ Xα and jα∶ Uα ↪ X be the inclusions. As J is cofiltered with respect to inclusion, there is a cofiltered system of short exact sequences of τ-sheaves

(2.5) 0 → (jα)!(FSUα )→F→(ια)∗(FSSα ) → 0.

It is easily seen from our assumption that lim (j ) (FS ) = 0 so that the map F → Ð→ α ! Uα α∈J lim (ι ) (FS ) is an isomorphism. Since H∗(S , FS ) ≅ H∗(X, (ι ) (FS )), we are Ð→ α ∗ Sα τ α Sα τ α ∗ Sα α∈J done by [8, Proposition 58.89.6].

Lemma 2.6. Let k be a ﬁeld and (X, D) a reduced modulus pair in Schk. Assume that X is of pure dimension d ≥ 1 and τ ∈ {nis, et´ }. Let n ≥ 1 be an integer and

d ̃M d ̂M (2.6) Hτ (X, Kr,(X,D)) → Hτ (X, Kr,(X,D));

d ̃M d ̂M (2.7) Hτ (X, Kr,(X,D)~n) → Hτ (X, Kr,(X,D)~n) the canonical maps. Then the following hold. (1) If k is infinite, then both maps are isomorphisms. (2) If d = 1 and r ≤ 1, then both maps are isomorphisms. (3) If d ≥ 2 and τ = nis, then both maps are isomorphisms. (4) If k is finite, d ≥ 3 and τ = et´ , then (2.7) is an isomorphism. (5) If k ≠ F2 is finite, d = 2, r ≤ 2 and τ = et´ , then (2.7) is an isomorphism.

Proof. The items (1) and (2) are clear from Lemma 2.2. To prove (3) and (4), we let n ≥ 1 and consider the commutative diagram

d−1 ̃M / d−1 ̃M / d ̃M / d ̃M / d ̃M Hτ (X, Kr,X ~n) Hτ (D, Kr,D~n) Hτ (X, Kr,(X,D)~n) Hτ (X, Kr,X ~n) Hτ (D, Kr,D~n)

d−1 ̂M / d−1 ̂M / d ̂M / d ̂M / d ̂M Hτ (X, Kr,X ~n) Hτ (D, Kr,D~n) Hτ (X, Kr,(X,D)~n) Hτ (X, Kr,X ~n) Hτ (D, Kr,D~n) with d ≥ 2. The two rows are exact by (2.4). It suffices to show that all vertical arrows except possibly the middle one are isomorphisms. But this follows easily from Lemmas 2.2 and 2.5 if we observe that cdnis(k) = 0 for k arbitrary and cdet´ (k) = 1 for k finite (see [2, ExposéX]). The proof of (2.6) for τ = nis is identical if we observe that Lemma 2.5 holds even if the sheaf F is not torsion when τ = nis. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 9

We now prove (5). We only consider the case r = 2 as the other cases are trivial. We look at the commutative diagram of Nisnevich (or ´etale) sheaves

/ ̃M / ̃M / ̃M / (2.8) 0 Kr,(X,D)~n Kr,X ~n Kr,D~n 0

αX αD / ̂M / ̂M / ̂M / 0 Kr,(X,D)~n Kr,X ~n Kr,D~n 0. We now recall from [27, Proposition 10(6)] that for a local ring A containing k, the ̂ M map K2 (A) → K2(A) is an isomorphism. Using this, it follows from [31, Corollary 3.3] ̂ M that K2 (A) is generated by the Milnor symbols {a, b} subject to bilinearity, Steinberg ̃M ̂ M relation, and the relation {a, b} =−{b, a}. Since Ker(K2 (A) ↠ K2 (A)) surjects onto ̃ M ̂ M Ker(K2 (A)~n ↠ K2 (A)~n), it follows that the latter is generated by the images of the symbols of the type {a, b} + {b, a}. Since A× → (A~I)× is surjective for any ideal I ⊂ A, we conclude that the map ̃ M ̂ M ̃M ̂ M (2.9) Ker(K2 (A)~n ↠ K2 (A)~n) → Ker(K2 (A~I)~n ↠ K2 (A~I)~n) is surjective. We have therefore shown that Ker(αX ) ↠ Ker(αD) in (2.8). Subsequently, a diagram chase shows that ̃M ̂M (2.10) Kr,(X,D)~n → Kr,(X,D)~n is a surjective map of Nisnevich (or ´etale) sheaves. We let F denote the kernel of this M ̂M map. A diagram similar to (2.2) (with Kr,X replaced by Kr,X ) and Lemma 2.2 together show that Fx = 0 for every x ∈ X(q) with q > 0. We now apply Lemma 2.5 to ﬁnish the proof. Combining Lemmas 2.3, 2.4 and 2.6, we get the following key result that we shall use.

Proposition 2.7. Let k be a ﬁeld and (X, D) a reduced modulus pair in Schk. Assume that X is of pure dimension d ≥ 2. Then there are natural isomorphisms d M ≅ d ̂M Hnis(X, Kr,(X,D)) Ð→ Hnis(X, Kr,(X,D));

d M ≅ d ̂M Hnis(X, Kr,(X,D)~n) Ð→ Hnis(X, Kr,(X,D)~n) for every integer n ≥ 1. If d = 1, these isomorphisms hold if either k is inﬁnite or r ≤ 1.

3. The idele class group of Kerz-Zhao Let A be a local domain with fraction field F and let I = (f) be a principal ideal, −1 where f ∈ A is a non-zero element. Let Af denote the localization A f obtained by [ ] inverting the powers of f so that there are inclusions of rings A ↪ Af ↪ F . We let KM A I = KM A, I and for r ≥ 2, we let KM A I denote the image of the canonical ̂1 1 ̂r map( ofS abelian) groups( ) ( S ) M × × M (3.1) K A, I ⊗ Af ⊗⋯⊗ Af → K F , 1 ( ) ( ) ( ) r ( ) induced by the product in Milnor K-theory, where the tensor product is taken r times. These groups coincide with the relative Milnor K-groups of Rülling-Saito (see [42, Def- inition 2.4 and Lemma 2.1]) when A is regular and Spec A f red is a normal crossing divisor on Spec A . We shall denote the associated sheaf( on~ an) integral scheme X with an effective Cartier( ) divisor D by KM . ̂r,XSD 10 RAHULGUPTA,AMALENDUKRISHNA

M M We let Er A, I be the image of the canonical map Kr A, I → Kr F . This is same as the image( ) of the map KM A, I → KM F by Lemma( 2.1.) The following( ) result ̃r r will be important for us. ( ) ( ) Lemma 3.1. We have the following. (1) The composite map

M M ∂ M Kr A I → Kr F Ð→ Kr−1 k p ̂ ( S ) ( ) ht(⊕p)=1 ( ( )) is zero. In particular, KM A I ⊂ KM A if A is regular and equicharacteristic ̂r ̂r or a Henselian discrete valuation( S ) ring.( ) (2) There is an inclusion E A, I ⊆ KM A I which is an equality if I is a prime r ̂r ideal. ( ) ( S ) Proof. We let p ⊂ A be a prime ideal of height one such that f ∈ p. To prove (1), it ∂ suffices to show that the composite KM A I → KM F Ð→ KM k p is zero, where ̂r p p r r−1 ∂ is the boundary map defined in [23, § 1].( WeS can) therefore( ) assume( that( )) dim A = 1 and f ∉ A×. By the definition of ∂, we can assume that A is normal. But one easily( ) checks in this case that KM A I ⊂ KM A (see [42, Proposition 2.8]) and ∂ KM A = 0 (see ̂r ̂r ̂r [7, Proposition 2.7]).( TheS ) second( part) of (1) follows from [27, Proposition( ( 10])) and [35, Theorem 5.1]. M We now prove (2). One checks using Lemma 2.1 that Er A, I ⊆ K A I . Suppose ( ) ̂r ( S ) now that I is prime and consider an element α = 1 af, u2, . . . , ur , where a ∈ A and × { + } ni × ui ∈ Af for i ≥ 2. Since I is prime, it is easy to check that ui = aif for ai ∈ A ( ) and ni ≥ 0. Using the bilinearity and the Steinberg relations, it easily follows from [23, Lemma 1] that α ∈ Er A, I . ( ) Suppose that k is any field and X ∈ Schk is regular. Let D ⊂ X be an effective Cartier divisor. It follows from Lemma 3.1 that there are inclusions KM ⊂ KM KM ̂r,XSnD ̂r,X ⊃ ̂r,(X,nD) for n 1. We let KM m = KM KM mKM . ≥ ̂r,XSD~ ̂r,XSD~( ̂r,XSD ∩ ̂r,X ) Proposition 3.2. Let k be a field and X ∈ Schk a regular scheme of dimension d ≥ 1. Let D ⊂ X be a simple normal crossing divisor. Let τ ∈ nis, et´ and m ≥ 1. Then the conclusions (1) (5) of Lemma 2.6 hold for the maps { } ∼ d M d M H X, K n → H X, K n; { τ ( ̃r,(X,nD))} { τ ( ̂r,XSnD)} d M d M H X, K m n → H X, K m n { τ ( ̃r,(X,nD)~ )} { τ ( ̂r,XSnD~ )} of pro-abelian groups. Proof. We let E = Image KM → KM and E m = Image KM → r,(X,nD) ̃r,(X,nD) ̂r,X r,(X,nD) ̃r,(X,nD) M ( ) ~ ( K m . It is easy to check using [42, Proposition 2.8] that the inclusions Er, X,nD n ↪ ̂r,X ~ ) { ( )} M M K n and Er, X,nD m n ↪ K m n are pro-isomorphisms. The rest of the { ̂r,XSnD} { ( )~ } { ̂r,XSnD~ } proof is now completely identical to that of Lemma 2.6.

Let us now assume that k is any field and X ∈ Schk is an integral scheme of dimension d ≥ 1. We shall endow X with its canonical dimension function dX (see [28]). Assume that D ⊂ X is an effective Cartier divisor. Let F denote the function field of X and U = X D. We refer the reader to [26, § 1] or [15, § 2] for the definition and all properties∖ of Parshin chains and their Milnor K-groups that we shall need. max We let PU X denote the set of Parshin chains on the pair U ⊂ X and let P be ~ ( ) U~X the subset of PU~X consisting of maximal Parshin chains. If P is a Parshin chain on RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 11

M X of dimension dX P , then we shall consider K k P as a topological abelian ( ) dX (P )( ( )) group with its canonical topology if P is maximal (see [15, § 2,5]). Otherwise, we shall consider KM k P as a topological abelian group with its discrete topology. If dX (P ) ( ( )) ′ P = p0,...,pd is a maximal Parshin chain on X, we shall let P = p0,...,pd−1 . Recall( from) [15, Deﬁnition 3.1] that the idele group of U ⊂ X is( deﬁned as ) ( ) (3.2) I = KM k P . U~X ? dX (P ) P ∈PU~X ( ( ))

We consider IU~X as a topological group with its direct sum topology. We let

⎛ M h ⎞ (3.3) I X D = Coker ? Kd P OX,P ′ ID → IU~X , max ̂ X ( ) ( S ) ⎜P ∈P ( S ) ⎟ ⎝ U~X ⎠ h where ID is the extension of the ideal sheaf ID ⊂ OX to OX,P ′ and the map on the right is M h M max induced by the composition K O ′ ID ↪ K k P → IU~X for P ∈ P . ̂dX (P )( X,P S ) dX (P )( ( )) U~X We consider I X D a topological group with its quotient topology. Recall from [15, ( S ) M h § 3.1] that I X, D is deﬁned analogous to I X D , where we replace K O ′ ID ̂dX (P ) X,P M ( h ) ( S ) ( S ) by K O ′ ,ID . dX (P )( X,P ) We let QU~X denote the set of all Q-chains on U ⊂ X . Recall that the idele class group of the pair U ⊂ X is the topological abelian( group) ( )

∂U~X (3.4) C = Coker KM k Q ÐÐÐ→ I U~X ⎛ ? dX (Q) U~X ⎞ Q ( ( )) ⎝ ∈QU~X ⎠ with its quotient topology. We let

(3.5) C X D = Coker KM k Q → I X D . ⎛ ? dX (Q) ⎞ ( S ) Q∈Q ( ( )) ( S ) ⎝ U~X ⎠ This is a topological group with its quotient topology. This idele class group was introduced by Kerz-Zhao in [30, Deﬁnition 2.1.4]. We let C X, D be the cokernel of the map KM k Q → I X, D . One checks using part( (2)) of Lemma 3.1 that there is dX (Q) Q∈Q⊕U~X ( ( )) ( ) a commutative diagram / / (3.6) IU X / I X, D / I X D ~ ( ) ( S )

/ / CU X / C X, D / C X D . ~ ( ) ( S ) If dim X = d, we let C X D = Hd X, KM and Cet´ X D = Hd X, KM . KS nis ̂d,XSD KS et´ ̂d,XSD ( ) d ( MS ) ( et´ ) d ( S M) ( ) We let CKS X, D = H X, K and C X, D = H X, K . ( ) nis( d,(X,D)) KS( ) et´ ( d,(X,D))

Lemma 3.3. The canonical map CKS X, D → CKS X D is surjective. It is an isomorphism if D ⊂ X is a reduced Cartier( divisor.) ( S )

Proof. It is clear that the stalks of the kernel and cokernel of the map KM → KM r,(X,D) ̂r,XSD vanish at the generic point of X. It follows moreover from Lemma 3.1 that these stalks vanish at all points of X with dimension d 1 or more if D is reduced. A variant of Lemma 2.5 therefore implies the desired result.− 12 RAHULGUPTA,AMALENDUKRISHNA

Let Z0 U be the free abelian group on U 0 . It is clear from the deﬁnitions of ( ) ( ) CKS X D and C X D that there is a diagram ( S ) ( S ) (3.7) Z0 U ( )❏❏ ❏❏ ❏❏ ❏❏ ❏% C X D / CKS X D , ( S ) ( S ) where the left vertical arrow is induced by the inclusion of the Milnor K-groups of the length zero Parshin chains on U ⊂ X into IU X , and the diagonal arrow is induced ( ) ~ by the coniveau spectral sequence for CKS X D . The following is [30, Theorem 3.1.1] whose proof is obtained by simply repeating( S the) proof of [28, Theorem 8.2] mutatis mutandis (see [15, Theorem 3.8]). Theorem 3.4. Assume that U is regular. Then one has an isomorphism ≅ φX D∶ C X D Ð→ CKS X D S ( S ) ( S ) such that (3.7) is commutative.

3.1. Degree map for CKS X D . Let k be any ﬁeld and X ∈ Schk an integral scheme ( S ) F of dimension d ≥ 1. Assume that D ⊂ X is an eﬀective Cartier divisor. Let CHr X denote the Chow group of r-dimensional cycles on X as in [9, Chapter 1]. It follows from( ) Lemma 3.1 and [23, Proposition 1] that M M M (3.8) Kr,XSD → ιx ∗ Kr k x → ιx ∗ Kr−1 k x → ⋯ ̂ x∈X⊕(0)( ) ( ( ( ))) x∈X⊕(1)( ) ( ( ( )))

M ∂ M ⋯ → ιx ∗ Kr−d+1 k x Ð→ ιx ∗ Kr−d k x → 0 x∈X⊕(d−1)( ) ( ( ( ))) x∈X⊕(d)( ) ( ( ( ))) is a complex of Nisnevich sheaves on X. An elementary cohomological argument (see [16, § 2.1]) shows that by taking the cohomology of the sheaves in this complex, one gets a canonical homomorphism F (3.9) νX D∶ CKS X D → CH X . S ( S ) 0 ( ) F If X is projective over k, there is a degree map deg∶ CH0 X → Z. We let deg∶ CKS X D → ν XSD F deg ( ) ( S ) Z be the composition CKS X D ÐÐÐ→ CH0 X ÐÐ→ Z. We let CKS X D 0 = Ker deg . ( S ) ( ) ( S ) deg( ) We let CKS X, D 0 be the kernel of the composite map CKS X, D ↠ CKS X D ÐÐ→ Z. ( ) ( ) ( S ) 4. Comparison theorem for cohomology of K-sheaves

Let k be a perfect ﬁeld of characteristic p > 0 and X ∈ Schk an integral regular scheme of dimension d. Let D ⊂ X be an eﬀective Cartier divisor. Let U = X ∖ D and F = k X . Let m, r ≥ 1 be integers. A combination of [22, Theorem 1.1.5] and [30, Theorems 2.2.2,( ) 3.3.1] implies the following.

Theorem 4.1. Assume that k is ﬁnite and Dred is a simple normal crossing divisor. Then the canonical map Hd X, KM pm → Hd X, KM pm nis( ̂d,XSD~ ) et´ ( ̂d,XSD~ ) is an isomorphism. Corollary 4.2. Under the assumptions of Theorem 4.1, the canonical map

d M m d M m H X, K p n → H X, K p n { nis( ̂d,(X,nD)~ )} { et´ ( ̂d,(X,nD)~ )} is an isomorphism if either k ≠ F2 or d ≠ 2. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 13

Proof. Combine Lemma 2.6, Proposition 3.2 and Theorem 4.1. Our goal in this section is to prove an analogue of Theorem 4.1 for the cohomology of M Kd,(X,D) when Dred is regular. This will be used in the proof of Theorem 1.2. 4.1. Filtrations of Milnor K-theory. Let A be a regular local ring essentially of ﬁnite type over k and I = f a principal ideal, where f ∈ A is a non-zero element such ( ) −1 that R = A I is regular. Let Af denote the localization A f obtained by inverting the powers~ of f. Let F be the quotient ﬁeld of A so that there[ ] are inclusions of rings n n A ↪ Af ↪ F . For an integer n ≥ 1, let I = f . ( ) We let E A, In be the image of the canonical map KM A, In → KM A ⊂ KM F . r ̃r ̂r r One easily checks( that) there is a commutative diagram (see( (2.3))) ( ) ( )

̂ M n m ̂ M (4.1) KM A, In / / KM A, In pm ≅ / Kr (A,I )+p Kr (A) ̂r O ̂r O pmK̂ M (A) ( O ) ( O )~ 7 r _ ♥♥♥ ♥♥♥ ♥♥♥ )Ù ♥♥♥ n m ̂ M KM A, In / / KM A, In pm / / Er(A,I )+p Kr (A) ̃r ̃r pmK̂ M (A) ( ) ( )~ r _

̂ M n m ̂ M ̂ M M n / / M n m ≅ / Kr (ASI )+p Kr (A) / Kr (A) Kr A I Kr A I p m ̂ M m ̂ M , ̂ ( S ) ̂ ( S )~ p Kr (A) p Kr (A) where the existence of the left-most bottom vertical arrow and the right-most bottom horizontal inclusion follows from [42, Proposition 2.8] (see also [14, Lemma 3.8]). We consider the ﬁltrations F ● A and G● A of KM A pm by letting F n A = m,r m,r ̂r m,r n m ̂ M ( ̂)M n m(̂ M) ( )~ ( ) Er(A,I )+p Kr (A) n Kr (ASI )+p Kr (A) m ̂ M and Gm,r A = m ̂ M . We then have a commutative p Kr (A) ( ) p Kr (A) diagram of ﬁltrations

● M ● M (4.2) Er A, I / K A I / K A ( ) ̂r ( S ) ̂r ( )

F ● A / G● A / KM A pm. m,r( ) m,r( ) ̂r ( )~ Lemma 4.3. One has KM A In+1 ⊆ E A, In and Gn+1 A ⊆ F n A . Furthermore, ̂r r m,r m,r M (1 S ) 1 ( ) ( ) ( ) Er A, I = K A I and F A = G A . ( ) ̂r ( S ) m,r( ) m,r( ) Proof. This is immediate from [42, Proposition 2.8].

M M M We let K Af be the image of the canonical map K Af → K F . We let ̃r ( ) ̃r ( ) r ( ) K̂ M A In pmK̃ M A Hn A = r ( S )+ r ( f ) . Note that the canonical map KM A ↠ KM A is m,r m ̃ M r r ( ) p Kr (Af ) ̃ ( ) ̂ ( ) surjective. We therefore have a surjective map Gn A ↠ Hn A . m,r( ) m,r( ) Lemma 4.4. The canonical map n n Gm,r A Hm,r A ( ) → ( ) Gn+1 A Hn+1 A m,r ( ) m,r ( ) is an isomorphism. Proof. The map in question is clearly surjective since its both sides are quotients of KM A In . So we need to prove only the injectivity. Using Lemma 4.3 and an easy ̂r ( S ) 14 RAHULGUPTA,AMALENDUKRISHNA reduction, one can see that it suﬃces to show that the map

M M K Af pm K Af ̃r ( ) Ð→ ̃r ( ) KM A KM A ̂r ( ) ̂r ( ) is injective. M Since K Af has no p-torsion, a snake lemma argument shows that the previous ̃r ( ) M m M m injectivity is equivalent to the injectivity of the map K A p → K Af p . We ̂r ( )~ ̃r ( )~ now look at the composition KM A pm → KM A pm → KM F pm. It suﬃces ̂r ̃r f r to show that this composition is( injective.)~ But( this)~ is an easy( consequence)~ of [27, Proposition 10(8)] and [11, Theorem 8.1].

We now consider the map (4.3) ρ∶ Ωr−1 Ωr−2 → KM A In KM A In+1 R ⊕ R ̂r ( S )~ ̂r ( S ) having the property that n ρ adlogb1 ⋯∧ dlogbr−1, 0 = 1 + af , b1,..., br−1 ; ( ∧ ) { ̃ ̃ ̃ } n ρ 0, adlogb1 ∧⋯∧ dlogbr−2 = 1 + af , b1,..., br−2,f , ( ) { ̃ ̃ ̃ } a b a b A ↠ R where and ̃i’s are arbitrary lifts of and i’s, respectively, under the surjection . It follows̃ from [5, §4, p. 122] that this map is well deﬁned. It is clear from (4.2) and Lemma 4.3 that ρ restricts to a map r−1 n M n+1 (4.4) ρ∶ Ω → Er A, I K A I . R ( )~ ̂r ( S ) This map is surjective by Lemma 2.1. r−1 Let ψ∶ R → R be the absolute Frobenius morphism. Let ZiΩR be the unique R- i r−1 submodule of ψ∗ΩR such that the inverse Cartier operator (see [20, Chap. 0, § 2]) −1 r−1 ≅ r−1 r−2 r−1 induces an R-linear isomorphism C ∶ Zi−1ΩR Ð→ ZiΩR dΩR , where we let Z1ΩR = r−1 r r−1 r−2 ~r−1 Ker ΩR → ΩR . We let B1ΩR = dΩR and let BiΩR (i ≥ 2) be the unique R- ( i) r−1 −1 r−1 r−1 r−2 submodule of ψ∗ΩR such that C ∶ Bi−1ΩR → BiΩR dΩR is an isomorphism of R-modules. ~ s Lemma 4.5. Let n = n1p , where s ≥ 0 and p ∤ n1. Then ρ induces isomorphisms m r−1 n ψ Ω ≅ F A ρn ∗ R m,r( ) m s m,r∶ r−1 Ð→ n+1 if ≤ ; ZmΩ G A R m,r ( ) s r−1 n ψ Ω ≅ F A ρn ∗ R m,r( ) m s. m,r∶ r−1 Ð→ n+1 if > BsΩ G A R m,r ( ) n Proof. Once it exists, ρm,r is clearly surjective in both cases. We therefore need to prove its existence and injectivity. These are easy consequences of various lemmas in [5, § 4] (see also [22, Proposition 1.1.9]) once we have Lemma 4.4. The proof goes as follows. When m ≤ s, we look at the diagram

1 n ψmΩr− ρm,r F n (A) ∗ R / m,r (4.5) r−1 / n+1 ZmΩ G (A) R_ m,r _

ψmΩr−1 ψmΩr−2 ρ Gn (A) Hn (A) ∗ R ∗ R / m,r ≅ / m,r Z Ωr−1 Z Ωr−2 Gn+1(A) Hn+1(A) , m R ⊕ m R m,r m,r RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 15 where the bottom right isomorphism is by Lemma 4.4. The right vertical arrow is clearly injective. The composite arrow on the bottom is an isomorphism by [5, (4.7), Remark 4.8]. It follows that the top horizontal arrow exists and it is injective. When m > s, we look at the commutative diagram

1 n ψsΩr− ρm,r F n (A) ∗ R / m,r (4.6) r−1 / n+1 BsΩR Gm,r(A) ✉ l _ ✉✉ L ✉✉ ✉✉ z✉✉ s r−1 s r−2 ρ n n ψ∗ΩR ψ∗ΩR Gm,r(A) ≅ Hm,r(A) 1 2 / / θ / 1 / 1 , B Ωr− B Ωr− Coker Gn+ (A) Hn+ (A) s R ⊕ s R ( ) m,r m,r where θ is defined in [5, Lemma 4.5]. The map ρ on the bottom is again an isomorphism. Hence, it suffices to show that the left vertical arrow is injective. But this is immediate from the definition of θ (see loc. cit.) since C−s is an isomorphism.

4.2. The comparison theorem. Let k be a perfect field of characteristic p > 0. Let X ∈ Schk be a regular scheme of pure dimension d ≥ 1 and let D ⊂ X be a regular n closed subscheme of codimension one. Let Fm,r,X be the sheaf on Xnis such that for n n h h any point x ∈ X, the stalk Fm,r,X x coincides with Fm,r OX,x , where I = IDOX,x. ( ) ( ) h The corresponding étale sheaf is defined similarly by replacing the Henselization OX,x sh n by the strict Henselization OX,x. We define the Nisnevich and étale sheaves Gm,r,X in an identical way, using the groups Gn A for stalks. m,r( ) Lemma 4.6. The change of topology map Hi X, F 1 F n → Hi X, F 1 F n nis( m,r,X ~ m,r,X ) et´ ( m,r,X ~ m,r,X ) is an isomorphism for every i ≥ 0. Proof. We consider the exact sequence of Nisnevich (or étale) sheaves n 1 1 Fm,r,X Gm,r,X Fm,r,X (4.7) 0 → n+1 → n+1 → n → 0, Gm,r,X Gm,r,X Fm,r,X 1 1 where we have replaced Gm,r,X by Fm,r,X in the quotient on the right using Lemma 4.3. Working inductively on n, it follows from [22, Proposition 1.1.9] that the middle term in (4.7) has identical Nisnevich and étale cohomology. It suffices therefore to show that the same holds for the left term in (4.7) too. Since this isomorphism is obvious for i = 0, it suffices to show using the Leray spectral i n n+1 sequence that R ǫ∗ Fm,r,X Gm,r,X = 0 for i > 0. Equivalently, we need to show that if A = h ( ~ ) i n n+1 OX,x for some x ∈ X and f ∈ A defines D locally at x, then Het´ A, Fm,r A Gm,r A = 0 for i > 0. But this is immediate from Lemma 4.5. ( ( )~ ( )) Lemma 4.7. Assume that k is finite. Then the change of topology map Hd X, F n → Hd X, F n nis( m,d,X ) et´ ( m,d,X ) is an isomorphism.

1 1 Proof. We ﬁrst assume n = 1. Using Lemma 4.3, we can replace Fm,d,X by Gm,d,X . By 1 [22, Theorem 1.1.5] and [30, Theorems 2.2.2, Proposition 2.2.5], the dlog map Gm,d,X → d WmΩX,log is an isomorphism in Nisnevich and ´etale topologies. We conclude by [30, Theorem 3.3.1]. 16 RAHULGUPTA,AMALENDUKRISHNA

We now assume n ≥ 2 and look at the exact sequence 1 n 1 Fm,d,X 0 → Fm,d,X → Fm,d,X → n → 0. Fm,d,X d We saw above that the middle term can be replaced by WmΩX,log. We can now conclude the proof by Lemmas 4.6 and the independent statement Lemma 5.6 via the 5-lemma. Lemma 4.8. Assume that k is ﬁnite. Then the change of topology map Hd X, KM pm → Hd X, KM pm nis( ̃d,(X,nD)~ ) et´ ( ̃d,(X,nD)~ ) is an isomorphism if either d ≠ 2 or k ≠ F2. Proof. In view of Lemma 4.7, it suﬃces to show that the canonical map (see (4.1))

(4.8) Hd X, KM pm → Hd X, F n τ ( ̃d,(X,nD)~ ) τ ( m,d,X ) is an isomorphism for τ ∈ nis, et´ . Using Lemma 2.5 and (4.1), the proof of this is completely identical to that{ of Lemma} 2.6. The following is the main result we were after in this section. We restate all assumptions for convenience.

Theorem 4.9. Let k be a ﬁnite ﬁeld of characteristic p. Let X ∈ Schk be a regular scheme of pure dimension d ≥ 1 and let D ⊂ X be a regular closed subscheme of codimension one. Assume that either d ≠ 2 or k ≠ F2. Then there is a natural isomorphism ≅ Hd X, KM pm Ð→ Hd X, KM pm nis( d,(X,nD))~ et´ ( ̂d,(X,nD)~ ) for every m,n ≥ 1. Proof. Combine Lemmas 2.3, 2.4, 2.6 and 4.8. Corollary 4.10. Under the assumptions of Theorem 4.9, there is a commutative diagram

(4.9) Hd X, KM pm / Hd X, KM pm nis( d,(X,nD))~ et´ ( ̂d,(X,nD)~ )

Hd X, KM pm / Hd X, KM pm nis( ̂d,XSnD~ ) et´ ( ̂d,XSnD~ ) for m,n ≥ 1 in which the horizontal arrows are isomorphisms. Proof. In view of Theorems 4.1 and 4.9, we only have to explain the vertical arrows. But their existence follows from Lemmas 2.3, 2.4 and 2.6, and the diagram (4.1).

Remark 4.11. The reader may note that Theorem 4.1 holds whenever Dred is a simple normal crossing divisor, but this is not the case with Theorem 4.9. The main reason for this is that while KM satisﬁes cdh-descent for closed covers [22], the same is not ̂d,XSD true for KM . This prevents us from using an induction argument on the number of ̂d,(X,D) irreducible components of D.

5. Relative logarithmic Hodge-Witt sheaves In this section, we shall recall the relative logarithmic Hodge-Witt sheaves and show that they coincide with the Kato-Saito relative Milnor K-theory with Z pm coefficients ~ in a pro-setting. We fix a field k of characteristic p > 0 and let X ∈ Schk. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 17

● 5.1. The relative Hodge-Witt sheaves. Let WmΩX m≥1 denote the pro-complex of de Rham-Witt (Nisnevich) sheaves on X. This{ is a pro-complex} of differential graded algebras with the structure map R and the differential d. Let m∶ OX → WmOX be the [−] ● multiplicative Teichmüller homomorphism. Recall that the pro-complex WmΩX m≥1 is {r } r equipped with the Frobenius homomorphism of graded algebras F ∶ WmΩX → Wm−1ΩX r r and the additive Verschiebung homomorphism V ∶ WmΩX → Wm+1ΩX . These homomorphisms satisfy the following properties which we shall use frequently. (1) FV = p = V F . (2) F dV = d, dF = pF d and pdV = V d. p−1 (3) F d a = a d a for all a ∈ OX . ( [ ]) [ ] [ ] i j (4) V F x y = xV y and V xdy = VxdVy for all x ∈ WmΩX , y ∈ WmΩX . i ( ( )r ) i ( ) r ( i ) r−1 i r r−i i r Let fil WmΩ = V Wm−iΩ +dV Wm−iΩ , fil WmΩ = Ker R and fil WmΩ = V X X X R X ( ) p X Ker pr−i for 0 ≤ i ≤ r. By [19, Lemma 3.2.4] and [20, Proposition I.3.4], we know that i ( )r i r i r filV WmΩX = filRWmΩX ⊆ filpWmΩX and the last inclusion is an equality if X is regu- r r r+1 r r−1 lar. We let ZWmΩX = Ker d∶ WmΩX → WmΩX and BWmΩX = Image d∶ WmΩX → r r ●( r r) ( WmΩX . We set H WmΩX = ZWmΩX BWmΩX . Let ψ) ∶ X → X denote( the) absolute~ Frobenius morphism. One then knows that r r+1 r r d∶ ψ∗W1ΩX → ψ∗W1ΩX is OX -linear. In particular, the sheaves ψ∗ZW1ΩX , ψ∗BW1ΩX r ● and ψ∗H W1ΩX are quasi-coherent on Xnis. If X is F -finite, then ψ is a finite mor- ( ) r r r ● phism, and therefore, ψ∗ZW1Ω , ψ∗BW1Ω and ψ∗H W1Ω are coherent sheaves X X ( X ) on Xnis. Let D ⊂ X be a closed subscheme defined by a sheaf of ideals ID and let nD ⊂ X n ● ● be the closed subscheme defined by ID for n ≥ 1. We let WmΩ(X,D) = Ker WmΩX ↠ ● ( ● ι∗WmΩ , where ι∶ D ↪ X is the inclusion. We shall often use the notation WmΩ D) (A,I) if X = Spec A and D = Spec A I . If D is an effective Cartier divisor on X and ( ) ( ~ ) n ∈ Z, we let WmOX nD be the Nisnevich sheaf on X which is locally defined as −n ( ) WmOX nD = f ⋅ WmOX ⊂ j∗WmOU , where f ∈ OX is a local equation of D and ( ) [ ] j∶ U ↪ X is the inclusion of the complement of D in X. One knows that WmOX nD is a ● ● ( ) sheaf of invertible WmOX -modules. We let WmΩX nD = WmΩX ⊗WmOX WmOX nD . ( ) ● ● ( ) It is clear that there is a canonical WmOX -linear map WmΩ −D → WmΩ . The X ( ) (X,D) reader is invited to compare the following with a weaker statement [10, Proposition 2.5]. Lemma 5.1. If X, D is a modulus pair and m ≥ 1 an integer, then the canonical map of pro-sheaves ( ) ● ● WmΩX −nD n → WmΩ(X,nD) n { ( )} { } ● ● on Xnis is surjective. If X is furthermore regular, then the map WmΩ −D → WmΩ X ( ) (X,D) is injective. Proof. The first part follows directly from Lemma 5.2 below. We prove the second part. Let U = X ∖D and j∶ U ↪ X the inclusion. Using the Néron-Popescu approximation and ● the Gersten resolution of WmΩX (when X is smooth), proven in [12, Proposition II.5.1.2], ● ● it follows that the canonical map WmΩX → j∗WmΩU is injective (see [33, Proposi- tion 2.8]). Since WmOX −D is invertible as a WmOX -module, there are natural isomor- ● ( ) ● ∗ ● phisms j∗WmΩ ⊗W O WmOX −D → j∗ WmΩ ⊗W O j WmOX −D ≅ j∗WmΩ U m X ( ) ( U m U ( )) U (see [17, Exercise II.5.1(d)]). Using the invertibility of WmOX −D again, we conclude ● ● ( ) that the canonical map WmΩX −D → j∗WmΩU is injective. Since this map factors ● ● ( ) ● through WmΩ −D → WmΩ ↪ WmΩ , the lemma follows. X ( ) (X,D) X Lemma 5.2. Let A be a commutative Fp-algebra and I = f ⊆ A a principal ideal. Let m ( r) r m ≥ 1 be any integer and n = p . Then the inclusion WmΩ(A,In) ⊆ WmΩA of Wm A - r r ( ) modules factors through WmΩ n ⊆ f ⋅ WmΩ for every r ≥ 0. (A,I ) [ ] A 18 RAHULGUPTA,AMALENDUKRISHNA

r r Proof. By [18, Lemma 1.2.2], WmΩ is the Wm A -submodule of WmΩ generated (A,I) ( ) A by the de Rham-Witt forms of the type a0da1 ∧⋯∧ dar, where ai ∈ Wm A for all r ( ) i and ai ∈ Wm I for some i. We let ω = a0da1 ∧⋯∧ dar ∈ WmΩ(A,In) such that n ( ) ai ∈ Wm I for some i ≠ 0. We can assume after a permutation that i = 1. We let ( ) m−1 ′ i n ′ ′ ω = da2 ∧⋯∧ dar. We can write a1 = ∑ V f m−i ai m−i for some ai ∈ A. It follows i=0 ([ ] [ ] ) m−1 i n ′ ′ that ω = ∑ a0dV f m−i ai m−i ∧ ω . i=0 ([ ] [ ] ) i n ′ ′ We fix an integer 0 ≤ i ≤ m − 1 and let ωi = a0dV f m−i ai m−i ∧ ω . It suffices to r ([ ] [ ] ) show that each ωi ∈ WmΩX −D , where X = Spec A and D = Spec A I . However, we have ( ) ( ) ( ~ ) i n ′ ′ ωi = a0dV f m−i ai m−i ∧ ω ([pm−i] [i ]′ ) ′ = a0d f mV ai m−i ∧ ω pm([−i ] i [′ ] ) ′ m−i pm−i−1 i ′ ′ = f ma0dV a m−i ∧ ω + p f ma0V a m−id f m ∧ ω . [ ] [ i] [ ] [ i] [ ] r It is clear that the last term lies in WmΩ −D . This finishes the proof. X ( ) The following result will play a key role in our exposition. Corollary 5.3. If X, D is a regular modulus pair, then the canonical map of pro- sheaves ( ) ● ● WmΩ −nD n → WmΩ n { X ( )} { (X,nD)} on Xnis is an isomorphism for every m ≥ 1. 5.2. The relative logarithmic sheaves. In the remaining part of § 5, our default topology will be the étale topology. All other topologies will be mentioned specifically. Let k be a field of characteristic p > 0 and D ⊆ X a closed immersion in Schk. Recall r r from [20] that WmΩX,log is the étale subsheaf of WmΩX which is the image of the map M m r dlog∶ K p → WmΩ , given by dlog a1, . . . , ar = dlog a1 m ∧⋯∧ dlog ar m. It is ̃r,X ~ X ({ }) [ ] [ ] easily seen that this map exists. One knows (e.g., see [36, Remark 1.6]) that this map in fact factors through dlog∶ KM pm ↠ W Ωr . Moreover, this map is multiplicative. ̂r,X m X,log The naturality of the dlog map~ gives rise to its relative version

M m r r r (5.1) dlog∶ K p → WmΩ ∶= Ker WmΩ ↠ WmΩ ̂r,(X,D)~ (X,D),log ( X,log D,log) M m which is a morphism of étale sheaves of K∗,X p -modules as r ≥ 0 varies. ̂ r ~ r Recall that the Frobenius map F ∶ Wm+1ΩX → WmΩX sends Ker R into the subsheaf m−1 r−1 r r ( ) m−1 r−1 dV ΩX so that there is an induced map F ∶ WmΩX → WmΩX dV ΩX . We let r r m−1 r−1 ~ r π∶ WmΩX → WmΩX dV ΩX denote the projection map. Since R − F ∶ Wm+1ΩX → r ~ WmΩX is surjective by [20, Proposition I.3.26], it follows that π − F is also surjective. Indeed, this surjectivity is proven for smooth schemes over k in loc. cit., but then the claim follows because X can be seen locally as a closed subscheme of a regular scheme and a regular scheme is a filtered inductive limit of smooth schemes over k, by Néron–Popescu desingularisation. We thus get a commutative diagram

π F W Ωr / r / r − / m X / (5.2) 0 WmΩX,log WmΩX m−1 r−1 0 dV ΩX

π F W Ωr / r / r − / m D / 0 WmΩD,log WmΩD m−1 r−1 0. dV ΩD The middle and the right vertical arrows are surjective by deﬁnition and the two rows are exact by loc. cit. and [40, Corollary 4.2]. The surjectivity of the left vertical arrow RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 19 follows by applying the dlog map to the surjection KM ↠ KM . Taking the kernels of ̂r,X ̂r,D the vertical arrows, we get a short exact sequence r WmΩ X,D W r W r ( ) . (5.3) 0 → mΩ(X,D),log → mΩ(X,D) → r m−1 r−1 → 0 WmΩ(X,D) ∩ dV ΩX The following property of the relative dlog map will be important to us. Lemma 5.4. Assume that k is perfect and X is regular. Then the map of ´etale pro- sheaves M m r dlog∶ K p n → WmΩ n { ̂r,(X,nD)~ } { (X,nD),log} is an isomorphism for every m, r ≥ 1. Proof. We consider the commutative diagram of pro-sheaves:

M m M m M m (5.4) 0 / K p n / K p / K p n / 0 { ̂r,(X,nD)~ } ̂r,X ~ { ̂r,nD~ } dlog ≅ dlog dlog r r r 0 / WmΩ n / WmΩ / WmΩ n / 0. { (X,nD),log} X,log { nD,log} The middle vertical arrow is an isomorphism by the Bloch-Gabber-Kato theorem for fields [5, Corollary 2.8] and the proof of the Gersten conjecture for Milnor K-theory ([27, Proposition 10]) and logarithmic Hodge-Witt sheaves ([13, Thèoréme 1.4]). The right vertical arrow is an isomorphism by [36, Theorem 0.3]. Since the rows are exact, it follows that the left vertical arrow is also an isomorphism. This finishes the proof. We now prove a Nisnevich version of Lemma 5.4. For any k-scheme X, we let r M m r WmΩ be the image of the map of Nisnevich sheaves dlog∶ K p → WmΩ , X,log,nis ̂r,X ~ X given by dlog a1, . . . , ar = dlog a1 m ∧⋯∧ dlog ar m (see [36, Remark 1.6] for the existence of this({ map). The}) naturality[ ] of this map gives[ ] rise to its relative version

M m r r r (5.5) dlog∶ K p → WmΩ ∶= Ker WmΩ ↠ WmΩ ̂r,(X,D)~ (X,D),log,nis ( X,log,nis D,log,nis) if D ⊆ X is a closed immersion. Lemma 5.5. Assume that k is perfect and X is regular. Then the map of Nisnevich pro-sheaves M m r dlog∶ K p n → WmΩ n { ̂r,(X,nD)~ } { (X,nD),log,nis} is an isomorphism for every m, r ≥ 1. Proof. The proof is completely identical to that of Lemma 5.4, where we only have to observe that the middle and the right vertical arrows are isomorphisms even in the Nisnevich topology, by the same references that we used for the étale case. Lemma 5.6. Assume that k is a finite field and X is regular of pure dimension d ≥ 1. Then the canonical map Hi X, KM pm → Hi X, KM pm nis( ̂d,X ~ ) et´ ( ̂d,X ~ ) is an isomorphism for i = d and surjective for i = d − 1. Proof. We have seen in the proofs of Lemmas 5.4 and 5.5 that for every r ≥ 0, there are natural isomorphisms M m r M m r (5.6) K p ≅ WmΩ and K p ≅ WmΩ . ( ̂r,X ~ )nis X,log,nis ( ̂r,X ~ )et´ X,log Using these isomorphisms, the lemma follows from [30, Propositions 3.3.2, 3.3.3]. The b d (a) only additional input one has to use is that H Xet´ ,WmΩ = 0 if x ∈ X and b < a x( X,log) 20 RAHULGUPTA,AMALENDUKRISHNA by [39, Corollary 2.2] so that the right hand side of [loc. cit., (3.3.3)] is zero. This is needed in the proof of the i = d − 1 case of the lemma.

Assume now that k is a perfect ﬁeld, X ∈ Schk is a regular scheme of pure dimension d ≥ 1 and D ⊂ X is a nowhere dense closed subscheme. We ﬁx integers m ≥ 1 and i, r ≥ 0. Under these assumptions, we shall prove the following results. Lemma 5.7. There is a short exact sequence

pi m−i r r R r 0 → Wm−iΩX,log Ð→ WmΩX,log ÐÐÐ→ WiΩX,log → 0 of sheaves on X in Nisnevich and étale topologies. Proof. The étale version of the lemma is already known by [6, Lemma 3]. To prove its Nisnevich version, we can use the first isomorphism of (5.6). This reduces the proof to showing that tensoring the exact sequence pi π 0 → Z pm−i Ð→ Z pm ÐÐÐ→m−i Z pi → 0 ~ ~ ~ with KM yields an exact sequence ̂r,X pi π (5.7) 0 → KM pm−i Ð→ KM pm ÐÐÐ→m−i KM pi → 0 ̂r,X ~ ̂r,X ~ ̂r,X ~ of Nisnevich sheaves on X. But this follows directly from the fact that KM has no ̂r,X pi-torsion (see [11, Theorem 8.1]). Lemma 5.8. There is a short exact sequence

pi m−i r r R r 0 → Wm−iΩ n Ð→ WmΩ n ÐÐÐ→ WiΩ n → 0 { nD,log} { nD,log} { nD,log} of pro-sheaves on D in Nisnevich and ´etale topologies. Proof. The argument below works for either of Nisnevich and ´etale topologies. We shall prove the lemma by modifying the proof of [40, Theorem 4.6]. The latter result says that there is an exact sequence

i m−i r p r R r (5.8) WmΩ n Ð→ WmΩ n ÐÐÐ→ WiΩ n → 0. { nD,log} { nD,log} { nD,log} It suﬃces therefore to show that the ﬁrst arrow in this sequence has a factorization

i pi r R r r (5.9) WmΩ n ↠ Wm−iΩ n ↪ WmΩ n. { nD,log} { nD,log} { nD,log} To prove this factorization, we look at the commutative diagram

pi r / r (5.10) WmΩnD,log n WmΩnD,log n { _ } { _ }

i r p r WmΩ n / WmΩ n. { nD} { nD} It follows from [40, Proposition 2.14] that the map pi at the bottom has a factorization ● (see § 5.1 for the deﬁnitions of various ﬁltrations of WmΩnD)

i pi r R r r (5.11) WmΩ n ↠ Wm−iΩ n ↪ WmΩ n. { nD} { nD} { nD} r i r Since the image of WmΩ n under R is Wm−iΩ n (note the surjectivity of { nD,log} { nD,log} the second arrow in (5.8)), it follows at once from (5.11) that the map pi on the top in (5.10) has a factorization as desired in (5.9). This concludes the proof. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 21

Proposition 5.9. There is a short exact sequence

pi m−i r r R r 0 → Wm−iΩ n Ð→ WmΩ n ÐÐÐ→ WiΩ n → 0 { (X,nD),log} { (X,nD),log} { (X,nD),log} of pro-sheaves on X in Nisnevich and ´etale topologies. r r Proof. Combine the previous two lemmas and use that the maps WmΩX,log → WmΩnD,log r r and WmΩX,log,nis → WmΩnD,log,nis are surjective. Applying the cohomology functor, we get Corollary 5.10. There is a long exact sequence of pro-abelian groups pi j r j r ⋯ → H X,Wm−iΩ n Ð→ H X,WmΩ n { et´ ( (X,nD),log)} { et´ ( (X,nD),log)} m−i i R j r ∂ ÐÐÐ→ H X,WiΩ n Ð→ ⋯ { et´ ( (X,nD),log)} The same also holds in the Nisnevich topology.

5.3. Some cohomology exact sequences. Let us now assume that k is a finite field m and X ∈ Schk is regular of pure dimension d ≥ 1. For any p -torsion abelian group V , we ⋆ m let V = HomZ~pm V, Z p . Let D ⊂ X be an effective Cartier divisor with complement j ( ~ j ) r j j U. We let Fm,r n = H X,WmΩ and Fm,r U = lim Fm,r n . Each group et´ (X,nD),log ←Ðn j ⋆ ( ) ( ) ( ) ( ) Fm,r n is a profinite abelian group (see [41, Theorem 2.9.6]). ( ( )) Lemma 5.11. The sequence ⋯ → F j U → F j U → F j U → F j+1 U → ⋯ m−1,r( ) m,r( ) 1,r( ) m−1,r( ) is exact. 1 j Proof. We first prove by induction on m that lim Fm,r n = 0. The m = 1 case follows ←Ðn ( ) from Lemma 8.4. In general, we break the long exact sequence of Corollary 5.10 into short exact sequences. This yields exact sequences of pro-abelian groups 1 j m−1 (5.12) 0 → Image ∂ → F n n → Ker R → 0; ( ) { m−1,r( )} ( ) m−1 j m−1 (5.13) 0 → Ker R → F n n → Image R → 0; ( ) { m,r( )} ( ) m−1 j 1 (5.14) 0 → Image R → F n n → Image ∂ → 0. ( ) { 1,r( )} ( ) j Lemma 8.4 says that F n n is a pro-system of finite abelian groups. This implies { 1,r( )} by (5.14) that each of Image Rm−1 and Image ∂1 is isomorphic to a pro-system of finite abelian groups. In particular,( the) pro-systems( of) (5.14) do not admit higher derived lim functors. On the other hand, lim1 F j n = 0 by induction. It follows from (5.12) that ←Ðn m−1,r 1 m−1 ( ) 1 j lim Ker R = 0. Using (5.13), we get lim Fm,r n = 0. ←Ðn ( ) ←Ðn ( ) It follows from what we have shown is that the above three short exact sequences of pro-abelian groups remain exact after we apply the inverse limit functor. But this implies that the long exact sequence of Corollary 5.10 also remains exact after applying the inverse limit functor. This proves the lemma. Lemma 5.12. There is a long exact sequence

i ⋆ ⋆ (Rm−i)⋆ ⋆ (p ) ⋆ ⋯ → lim F j n ÐÐÐÐ→ lim F j n ÐÐÐ→ lim F j n → ⋯ Ð→n( i,r( )) Ð→n( m,r( )) Ð→n( m−i,r( )) of abelian groups for every i ≥ 0. 22 RAHULGUPTA,AMALENDUKRISHNA

Proof. This is an easy consequence of Corollary 5.10 using the fact that the Pontryagin dual functor (recalled in § 7.4) is exact on the category of discrete torsion abelian groups (see [41, Theorem 2.9.6]) and the direct limit functor is exact on the category of ind- abelian groups. We also have to note that the Pontryagin dual of a discrete pm-torsion abelian group V coincides with V ⋆. Lemma 5.13. Assume further that D ⊂ X is a simple normal crossing divisor. Then the group lim F d n is profinite and the canonical map ←Ðn m,d( ) ⋆ lim F d n → lim F d n ⋆ Ð→n( m,d( )) (←Ðn m,d( )) is an isomorphism of topological abelian groups if either k ≠ F2 or d ≠ 2. d Proof. By [26, Theorem 9.1], Proposition 2.7, Corollary 4.2 and Lemma 5.4, F n n { m,d( )} is isomorphic to a pro-abelian group En n such that each En is finite and the map { } En+1 → En is surjective. We can therefore apply Lemma 7.3. 6. Cartier map for twisted de Rham-Witt complex In this section, we shall prove the existence and some properties of the Cartier homomorphism for the twisted de Rham-Witt complex. We fix a perfect field k of characteristic p > 0 and a modulus pair X, D in Schk such that X is connected and regular of dimension d ≥ 1. Let ι∶ D ↪ X and( j∶ U) ↪ X be the inclusions, where U = X ∖ D. Let F denote the function field of X. We also fix integers m ≥ 1, n ∈ Z and r ≥ 0. Let WmOX nD be the Nisnevich sheaf on X defined in § 5.1. We begin with following result describing( ) the behavior of the Frobenius and Verschiebung operators on W●OX nD . We shall then use this result to describe these operators on the full twisted de( Rham-Witt) complex. Until we talk about the topology of X again in this section, we shall assume it to be the Nisnevich topology. Lemma 6.1. We have

(1) F Wm+1OX nD ⊆ WmOX pnD . ( ( )) ( ) (2) V WmOX pnD ⊆ Wm+1OX nD . ( ( )) ( ) (3) R Wm+1OX nD ⊆ WmOX nD . ( ( )) ( ) h Proof. We can check the lemma locally at a point x ∈ X. We let A = OX,x and let f ∈ A define D at x. The lemma is easy to check when n ≤ 0. We therefore assume n ≥ 1. For −n −n −pn h w ∈ Wm+1 A , we have F w f = F w ⋅ F f = F w ⋅ f ∈ WmO pnD . ( ) ( [ ]) ( ) ([ ]) ( ) [ ] X,x( ) This proves (1). For (2), we use the projection formula to get V w′ f −pn = V w′ ⋅ −n ′ −n h ′ ( [ ]) ( F f = V w ⋅ f ∈ Wm+1O nD for w ∈ Wm A . The part (3) is obvious. ([ ])) ( ) [ ] X,x( ) ( ) Lemma 6.2. For i ≥ 1, there is a short exact sequence i m−i i V R 0 → Wm−iOX p nD Ð→ WmOX nD ÐÐÐ→ WiOX nD → 0. ( ) ( ) ( ) Proof. Since R is a ring homomorphism on j∗WmOU on which it is surjective, it follows that it is surjective on WmOX ∗D . Since the sequence ( ) V i Rm−i (6.1) 0 → Wm−iOX Ð→ WmOX ÐÐÐ→ WiOX → 0 is anyway exact, and since V i and R are defined by Lemma 6.1, we only have to show i i i that V j∗Wm−iOU ∩ WmOX nD = V Wm−iOX p nD . Arguing by induction on i ≥ 1, it( suffices to check) this for( i =)1. ( ( )) Now, we let w = a1, . . . , am−1 ∈ j∗Wm−1OU be such that V w = 0, a1, . . . , am−1 = −n ( −n −)pn −pm−1n ( ) ( ) b1, . . . , bm ⋅ f = b1f , b2f , . . . , bmf , where bi ∈ OX . This implies that ( ) [ ] −(pin ) −pn −pm−1n b1 = 0 and ai = bi+1f for i ≥ 1. Hence, w = b2f , . . . , bmf = b2, . . . , bm ⋅ −pn ( ) ( ) f ∈ Wm−1OX pnD . [ ] ( ) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 23

We shall now extend Lemma 6.1 to the de Rham-Witt forms of higher degrees. One knows using the Gersten conjecture for the de Rham-Witt sheaves due to Gros [12] that r r r the map WmΩX → WmΩK is injective. It follows that the canonical map WmΩX → r j∗WmΩU is injective too. Using the invertibility of the WmOX -module WmOX nD , we get an injection ( ) r r r (6.2) WmΩ nD WmOX nD ⊗W WmΩ / WmOX nD ⊗W j WmΩ X ❙v mOX X mOX ∗ U ( )❙❙❙❙ ( ) ( ) ❙❙❙❙ ❙❙❙❙ ❙❙) r r j∗WmΩ WmOX nD ⋅ j∗WmΩ . U ( ) U Lemma 6.3. Let q ≥ 1 be a positive integer. Then the inclusions WmOX qnD ↪ ( ) WmOX q n + 1 D induce an isomorphism ( ( ) ) ≅ lim W O qnD Ð→ j W O . Ð→ m X ∗ m U n≥1 ( ) Proof. We shall prove the lemma by induction on m. For m = 1, the statement is clear. We now assume m ≥ 2 and use the commutative diagram

V Rm−1 (6.3) 0 / lim W O qpnD / lim W O qnD / lim W O qnD / 0 Ð→ m−1 X Ð→ m X Ð→ 1 X n≥1 ( ) n≥1 ( ) n≥1 ( )

m−1 / V / R / / 0 / j∗Wm−1OU / j∗WmOU / j∗W1OU / 0. It is easy to check that the bottom row is exact. The top row is exact by Lemma 6.2. The right vertical arrow is an isomorphism by m = 1 case and the left vertical arrow is an isomorphism by induction on m (with q replaced by qp). The lemma follows. r r Corollary 6.4. For q ≥ 1, the canonical map WmΩX qnD → WmΩX q n + 1 D is injective and the map lim W Ωr qnD → j W Ωr is an( isomorphism) of(W( O )-modules.) Ð→ m X ∗ m U m X n≥1 ( ) Proof. The injectivity claim follows from (6.2). To prove the isomorphism, we take the r tensor product with the WmOX -module WmΩX on the two sides of the isomorphism in Lemma 6.3. This yields an isomorphism lim W Ωr qnD ≅ j W O ⊗ W Ωr . Ð→ m X ∗ m U WmOX m X n≥1 ( ) ( ) On the other hand, the ´etale descent property (see [20, Proposition I.1.14]) of the de r r Rham-Witt sheaves says that the canonical map j∗WmOU ⊗WmOX WmΩX → j∗WmΩU is an isomorphism. The corollary follows. ( ) r In view of Corollary 6.4, we shall hereafter consider the sheaves WmΩX nD as r ( ) WmOX -submodules of j∗WmΩU . r Lemma 6.5. The V, F and R operators of j∗WmΩU satisfy the following. r r (1) F WmΩX nD ⊆ Wm−1ΩX pnD . ( r ( )) r( ) (2) V WmΩX pnD ⊆ Wm+1ΩX nD . ( r ( )) r ( ) (3) R WmΩ nD ⊆ WmΩ nD . ( X ( )) X ( ) −n −n r r Proof. We have F f ω = F ω F f ∈ Wm−1ΩX ⋅Wm−1OX pnD = Wm−1ΩX pnD , where the ﬁrst inclusion([ ] holds) ( by) Lemma([ ]) 6.1. We have V f −pn( ω =)V F f −n (ω = ) −n r r ([ ] ) ( ([ ]) ) f V ω ∈ Wm+1OX nD ⋅Wm+1ΩX = Wm+1ΩX nD , where the second inclusion holds [again] by( Lemma) 6.1. The( ) last assertion is clear. ( ) r r Lemma 6.6. The multiplication by p map p∶ Wm+1ΩX nD → Wm+1ΩX nD has a factorization ( ) ( ) p r R r r Wm+1Ω nD ↠ WmΩ nD ↪ Wm+1Ω nD X ( ) X ( ) X ( ) 24 RAHULGUPTA,AMALENDUKRISHNA in the category of sheaves of Wm+1OX -modules. This factorization is natural in X.

Proof. Since R is Wm+1OX -linear, we get an exact sequence (see § 5.1) (6.4) m r m r−1 r R r 0 → V W1Ω + dV W1Ω ⋅ Wm+1OX nD → Wm+1Ω nD Ð→ WmΩ nD → 0. ( X X ) ( ) X ( ) X ( ) r r On the other hand, we note that p∶ Wm+1ΩX → Wm+1ΩX is also a Wm+1OX -linear homomorphism. Since Wm+1OX nD is an invertible Wm+1OX -module, we get an exact sequence ( ) r p r (6.5) 0 → Ker p ⋅ Wm+1OX nD → Wm+1Ω nD Ð→ Wm+1Ω nD . ( ) ( ) X ( ) X ( ) Since Ker p = Ker R as X is regular (see § 5.1), we get Ker p ⋅ Wm+1OX nD = m r( ) m ( )r−1 ( ) ( ) V W1ΩX + dV W1ΩX ⋅ Wm+1OX nD . The ﬁrst part of the lemma now follows. (The naturality is clear from) the proof.( ) Lemma 6.7. The square p r / r (6.6) WmΩX Wm+1ΩX

j∗ j∗ p r / r j∗WmΩU j∗Wm+1ΩU is Cartesian. r r Proof. We can check this locally. So let x ∈ j∗WmΩ be such that p x ∈ Wm+1Ω . Since U ( ) X r r j∗R r j is affine and WmΩX is an WmOX -module, it follows that j∗Wm+1ΩU ÐÐ→ j∗WmΩU r is surjective. We can therefore findx ˜ ∈ j∗Wm+1ΩU such that R x˜ = x. In particular, r r ( ) px˜ = p x ∈ Wm+1Ω . We thus get V F x˜ = px˜ ∈ Wm+1Ω . Since ( ) X ( ) X r r m r r m r r j∗VWmΩ ∩ Wm+1Ω = Ker F ∶ Wm+1Ω → j∗Ω = Ker F ∶ Wm+1Ω → Ω ( U ) X ( X U ) ( X X ) r = VWmΩX , ′ r ′ it follows that there exists y ∈ WmΩX such that V F x˜ = Vy . r r m (r−)1 Since Ker V ∶ j∗WmΩU → j∗Wm+1ΩU = j∗F dV ΩU (see [20, I.3.21.1.4]), it follows ( ′ r−1 ) m ′ ′ that there exists z ∈ j∗ΩU such that F dV z = F x˜ − y . Equivalently, F x˜ − m ′ ′ r ( ) ( ) ( dV z = y ∈ WmΩ . Since ( )) X r r m−1 r r+1 m−1 r r+1 j∗FWm+1Ω ∩ WmΩ = Ker F d∶ WmΩ → j∗Ω = Ker F d∶ WmΩ → Ω U X ( X U ) ( X X ) r = FWm+1ΩX , ′′ r m ′ ′′ we can find y ∈ Wm+1ΩX such that F x˜ − dV z = F y . r r( (m ))r ( ) Since Ker F ∶ j∗Wm+1ΩU → j∗WmΩU = j∗V ΩU (see [20, I.3.21.1.2]), we can find ′′ r ( m ′′ m )′ ′′ ′′ m ′′ m ′ z ∈ j∗ΩU such that V z = x˜−dV z −y . Equivalently,x ˜−y = V z +dV z . On the other hand, we have( ) ( ) ( ) ( ) m r m r−1 r r V j∗Ω + dV j∗Ω = Ker R∶ j∗Wm+1Ω → j∗WmΩ . U U ( U U ) ′′ ′′ r r We thus get x = R x˜ = R y . Since y ∈ Wm+1ΩX , we get x ∈ WmΩX . This proves the lemma. ( ) ( ) r r+1 It is easy to see that for n ∈ Z, the differential d∶ j∗WmΩX → j∗WmΩX restricts r r+1 to a homomorphism d∶ WmΩX nD → WmΩX n + 1 D such that the composite map 2 r r+2 ( ) (( ) ) r m d ∶ WmΩX nD → WmΩX n+2 D is zero by Corollary 6.4. The map d∶ WmΩX p nD → r+1 ( m ) (( ) ) r m r+1 m( ) WmΩX p n + 1 D actually factors through d∶ WmΩX p nD → WmΩX p nD as (( ) ) ● m ( ) ( ) one easily checks. In particular, WmΩ p nD is a complex for every m ≥ 1 and n ∈ Z. X ( ) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 25

Let r r r m−1 r Z1WmΩ nD = j∗Z1WmΩ ∩ WmΩ nD = j∗Ker F d ∩ WmΩ nD X ( ) ( U ) X ( ) ( ) X ( ) m−1 r m−1 r r+1 m−1 = Ker F d∶ WmΩ nD → j∗ΩU = Ker F d∶ WmΩ nD → Ω p n + 1 D , ( X ( ) ) ( X ( ) X ( ( ) )) where the third equality is by the left exactness of j∗ and the last equality by Lemma 6.5. If m = 1, we get r r r r+1 (6.7) Z1W1Ω nD = ZΩ nD = Ker d∶ Ω nD → Ω n + 1 D . X ( ) X ( ) ( X ( ) X (( ) )) We let BΩr nD = Image d∶ Ωr n − 1 D → Ωr nD . X( ) ( X (( ) ) X ( )) r r Proposition 6.8. There exists a homomorphism C∶ Z1WmΩX pnD → WmΩX nD such that the diagram ( ) ( ) r V / r (6.8) Z1WmΩ pnD / Wm+1Ω nD X ❖ 7 X ( ) ( ❖❖❖) ♣♣♣ ❖❖❖ ♣♣♣ ❖❖ ♣♣ p C ❖❖' ♣♣♣ r WmΩ nD X ( ) is commutative. The map C induces an isomorphism of OX -modules r ● ≅ r C∶ H ψ∗W1Ω pnD Ð→ W1Ω nD . ( X ( )) X ( ) Proof. We consider the diagram V

p & r / r / r (6.9) Z1WmΩ pnD / WmΩ nD / Wm+1Ω nD X ( ) X ( ) X ( )

p r C / r / r j∗Z1WmΩ / j∗WmΩ / j∗Wm+1Ω , U U 8 U V where the vertical arrows are inclusions from Corollary 6.4. The right square exists and commutes by Lemma 6.6. The big outer square clearly commutes. It suﬃces therefore to show that the right square is Cartesian. We have a commutative diagram of Wm+1OX -modules r p / r / r / (6.10) Wm+1Ω nD / Wm+1Ω nD / Wm+1Ω nD p / 0 X ( ) X ( ) X ( )~

r p // r / r / j∗Wm+1Ω // j∗Wm+1Ω / j∗Wm+1Ω p / 0. U U U ~ The top sequence is clearly exact and the bottom sequence is exact by the Serre vanishing because all sheaves are Wm+1OX -modules and j is an aﬃne morphism. By Lemma 6.6, it suﬃces to show that the right vertical arrow is injective. To prove this injectivity, we note that the top row of (6.10) is same as the sequence

r p⊗1 r r Wm+1Ω ⊗O O nD ÐÐ→ Wm+1Ω ⊗O O nD → Wm+1Ω p ⊗O O nD → 0, X ( ) X ( ) X ~ ( ) where O = Wm+1OX . Similarly, we have r r ∗ j∗Wm+1ΩU p ⊗O O nD ≅ j∗ Wm+1ΩU p ⊗Wm+1OU j O nD ≅ ~ r ( ) ( ~ r ( )) j∗ Wm+1Ω p ⊗W 1O Wm+1OU ≅ j∗Wm+1Ω p, ( U ~ m+ U ) U ~ where the ﬁrst isomorphism follows from the projection formula for O-modules using the fact that O nD is an invertible O-module (see [17, Exercise II.5.1]). ( ) 26 RAHULGUPTA,AMALENDUKRISHNA

Moreover, it is clear that the right vertical arrow in (6.10) is the map

∗ r j ⊗1 r Wm+1Ω p ⊗O O nD ÐÐ→ j∗Wm+1Ω p ⊗O O nD . X ~ ( ) ( U ~ ) ( ) ∗ r Since O nD is an invertible O-module, it suﬃces to show that the map j ∶ Wm+1ΩX p → ( r ) ∗ ~ j∗Wm+1ΩU p is injective. But this follows from Lemma 6.7 since j in (6.6) is injective. This proves~ the ﬁrst part of the lemma. We now prove the second part for which we can assume m = 1. We know classically r r ● ≅ r that the map C on ZΩ induces an OX -linear isomorphism C∶ H ψ∗Ω Ð→ Ω . Taking X ( X ) X its inverse, we get an OX -linear isomorphism

−1 r ≅ r ● C ∶ Ω Ð→H ψ∗Ω . X ( X ) ● + r ● Since ψ∗ΩX ∈ D CohX , we see that H ψ∗ΩX is a coherent OX -module. In particular, we get an isomorphism( ) ( )

−1 r ≅ r ● C ∶ Ω nD Ð→H ψ∗Ω nD . X ( ) ( X )( ) On the other hand, we have r ● r ● r ● ∗ r ● H ψ∗Ω nD ≅ H ψ∗Ω nD ≅ H ψ∗ Ω ⊗O ψ O nD ≅ H ψ∗ Ω pnD . ( X )( ) (( X )( )) ( ( X X ( ( )))) ( ( X ( ))) This proves the second part. Lemma 6.9. We have the following. m−1 r r m−1 r (1) Ker F ∶ WmΩX nD → ΩX p nD = VWm−1ΩX pnD . ( r ( ) r ( )) ( ) (2) Z1WmΩ pnD = FWm+1Ω nD . X ( ) X ( ) Proof. We first prove (1). It is clear that the right hand side is contained in the left hand side. We prove the other inclusion. It suffices to show that r r r j∗VWm−1Ω ∩ WmΩ nD ⊂ VWm−1Ω pnD . ( U ) X ( ) X ( ) −n We can check this locally. So let D be defined by f ∈ OX and let y = Vx = f ω, r r n n[ ] where x ∈ j∗Wm−1ΩU and ω ∈ WmΩX . This yields ω = f Vx = V F f x = pn r r [ ] ( ([ ]) ) V f x . This implies that ω ∈ WmΩ ∩ j∗VWm−1Ω . On the other hand, we have ([ ] ) X U r r m−1 r r m−1 r r WmΩ ∩ j∗VWm−1Ω = Ker F ∶ WmΩ → j∗Ω = Ker F ∶ WmΩ → Ω X U ( X U ) ( X X ) r = VWm−1ΩX . r ′ r ′ We thus get ω ∈ VWm−1ΩX . Let y ∈ Wm−1ΩX be such that ω = Vy . This yields −n −n ′ −n ′ −pn ′ r y = f ω = f Vy = V F f y = V f y ∈ VWm−1ΩX pnD . This proves (1).[ ] [ ] ( ([ ]) ) ([ ] ) ( ) r r r We now prove (2). Since FWm+1ΩX nD ⊂ j∗FWm+1ΩU ∩WmΩX pnD by Lemma 6.5, r ( r) ( ) it follows that FWm+1ΩX nD ⊂ Z1WmΩX pnD . We show the other inclusion. r ( ) ( ) −pn r We let z ∈ Z1WmΩX pnD so that z = Fx = f ω for some x ∈ j∗Wm+1ΩU and r ( ) pn [ n ] r ω ∈ WmΩX . We can then write ω = f Fx = F f x . This implies that ω ∈ WmΩX ∩ r [ ] ([ ] ) j∗FWm+1ΩU . On the other hand, we have r r m−1 r r+1 m−1 r r+1 WmΩ ∩ j∗FWm+1Ω = Ker F d∶ WmΩ → j∗Ω = Ker F d∶ WmΩ → Ω X U ( X U ) ( X X ) r = FWm+1ΩX . ′ ′ r −pn −pn ′ We can thus write ω = Fx for some x ∈ Wm+1ΩX . This gives z = f ω = f Fx = −n ′ −n ′ r [ ] [ ] F f Fx = F f x ∈ FWm+1Ω nD . This proves (2). ([ ]) ([ ] ) X ( ) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 27

7. The pairing of cohomology groups We fix a finite field k of characteristic p and an integral smooth projective scheme X of dimension d ≥ 1 over k. Let D ⊂ X be an effective Cartier divisor with complement U. Let ι∶ D ↪ X and j∶ U ↪ X be the inclusions. In this section, we shall establish the pairing for our duality theorem for the p-adic étale cohomology of U. We fix integers m,n ≥ 1 and r ≥ 0. We shall consider the étale topology throughout our discussion of duality. 7.1. The complexes. We consider the complex of étale sheaves

r,● r 1−C r (7.1) WmF = Z1WmΩ nD ÐÐ→ WmΩ nD . n [ X ( ) X ( )] The differential of this complex is induced by the composition r r C r Z1WmΩ nD ↪ Z1WmΩ pnD Ð→ WmΩ nD , X ( ) X ( ) X ( ) where the last map is defined by virtue of Proposition 6.8. r r We now consider the map F ∶ Wm+1ΩX −nD → WmΩX −pnD whose existence is ( ) ( ) m r shown in Lemma 6.5. We have shown in (5.2) that F Ker R = F V W1ΩX + m r−1 r m−1 r−1 r ( ( )) (( dV W1ΩX ∩ Wm+1ΩX −nD ⊆ dV ΩX ∩ WmΩX −pnD . It follows that F in- ) r ( )) r m(−1 r−1 ) r duces a map F ∶ WmΩX −nD → WmΩX −pnD dV ΩX ∩ WmΩX −pnD . We denote the composition ( ) ( )~( ( )) W Ωr −pnD W Ωr −nD W r nD m X m X mΩX − → m−1 r−1 ( r ) ↪ m−1 r−1 ( r) ( ) dV Ω ∩ WmΩ −pnD dV Ω ∩ WmΩ −nD X X ( ) X X ( ) also by F and consider the complex of étale sheaves r π−F W Ω −nD W r,● W r nD m X , (7.2) mGn = mΩX − ÐÐ→ m−1 r−1 ( r) ( ) dV Ω ∩ WmΩ −nD X X ( ) where π is the quotient map. We let

d,● d 1−C d (7.3) WmH = WmΩ ÐÐ→ WmΩ , [ X X ] d d where the map C (see Proposition 6.8 for n = 0) is defined because Z1WmΩX = WmΩX . 7.2. The pairing of complexes. We consider the pairing r d−r ∧ d (7.4) ⟨, ⟩1∶ Z1WmΩX (nD)× WmΩX (−nD) Ð→ WmΩX by letting ⟨w1, w2⟩1 = w1 ∧ w2. This is defined by the definition of the twisted de Rham- Witt sheaves. r d−r d We define a pairing ⟨, ⟩2∶ Z1WmΩX (nD)× WmΩX (−nD) → WmΩX by ⟨w1, w2⟩2 = m−1 d−r−1 d−r −C(w1 ∧ w2). We claim that C(w1 ∧ w2) = 0 if w2 ∈ dV ΩX ∩ WmΩX (−nD). ′ ′ r To prove it, we write w1 = F (w1) for some w1 ∈ j∗Wm+1ΩU . This gives V (w1 ∧ ∗ ′ ∗ ′ ∗ j w2) = V (F (w1)∧ j w2) = w1 ∧ j V w2 = 0, where the last equality holds because m−1 d−r−1 m d−r−1 d−r−1 ∗ V dV ΩX ⊂ pdV ΩX and pΩX = 0. In particular, j (w1 ∧ w2) ∈ Ker(V ∶ d d m d−1 m−1 d−1 WmΩU → Wm+1ΩU ) = F dV ΩU = dV ΩU = Ker(CU ), where CU is the Cartier ∗ map for U (see [21, Chapitre III] for the last equality). It follows that j ○ C(w1 ∧ w2) = ∗ d d CU ○ j (w1 ∧ w2) = 0. Since WmΩX ↪ j∗WmΩU , the claim follows. Using the claim, we get a pairing d−r WmΩ (−nD) (7.5) ⟨, ⟩ ∶ Z W Ωr (nD)× X → W Ωd . 2 1 m X m−1 d−r−1 d−r m X dV ΩX ∩ WmΩX (−nD) We define our third pairing of étale sheaves r d−r d (7.6) ⟨, ⟩3∶ WmΩX (nD)× WmΩX (−nD) → WmΩX 28 RAHULGUPTA,AMALENDUKRISHNA by ⟨w1, w2⟩3 = w1 ∧ w2. Lemma 7.1. The above pairings of étale sheaves give rise to a pairing of two-term complexes of sheaves r,● d−r,● d,● (7.7) ⟨, ⟩∶ WmFn × WmGn → WmH . Proof. By [38, § 1, p.175], we only have to show that

(7.8) (1 − C)(w1 ∧ w2) = (1 − C)(w1)∧ w2 − C(w1 ∧ (π − F )w2) r r for all w1 ∈ Z1WmΩX (nD) and w2 ∈ WmΩX (−nD). But this follows from the equalities

(1 − C)(w1 ∧ w2) = w1 ∧ w2 − C(w1 ∧ w2) = w1 ∧ w2 − C(w1)∧ C(w2) = w1 ∧ w2 − C(w1)∧ w2 + C(w1)∧ w2 − C(w1)∧ C(w2) = (1 − C)(w1)∧ w2 − C(w1) ∧ (C − 1)(w2) † = (1 − C)(w1)∧ w2 − C(w1 ∧ (π − F )(w2)), where =† holds because C ○ F = id (see [44, Proposition 1.1.4]).

7.3. The pairing of ´etale cohomology. Using Lemma 7.1, we get a pairing of hypercohomology groups i r,● d+1−i d−r,● d+1 d,● (7.9) Het´ (X,WmFn )× Het´ (X,WmGn ) → Het´ (X,WmH ). By [44, Proposition 1.1.7] and [39, Corollary 1.12], there is a quasi-isomorphism d ≅ d,● d+1 d ≅ WmΩX,log Ð→ WmH , and a bijective trace homomorphism Tr∶ Het´ (X,WmΩX,log) Ð→ Z~pm. We thus get a pairing of Z~pm-modules i r,● d+1−i d−r,● m (7.10) Het´ (X,WmFn )× Het´ (X,WmGn ) → Z~p . Since this pairing is compatible with the change in values of m and n, we get a pairing of ind-abelian (in ﬁrst coordinate) and pro-abelian (in second coordinate) groups i r,● d+1−i d−r,● m (7.11) {Het´ (X,WmFn )}n × {Het´ (X,WmGn )}n → Z~p .

r,● ≅ r 1−C r It follows from Corollary 6.4 that lim WmFn Ð→ j∗ Z1WmΩ ÐÐ→ WmΩ . Since Ð→n ([ U U ]) j is aﬃne, we get

r,● ≅ r 1−C r r (7.12) lim WmF Ð→ Rj∗ Z1WmΩ ÐÐ→ WmΩ ≅ Rj∗ WmΩ . Ð→n n ([ U U ]) ( U,log) r,● In order to understand the pro-complex WmGn n, we let { } r ⎡ π−F WmΩ(X,nD) ⎤ W Gr,● = ⎢W Ωr ÐÐ→ ⎥ . m ̃n ⎢ m (X,nD) m−1 r−1 r ⎥ ⎢ dV ΩX ∩ WmΩ X,nD ⎥ ⎢ ( ) ⎥ ⎣ ⎦ This complex is deﬁned by the exact sequence (5.3). It follows then from Corollary 5.3 that the canonical inclusion W Gr,● ↪ W Gr,● is an isomorphism of pro-complexes m n n m ̃n n of sheaves. Using the exact{ sequence} (5.3){ and Lemma} 5.4, we get that there is a canonical isomorphism of pro-complexes

M m ≅ r ≅ r,● (7.13) dlog∶ K p n Ð→ WmΩ n Ð→ WmG n. { ̂r,(X,nD)~ } { (X,nD),log} { n } We conclude that (7.11) is equivalent to the pairing of ind-abelian (in ﬁrst coordinate) and pro-abelian (in second coordinate) groups i r,● d+1−i d−r m (7.14) H X,WmF n × H X,WmΩ n → Z p . { et´ ( n )} { et´ ( (X,nD),log)} ~ RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 29

7.4. Continuity of the pairing. Recall from § 5.3 that for a Z pm-module V , one ⋆ m ~ has V = HomZ~pm V, Z p . For any proﬁnite or discrete torsion abelian group A, let ∨ ( ~ ) A = Homcont A, Q Z denote the Pontryagin dual of A with the compact-open topology (see [15, § 7.4]).( If A~ is) discrete and pm-torsion, then we have A∨ = A⋆. We shall use the following topological results.

Lemma 7.2. Let An n be a direct system of discrete torsion topological abelian groups whose limit is A with{ the} direct limit topology. Then the canonical map λ∶ A∨ → lim A∨ ←Ðn n is an isomorphism of profinite groups. Proof. The fact that λ is an isomorphism of abelian groups is well known and elementary. We show that λ is a homeomorphism. Since its source and targets are compact Hausdorff, it suffices to show that it is continuous. We let Bn = Image An → A . Then, we have a ( ) surjection of finite ind-groups An n ↠ Bn n whose limits coincides with A. We thus λ′ λ′′ { } { } get maps A∨ Ð→ lim B∨ Ð→ lim A∨ such that λ = λ′′ ○ λ′. Since λ′′ is clearly continuous, n n n n ←Ð ←Ð′ we have to only show that λ is continuous. We can therefore assume that each An is a subgroup of A. Lemma now follows from [41, Lemma 2.9.3 and Theorem 2.9.6].

Lemma 7.3. Let An n be an inverse system of discrete topological abelian groups. Let { } ∨ A be the limit of An n with the inverse limit topology. Then any f ∈ A factors through { } η the projection λ ∶ A → A for some n ≥ 1. In particular, the canonical map lim A∨ Ð→ A∨ n n Ð→n n is continuous and surjective. This map is an isomorphism if the transition maps of An n are surjective. { } Proof. Since f∶ A → Q Z is continuous and its target is discrete, it follows that Ker f ~ ( ) is open. By the deﬁnition of the inverse limit topology, the latter contains Ker λn for ′ ′ ′ ( ) some n ≥ 1. Letting An = A Ker λn , this implies that f factors through fn∶ An → Q Z. ~ ( ∨ ) ′ ∨ ~ Since An is discrete, the map An → A is surjective by [15, Lemma 7.10]. Choosing ( ) ( n) ′ λn fn a lift fn of fn under this surjection, we see that f factors through A Ð→ An Ð→ Q Z. ∨ ∨ ~ The continuity of η is equivalent to the assertion that the map An → A is continuous for each n. But this is well known since λn is continuous. The remaining parts of the lemma are now obvious. Remark 7.4. The reader should note that the map lim A∨ → A∨ may not in general be Ð→n n injective even if each An is ﬁnite.

i r,● i d−r For n ≥ 1, we endow each of Het´ X,WmFn and Het´ X,WmΩ(X,nD),log with the i ( d−)r ( ) discrete topology. We endow lim H X,WmΩ with the inverse limit topology ←Ðn et´ (X,nD),log i r,● i ( r ) and lim H X,WmFn ≅ H U, WmΩ with the direct limit topology. Note that Ð→n et´ ( ) et´ ( U,log) the latter topology is discrete. i r,● ′ If we let x ∈ lim H X,WmFn , then we can find some n ≫ 0 such that x = fn x Ð→n et´ ′ i ( r,● ) i r,● i r,● ( ) for some x ∈ H X,WmFn , where fn∶ H X,WmFn → lim H X,WmFn is the et´ ( ) et´ ( ) Ð→n et´ ( ) canonical map to the limit. This gives a map d+1−i d−r m (7.15) ⟨x, ⋅⟩∶ lim H (X,WmΩ ) → Z~p ←Ðn et´ (X,nD),log which sends y to ⟨x′,π(y)⟩ under the pairing (7.10), where π is the composite map d+1−i d−r d+1−i d−r,● gn d+1−i d−r,● lim H (X,WmΩ ) ≅ lim H (X,WmG ) Ð→ H (X,WmG ). ←Ðn et´ (X,nD),log ←Ðn et´ n et´ n i r ∨ One checks that (7.15) is defined. Since Het´ (U, WmΩU,log) is profinite, this shows that the map (see Lemma 7.2) d+1−i d−r i r ∨ (7.16) θm∶ lim H (X,WmΩ ) → H (U, WmΩ ) ←Ðn et´ (X,nD),log et´ U,log 30 RAHULGUPTA,AMALENDUKRISHNA

i r is continuous. Since Het´ (U, WmΩU,log) is discrete, the map i r d+1−i d−r ∨ (7.17) ϑm∶ H (U, WmΩ ) → (lim H (X,WmΩ )) et´ U,log ←Ðn et´ (X,nD),log is clearly continuous. We have thus shown that after taking the limits, (7.14) gives rise to a continuous pairing of topological abelian groups i r d+1−i d−r H (U, WmΩ )× lim H (X,WmΩ ) → Q~Z. et´ U,log ←Ðn et´ (X,nD),log d+1−i d−r m Since each Het´ (X,WmΩ(X,nD),log) is p -torsion and discrete, we get the following. Proposition 7.5. There is a continuous pairing of topological abelian groups (7.18) Hi (U, W Ωr )× lim Hd+1−i(X,W Ωd−r ) → Z~pm. et´ m U,log ←Ðn et´ m (X,nD),log Equivalently, there is a continuous pairing of topological abelian groups

i r d+1−i M m m (7.19) H (U, WmΩ )× lim H (X, K̂ ~p ) → Z~p . et´ U,log ←Ðn et´ d−r,(X,nD) These pairings are compatible with respect to the canonical surjection Z~pm ↠ Z~pm−1 p and the inclusion Z~pm−1 ↪ Z~pm. It follows from (7.14) that the map ϑm in (7.17) has a factorization θ′ ϑ′ (7.20) Hi (U, W Ωr ) Ð→m lim Hd+1−i(X,W Ωd−r )∨ ↠m et´ m U,log Ð→n et´ m (X,nD),log d+1−i d−r ∨ (lim H (X,WmΩ )) , ←Ðn et´ (X,nD),log ′ where ϑm is the canonical map. This map is continuous and surjective by Lemma 7.3. ′ The map θm is continuous since its source is discrete. Our goal in the next section will ′ be to show that the maps θm and θm are isomorphisms. 8. Perfectness of the pairing We continue with the setting of § 7. Our goal in this section is to prove a perfectness statement for the pairing (7.14). To make our statement precise, it is convenient to have the following deﬁnition. m Deﬁnition 8.1. Let {An}n and {Bn}n be ind-system and pro-system of p -torsion discrete abelian groups, respectively. Suppose that there is a pairing m (⋆) {An}n × {Bn}n → Z~p . We shall say that this pairing is continuous and semi-perfect if the induced maps θ∶ lim B → (lim A )∨ and θ′∶ lim A → lim B∨ ←Ðn n Ð→n n Ð→n n Ð→n n are continuous and bijective homomorphisms of topological abelian groups. Recall here that (lim A )∨ ≅ lim A∨ by Lemma 7.2. We shall say that (⋆) is perfect if it is semi- n n n n Ð→ ←Ð ∨ ∨ perfect, the surjective map (see Lemma 7.3) lim B → (lim Bn) is bijective and θ is a Ð→n n ←Ðn homeomorphism. Note that perfectness implies that θ′ is also a homeomorphism. The following is easy to check.

Lemma 8.2. The pairing (⋆) is perfect if it is semi-perfect and {Bn}n is isomorphic to a surjective inverse system of compact groups. Our goal is to show that (7.14) is semi-perfect by induction on m ≥ 1. We ﬁrst consider ● m the case m = 1. We shall prove this case using our earlier observation that WmΩX (p nD) is a complex for every m ≥ 1 and n ∈ Z. r r r+1 In this case, we have Z1ΩX (pnD) = Ker(d∶ ΩX (pnD) → ΩX (pnD)) by (6.7). We r−1 r r−1 claim that the inclusion dΩX (−pnD) ↪ ΩX (−pnD)∩ dΩX is actually a bijection when RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 31

r r−1 r r−1 n ≥ 1. Equivalently, the map ΩX (−pnD)~dΩX (−pnD) → ΩX ~dΩX is injective. Since ψ is identity on the topological space X, the latter injectivity is equivalent to showing that the map r r r (ψ∗ΩX )(−nD) ψ∗ΩX (−pnD) ψ∗ΩX (8.1) r−1 ≅ r−1 → r−1 d(ψ∗ΩX )(−nD) dψ∗ΩX (−pnD) dψ∗ΩX r−1 is injective. Using a snake lemma argument, it suﬃces to show that the map ψ∗Z1ΩX → r−1 ψ∗Z1ΩX ⊗OX OnD is surjective. But this is obvious. r r r−1 Lemma 8.3. For n ∈ Z, the sheaves ψ∗(Z1ΩX (pnD)) and ψ∗(ΩX (pnD)~dΩX (pnD)) are locally free OX -modules. For n ≥ 1, the map r d−r d−r−1 d ψ∗(Z1ΩX (pnD)) →HomOX (ψ∗(ΩX (−pnD)~dΩX (−pnD)), ΩX ), induced by (7.5), is an isomorphism. Proof. We ﬁrst note that r r r+1 ψ∗(Z1ΩX (pnD)) ≅ ψ∗(Ker(ΩX (pnD) → ΩX (pnD))) r r+1 ≅ Ker(ψ∗ΩX (pnD) → ψ∗ΩX (pnD)) r r+1 (8.2) ≅ Ker((ψ∗ΩX )(nD) → (ψ∗ΩX )(nD)) r r+1 ≅ (Ker(ψ∗ΩX → ψ∗ΩX ))(nD) r ≅ (ψ∗Z1ΩX )(nD).

Since ψ∗ also commutes with Coker(d), we similarly get r r−1 r r−1 (8.3) ψ∗(ΩX (pnD)~dΩX (pnD)) ≅ (ψ∗ΩX ~dψ∗ΩX )(nD). In the exact sequence

r r d r+1 r+1 r 0 → ψ∗Z1ΩX → ψ∗ΩX Ð→ ψ∗ΩX → ψ∗ΩX ~dψ∗ΩX → 0 of OX -linear maps between coherent OX -modules, all terms are locally free by [38, Lemma 1.7]. It therefore remains an exact sequence of locally free OX -modules after tensoring with OX (nD) for any n ∈ Z. The first part of the lemma now follows by using (8.2) and (8.3) (for different values of r). To prove the second part, we can again use (8.2) and (8.3). Since OX (nD) is invertible, r d−r d−r−1 d it suffices to show that (ψ∗Z1ΩX )(nD) → HomOX (ψ∗ΩX ~dψ∗ΩX , ΩX (nD)) is an isomorphism. But the term on the right side of this map is isomorphic to the sheaf d−r d−r−1 d HomOX (ψ∗ΩX ~dψ∗ΩX , ΩX )(nD) via the canonical isomorphism ≅ HomOX (A, B)⊗OX E Ð→HomOX (A, B ⊗OX E), which locally sends (f ⊗ b) → (a ↦ f(a)⊗ b) if E is locally free. We therefore have to r d−r d−r−1 d only show that the map ψ∗Z1ΩX →HomOX (ψ∗ΩX ~dψ∗ΩX , ΩX ) is an isomorphism. But this follows from [38, Lemma 1.7]. i r,● i r Lemma 8.4. The groups Het´ (X,W1Fn ) and Het´ (X, Ω(X,nD),log) are finite for i, r ≥ 0. i r Proof. Using (5.3), the finiteness of Het´ (X, Ω(X,nD),log) is reduced to showing that the Ωr r i (X,nD) i ΩZ group Het´ (X, Ωr dΩr−1 ) is finite. By (5.2), it suffices to show that Het´ (Z, dΩr−1 ) is (X,nD)∩ X Z finite if Z ∈ {X,nD}. Since k is perfect, the absolute Frobenius ψ is a finite morphism. r i ψ∗ΩZ In particular, ψ∗ is exact. It suffices therefore to show that Het´ (Z, r−1 ) is finite. But ψ∗dΩZ r r−1 this is clear because k is finite, Z is projective over k and ψ∗ΩZ ~ψ∗dΩZ is coherent. i r,● r The finiteness of Het´ (X,W1Fn ) follows by a similar argument because ψ∗(Z1ΩX (nD)) is coherent. 32 RAHULGUPTA,AMALENDUKRISHNA

Lemma 8.5. The pairing i r,● d+1−i d−r,● {Het´ (X,W1Fn )}n × {Het´ (X,W1Gn )}n → Z~p is a perfect pairing of the ind-abelian and pro-abelian finite groups. Proof. The finiteness follows from Lemma 8.4. For perfectness, it suffices to prove that without using the pro-systems, the pairing in question is a perfect pairing of finite abelian groups if we replace n by pn. We shall show the latter. For an Fp-vector space V , we let ⋆ V = HomFp (V, Fp). Using the definitions of various pairings of sheaves and complexes of sheaves, we have a commutative diagram of long exact sequences (8.4) / i−1 r / i−1 r / i r,● / ⋯ Het´ (X,Z1ΩX (pnD)) Het´ (X, ΩX (pnD)) Het´ (X,W1Fpn ) ⋯

Ωd−r(−pnD) / d+1−i X ⋆ / d+1−i d−r ⋆ / d+1−i d−r,● ⋆ / ⋯ Het´ (X, d−r−1 ) Het´ (X, ΩX (−pnD)) Het´ (X,W1Gpn ) ⋯. dΩX (−pnD) The exactness of the bottom row follows from Lemma 8.4 and [15, Lemma 7.10]. Using Lemma 8.3 and Grothendieck duality for the structure map X → Spec (Fp), the map Ωd−r(−pnD) Hi X, ψ Z r pnD Hd−i X, ψ X ⋆ et´ ( ∗( 1ΩX ( ))) → et´ ( ∗( d−r−1 )) dΩX (−pnD) is an isomorphism. Since ψ∗ is exact, it follows that the map Ωd−r(−pnD) Hi X,Z r pnD Hd−i X, X ⋆ (8.5) et´ ( 1ΩX ( )) → et´ ( d−r−1 ) dΩX (−pnD) is an isomorphism. By the same reason, the map i r d−i d−r ⋆ (8.6) Het´ (X, ΩX (pnD)) → Het´ (X, ΩX (−pnD)) is an isomorphism. Using (8.4), (8.5) and (8.6), we conclude that the right vertical arrow in (8.4) is an isomorphism. An identical argument shows that the map d+1−i d−r,● i r,● ⋆ Het´ (X,W1Gpn ) → Het´ (X,W1Fpn ) is an isomorphism. This finishes the proof of the perfectness of the pairing of the hypercohomology groups. We can now prove the main duality theorem of this paper. Theorem 8.6. Let k be a finite field and X a smooth and projective scheme of pure dimension d ≥ 1 over k. Let D ⊂ X be an effective Cartier divisor with complement U. Let m ≥ 1 and i, r ≥ 0 be integers. Then i r,● d+1−i d−r,● m {Het´ (X,WmFn )}n × {Het´ (X,WmGn )}n → Z~p is a semi-perfect pairing of ind-abelian and pro-abelian groups. Proof. We have shown already (see (7.18)) that the pairing is continuous after taking lim- ′ its. We need to show that the maps θm (see (7.16)) and θm (see (7.20)) are isomorphisms of abelian groups. We shall prove this by induction on m ≥ 1. We first assume m = 1. Then Lemma 8.5 implies that the map d+1−i d−r,● i r,● ∨ θ1∶ {Het´ (X,W1Gn )}n → {Het´ (X,W1Fn ) }n is an isomorphism of pro-abelian groups. Taking the limit and using (7.13), we get an isomorphism d+1−i ≅ i r,● ∨ θ1∶ F (U) Ð→ lim H (X,W1F ) . 1,r ←Ðn et´ n RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 33

i r ∨ But the term on the right is same as Het´ (U, W1ΩU,log) by (7.12) and Lemma 7.2. ′ Lemma 8.5 also implies that θ1 is an isomorphism. This proves m = 1 case of the theorem. j j We now assume m ≥ 2 and recall the deﬁnitions of Fm,r(n) and Fm,r(U) from § 5.3. We consider the commutative diagram

/ d+1−i / d+1−i / d+1−i / (8.7) ⋯ Fm−1,r(U) Fm,r (U) F1,r (U) ⋯

θm−1 θm θ1 m−1 ∨ ∨ (p ) / i r ∨ R / i r ∨ / i r ∨ / ⋯ Het´ (U, Wm−1ΩU,log) Het´ (U, WmΩU,log) Het´ (U, W1ΩU,log) ⋯. The top row is exact by Lemma 5.11. The bottom row is exact by Lemma 5.7 and [15, Lemma 7.10]. The maps θi are isomorphisms for i ≤ m − 1 by induction. We conclude that θm is also an isomorphism. An identical argument, where we apply Lemma 5.12 ′ instead of Lemma 5.11, shows that θm is an isomorphism. This ﬁnishes the proof. Remark 8.7. In [22, Theorem 4.1.4] and [44, Theorem 3.4.2], a pairing using a logarithmic d+1−r d−r version of the pro-system {Het´ (X,WmΩ(X,nD),log)}n is given under the assumption that Dred is a simple normal crossing divisor. The authors in these papers say that their pairing is perfect. Although they do not explain their interpretations of perfectness, what they actually prove is the semi-perfectness in the sense of Deﬁnition 8.1, according to our understanding.

Corollary 8.8. The pairing of Theorem 8.6 is perfect if Dred is a simple normal crossing divisor, i = 1, r = 0 and one of the conditions {d ≠ 2, k ≠ F2} holds. Proof. Combine Theorem 8.6, Lemma 8.2, Corollary 4.2, Proposition 2.7, [26, Theo- rem 9.1] and the equivalence of (7.18) and (7.19).

et´ d ̂M m We let CKS(X, D; m) = Het´ (X, Kd,(X,D)~p ). We shall study the following special case in the next section. Corollary 8.9. Under the assumptions of Theorem 8.6, the map et´ ab m θm∶ lim C (X,nD; m) → π (U)~p ←Ðn KS 1 is a bijective and continuous homomorphism between topological abelian groups. This is an isomorphism of topological groups under the assumptions of Corollary 8.8. 8.1. The comparison theorem. We shall continue with the assumptions of Theo- ab rem 8.6. We fix an integer m ≥ 1. Let π1 (U) be the abelianized étale fundamental adiv group of U and π1 (X, D) the co-1-skeleton étale fundamental group of X with modulus D, introduced in [15, Definition 7.5]. The latter characterizes finite abelian covers of U whose ramifications are bounded by D at each of its generic point, where the bound is given by means of Matsuda’s Artin conductor. There is a natural surjection ab adiv π1 (U) ↠ π1 (X, D). d M d M m Let CKS(X, D) = Hnis(X, Kd,(X,D)) and CKS(X, D; m) = Hnis(X, Kd,(X,D)~p ) ≅ m CKS(X, D)~p . By Proposition 2.7, we have

≅ d ̂M m (8.8) CKS(X, D; m) Ð→ Hnis(X, Kd,(X,D)~p ). m We let C̃ = lim CKS(X, D). The canonical map C̃ ~p → lim CKS(X, D; m) is U~X ←Ðn U~X ←Ðn an isomorphism by [16, Lemma 5.9]. The groups CKS(X, D) and CKS(X, D; m) have ̃ ̃ m the discrete topology while CU~X and CU~X ~p have the inverse limit topology. The 34 RAHULGUPTA,AMALENDUKRISHNA

ab adiv groups π1 (U) and π1 (X, D) have the proﬁnite topology. By [15, Theorem 1.2], there is a commutative diagram of continuous homomorphisms of topological abelian groups ρ ̃ U~X / ab (8.9) CU~X π1 (U)

ρ XSD/ adiv C(X, D) / π1 (X, D). The horizontal arrows are injective with dense images. They become isomorphisms after tensoring with Z~pm by [16, Corollary 5.15]. We let C̃et´ (m) = lim Cet´ (X,nD; m). By Corollary 8.9, we have a bijective and con- U~X ←Ðn KS et´ ̃et´ ab m tinuous homomorphism between topological abelian groups ρU~X ∶ CU~X (m) → π1 (U)~p ≅ 1 m ⋆ Het´ (U, Z~p ) . We therefore have a diagram ̃ m (8.10) CU~X ~p ■■ ■■ρU~X η ■■ ■■ ρet´ ■$ ̃et´ U~X/ ab m CU~X (m) π1 (U)~p of continuous homomorphisms, where η is the change of topology homomorphism. We wish to prove the following. Proposition 8.10. The diagram (8.10) is commutative.

Proof. It follows from [16, Theorem 1.2] that ρU~X is an isomorphism of profinite groups. cyc ρ U~X ̃ m U~X ab m Since the image of the composite map Z0(U) ÐÐÐÐ→ CU~X ~p ÐÐÐ→ π1 (U)~p is dense by the generalized Chebotarev density theorem, it follows that the image of cycU~X is also ̃ m ab m dense in CU~X ~p . Since π1 (U)~p is Hausdorff, it suffices to show that ρU~X ○cycU~X = et´ ρU~X ○ η ○ cycU~X . Equivalently, we have to show that for every closed point x ∈ U, one has that (ρet´ ○ η ○ cyc x is the image of the Frobenius element under the map U~X U~X )([ ]) Gal k k x → πab U pm. But this is well known (e.g., see [30, Theorem 3.4.1]). ( ~ ( )) 1 ( )~ 1 8.2. A new filtration of Het´ U, Qp Zp . We keep the assumptions of Theorem 8.6. By (7.13) and Theorem 8.6, we( have the~ isomorphism) of abelian groups ≅ θ′ ∶ H1 U, Z pm Ð→ lim Cet´ X,nD; m ∨, m ( ~ ) Ð→n KS( ) where H1 U, Z pm denote the étale cohomology H1 U, Z pm . We let ( ~ ) et´ ( ~ ) filet´ H1 U, Z pm = θ′ −1 Image Cet´ X,nD; m ∨ → lim Cet´ X,nD; m ∨ . D ( ~ ) ( m) ( ( KS( ) Ð→n KS( ) )) We set et´ 1 et´ 1 m (8.11) fil H U, Qp Zp = lim fil H U, Z p . D ( ~ ) Ð→m D ( ~ ) et´ 1 1 It follows that filnDH U, Qp Zp n defines an increasing filtration of H U, Qp Zp . { ( ~ )} 1 ( ~ ) This is an étale version of the filtration filDH U, Qp Zp defined in [15, definition 7.12]. ( ~ ) 1 This new filtration is clearly exhaustive. We do not know if filDH U, Qp Zp and et´ 1 ( ~ ) fil H U, Qp Zp are comparable in general. We can however prove the following. D ( ~ ) Theorem 8.11. Let k be a finite field and X a smooth and projective scheme of pure dimension d ≥ 1 over k. Let D ⊂ X be an effective Cartier divisor with complement U RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 35 such that Dred is a simple normal crossing divisor. Assume that either d ≠ 2 or k ≠ F2. Then one has et´ 1 m 1 m et´ 1 1 fil H U, Z p ⊆ filDH U, Z p and fil H U, Qp Zp ⊆ filDH U, Qp Zp D ( ~ ) ( ~ ) D ( ~ ) ( ~ ) 1 m 1 as subgroups of H U, Z p and H U, Qp Zp , respectively. These inclusions are equal- ( ~ ) ( ~ ) ities if Dred is regular. Proof. The claim about the two inclusions follows directly from the definitions of the filtrations in view of Proposition 8.10 and Corollary 8.8. et´ Moreover, Corollary 8.9 yields that ρU~X is an isomorphism of profinite topological et´ ab m groups. Hence, there exists a unique quotient π1 X,nD; m of π1 U p such that the diagram ( ) ( )~

et´ ρU~X (8.12) Cet´ m / πab U pm ̃U~X ( ) 1 ( )~

et´ ρXSnD πet´ X,nD; m / πadiv X,nD pm 1 ( ) 1 ( )~ commutes and the horizontal arrows are isomorphisms of topological groups. One knows 1 m adiv m ∨ that ﬁlnDH U, Z p = π1 X,nD p . By Theorem 4.9, it is easy to see that et´ 1 ( m ~ et´) ( ( ∨ )~ ) ﬁl H U, Z p ≅ π X,nD; m when Dred is regular. nD ( ~ ) 1 ( ) More precisely, if Dred is a simple normal crossing divisor and either d ≠ 2 or k ≠ F2, we have the following diagram.

′ 1 m θ / et´ ∨ ≅ / et´ ∨ (8.13) H U, Z p lim CKS X,nD; m lim CKS X,nD; m O ~ ) ≅ Ð→n ( O ) (←Ðn ( ))

? ﬁlet´ H1 U, Z pm / Cet´ X,nD; m ∨ D ( _ ~ ) KS( )

1 m ≅ / ∨ filDH U, Z p / CKS X,nD; m . ( ~ ) ( ) Assume now that Dred is regular. By Theorem 4.9, it then follows that the transition et´ maps in the pro-system CKS X,nD; m n are surjective. In particular, the right top vertical arrow is injective{ and hence( the middle))} horizontal dotted arrow is a honest arrow such that the top square commutes. It follows from Proposition 8.10, that the bottom et´ 1 m square commutes as well. By the definition of filDH U, Z p , it is clear that the middle horizontal arrow is now surjective. To prove its injectivit( ~y, it) suffices to show that the right bottom vertical arrow is injective. But this follows from Theorem 4.9.

9. Reciprocity theorem for CKS X D ( S ) In this section, we shall prove the reciprocity theorem for the idele class group CKS X D . 1 ( S ) Before going into this, we recall the definition of some filtrations of Het´ K, Q Z for a Henselian discrete valuation field K. ( ~ )

9.1. The Brylinski-Kato and Matsuda filtrations. Let K be a Henselian discrete valuation field of characteristic p > 0 with ring of integers OK , maximal ideal mK ≠ 0 q q and residue field f. We let H A = Het´ A, Q Z q −1 for any commutative ring A. The generalized Artin-Schreier sequence( ) gives( rise~ to( an)) exact sequence

r 1−F ∂ 1 r (9.1) 0 → Z p → Wr K ÐÐ→ Wr K Ð→ H K, Z p → 0, ~ ( ) ( ) ´et( ~ ) 36 RAHULGUPTA,AMALENDUKRISHNA

p p where F ar−1, . . . , a0 = ar−1, . . . , a0 . We write the Witt vector ar−1, . . . , a0 as a in (( × )) ( ) ( ) short. Let vK∶ K → Z be the normalized valuation. Let δr denote the composite map ∂ 1 r 1 Wr K Ð→ Hét K, Z p ↪ H K . Then one knows that δr = δr+1 ○ V . For an integer ( ) ( ~ ) ( ) ′ m ≥ 1, let ordp m denote the p-adic order of m and let r = min r, ordp m . We let ( ) ( ( )) ordp 0 = −∞. For( )m ≥ 0, we let bk i (9.2) fil Wr K = a p vK ai ≥ −m ; and m ( ) { S ( ) } ms bk r−r′ bk (9.3) fil Wr K = fil Wr K + V fil Wr′ K . m ( ) m−1 ( ) ( m ( )) bk ms bk ms We have fil0 Wr K = fil0 Wr K = Wr OK . We let fil−1Wr K = fil−1 Wr K = 0. For m ≥ 1, we( let) ( ) ( ) ( ) ( ) bk 1 1 ′ bk (9.4) film−1H K = H K p ? δr film−1Wr K and ( ) ( ){ } r≥1 ( ( )) ms 1 1 ′ ms film H K = H K p ? δr film Wr K . ( ) ( ){ } r≥1 ( ( )) ms 1 bk 1 1 Moreover, we let fil0 H K = fil−1H K = H OK , the subgroup of unramified char- (bk ) 1 ( )ms 1 ( ) acters. The filtrations fil● H K and fil● H K are due to Brylinski-Kato [24] and Matsuda [37], respectively. We( refer) to [15, Theorem( ) 6.1] for the following. Theorem 9.1. The two filtrations defined above satisfy the following relations. 1 bk 1 ms 1 (1) H K = ⋃ film H K = ⋃ film H K . ( ) m≥0 ( ) m≥0 ( ) ms 1 bk 1 ms 1 (2) film H K ⊂ film H K ⊂ film+1H K for all m ≥ −1. ( ) ( ) ( )bk 1 ms 1 (3) If m ≥ 1 such that ordp m = 0, then fil H K = fil H K . In particular, ( ) m−1 ( ) m ( ) filbkH1 K = filmsH1 K , which is the subgroup of tamely ramified characters. 0 ( ) 1 ( ) M m × × For integers m ≥ 0 and r ≥ 1, let UmKr K be the subgroup 1 + mK, K ,...,K M ′ M ( ) m × { × M } of Kr K . We let UmKr K be the subgroup 1 + mK, OK ,..., OK of Kr K . It ( ) ( ) M { ′ M }M ( ) follows from [15, Lemma 6.2] that Um+1Kr K ⊆ UmKr K ⊆ UmKr K for every integer m ≥ 0. If K is a d-dimensional Henselian( ) local field( (see) [15, § 5.1]),( ) there is a pairing (9.5) , ∶ KM K × H1 K → Hd+1 K ≅ Q Z. { } d ( ) ( ) ( ) ~ The following result is due to Kato and Matsuda (when p ≠ 2). We refer to [15, Theorem 6.3] for a proof. Theorem 9.2. Let χ ∈ H1 K be a character. Then the following hold. ( ) bk 1 (1) For every integer m ≥ 0, we have that χ ∈ film H K if and only if α, χ = 0 for M ( ) { } all α ∈ Um+1Kd K . ( ) ms 1 (2) For every integer m ≥ 1, we have that χ ∈ film H K if and only if α, χ = 0 for all α ∈ U ′ KM K . ( ) { } m d ( ) We have the inclusions K ↪ Ksh ↪ K, where K is a fixed separable closure of K and Ksh is the strict Henselization of K. We shall use the following key result in the proof of our reciprocity theorem. Proposition 9.3. For m ≥ 0, the canonical square filbkH1 K / H1 K m ( ) ( )

filbkH1 Ksh / H1 Ksh m ( ) ( ) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 37 is Cartesian. bk 1 1 Proof. Since fil● H K is an exhaustive filtration of H K by Theorem 9.1(1), it suffices to show that( for) every m ≥ 1, the square ( )

ﬁlbk H1 K / ﬁlbkH1 K m−1 ( ) m ( )

filbk H1 Ksh / filbkH1 Ksh m−1 ( ) m ( ) is Cartesian. Equivalently, it suffices to show that for every m ≥ 1, the map φ∗ ∶ grbkH1 K → grbkH1 Ksh , m m ( ) m ( ) induced by the inclusion φ∶ K ↪ Ksh, is injective. We fix m ≥ 1. By [24, Corollary 5.2], there exists an injective (non-canonical) homo- bk 1 1 morphism rswπK ∶ grm H K ↪ Ωf f. By Theorem 5.1 of loc. cit., this refined swan ( ) ⊕ sh conductor rswπK depends only on the choice of a uniformizer πK of K. Since OK is sh unramified over OK , we can choose πK to be a uniformizer of K as well. It therefore follows that for all m ≥ 1, the diagram

rswπK grbkH1 K / Ω1 f m ( ) f ⊕

rsw πK grbkH1 Ksh / Ω1 f m ( ) f ⊕ is commutative, where f is a separable closure of f. Since the horizontal arrows in the above diagram are injective, it suﬃces to show that the natural map Ω1 → Ω1 is injective. f f But this is clear.

9.2. Logarithmic fundamental group with modulus. Let k be a finite field of characteristic p and X an integral projective scheme over k of dimension d ≥ 1. Let D ⊂ X be an effective Cartier divisor with complement U. Let K = k η denote the ( ) function field of X. We let C = Dred. We fix a separable closure K of K and let GK denote the absolute Galois group of K. Recall the following notations from [26, § 3.3] or [15, § 2.3]. Assume that X is normal. Let λ be a generic point of D. Let Kλ denote the Henseliza- tion of K at λ. Let P = p0,...,pd−2, λ, η be a Parshin chain on U ⊂ X . Let V ⊂ K ( ) ( ) be a d-DV which dominates P . Let V = V0 ⊂⋯⊂ Vd−2 ⊂ Vd−1 ⊂ Vd = K be the chain of valuation rings in K induced by V . Since X is normal, it is easy to check that for any such chain, one must have V = O . Let V ′ be the image of V in k λ . Let V d−1 X,λ ̃d−1 be the unique Henselian discrete valuation ring having an ind-étale local homomorphism( ) V → V such that its residue field E is the quotient field of V ′ h. Then V h is the d−1 ̃d−1 d−1 inverse image of V ′ h under the quotient map V ↠ E . It follows( ) that its function ̃d−1 d−1 field Q V h is a (d-dimensional) Henselian discrete valuation field whose ring of integers is V ((see) [26, § 3.7.2]). It then follows that there are canonical inclusions of discrete ̃d−1 valuation rings (9.6) O ↪ V ↪ Osh . X,λ ̃d−1 X,λ Moreover, we have (see the proof of [26, Proposition 3.3]) h (9.7) O ′ ≅ V , X,P M ̃d−1 V ∈V(P ) 38 RAHULGUPTA,AMALENDUKRISHNA where V P is the set of d-DV’s in K which dominate P . As an immediate consequence of Proposition( ) 9.3, we therefore get the following. Corollary 9.4. For every m ≥ 0, the square bk 1 / 1 fil H Kλ / H Kλ m ( ) ( )

ﬁlbkH1 Q V h / H1 Q V h m ( ( )) ( ( )) is Cartesian.

Let IrrC denote the set of all generic points of C and let Cλ denote the closure of an element λ ∈ IrrC . We write D = ∑ nλCλ. We can also write D = ∑ nx x , where 1 λ∈IrrC x∈X( ) { } nx = 0 for all x ∈ U. We allow D to be empty in which case we write D = 0. For any (1) x ∈ X ∩C, we let Kx denote the quotient field of the mx-adic completion OX,x of OX,x. sh ̂ sh Å Let OX,x denote the strict Henselization of OX,x and let Kx denote its quotient field. Then it is clear from the definitions that there are inclusions (9.8) K ↪ K ↪ Ksh ↪ K and K ↪ K ↪ K . x x x ̂x bk 1 1 Definition 9.5. Let filD H K denote the subgroup of characters χ ∈ H K such (1) ( ) ( 1 ) that for every x ∈ X , the image χx of χ under the canonical surjection H K ↠ 1 bk 1 bk 1 1 ( ) H Kx lies in fil H Kx . It is easy to check that fil H K ⊂ H U . We let ( ) nx−1 ( ) D ( ) ( ) filbkH1 U, Z m = H1 U, Z m ∩ filbkH1 K . D et´ ( ~ ) et´ ( ~ ) D ( ) Recall that H1 K is a torsion abelian group. We consider it a topological abelian ( ) 1 bk 1 group with discrete topology. In particular, all the subgroups H U (e.g., filD H K ) are also considered as discrete topological abelian groups. Recall( ) from [15, Defini-( ) 1 1 bk 1 tion 7.12] that filDH K is a subgroup of H U which is defined similar to filD H K , ( )bk 1 ms( 1) ( ) where we only replace fil H Kλ by fil H Kλ . nλ−1 ( ) nλ ( ) abk ab Definition 9.6. We define the quotient π1 X, D of π1 U to be the Pontryagin dual bk ( ) ( ) of fil H1 K ⊂ H1 U , i.e., D ( ) ( ) abk bk 1 bk 1 π1 X, D ∶= Homcont filD H K , Q Z = Hom filD H K , Q Z . 1 ( m) 1 ( ( ) ~ ) ( ( ) ~ ) Since H U, Z p = pm H U, Q Z , it follows that et´ ( ~ ) ( ~ ) abk m bk 1 m π X, D p ≅ Homcont fil H U, Z p , Q Z . 1 ( )~ ( D et´ ( ~ ) ~ ) bk 1 abk Since filD H K is a discrete topological group, it follows that π1 X, D is a profinite group. Moreover,( ) [41, Theorem 2.9.6] implies that ( ) (9.9) πabk X, D ∨ ≅ filbkH1 K and πabk X, D pm ∨ ≅ filbkH1 U, Z pm . 1 ( ) D ( ) ( 1 ( )~ ) D et´ ( ~ ) Lemma 9.7. The quotient map πab U → πabk X, D factors through πadiv X, D . 1 ( ) 1 ( ) 1 ( ) Proof. This is a straightforward consequence of Theorem 9.1 and [15, Theorem 7.16] once we recall that adiv 1 (9.10) π X, D = Hom filDH K , Q Z . 1 ( ) ( ( ) ~ ) Remark 9.8. Using [1, § 3], one can mimic the construction of [15, § 7] to show that abk π1 X, D is isomorphic to the abelianization of the automorphism group of the fiber functor( of) a certain Galois subcategory of the category of finite étale covers of U. Under abk this Tannakian interpretation, π1 X, D characterizes the finite étale covers of U whose ramifications are bounded at each generic( ) point of D by means of Kato’s Swan conductor. RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 39

9.3. Reciprocity for CKS X D . We shall continue with the setting of § 9.2. Before we ( S ) prove the reciprocity theorem for CKS X D , we recall the construction of the reciprocity ( S ) map for CU~X from [15, § 5.4]. We let P = p0,...,ps be any Parshin chain on X with ( ) the condition that ps ∈ U. Let X P = ps be the integral closed subscheme of X and ( ) { } h let U P = U ∩ X P . We let V ⊂ k ps be an s-DV dominating P . Then Q V is an s-dimensional( ) Henselian( ) local field.( By) [15, Proposition 5.1], we therefore have( ) the reciprocity map M h h ab h ab h (9.11) ρ h ∶ K Q V → Gal Q V Q V ≅ π Spec Q V . Q(V ) s ( ( )) ( ( ) ~ ( )) 1 ( ( ( ))) Taking the sum of these maps over V P and using [15, Lemma 5.10], we get a reciprocity map ( ) M ab ab (9.12) ρk(P )∶ K k P → π Spec k P → π U , s ( ( )) 1 ( ( ( ))) 1 ( ) where last map exists because ps ∈ U. Taking sum over all Parshin chains on the pair ab U ⊂ X , we get a reciprocity map ρU~X ∶ IU~X → π1 U . By [15, Theorem 5.13, Propo- (sition 5.15],) this descends to a continuous homomorphism( ) of topological groups ab (9.13) ρU X ∶ CU X → π U . ~ ~ 1 ( ) adiv adiv ab Recall that π X, D 0 is the kernel of the composite map π X, D ↠ π X → 1 ( ) 1 ( ) 1 ( ) Z. One defines πabk X, D similarly. One of the main results of this paper is the ̂ 1 0 following. ( ) Theorem 9.9. There is a continuous homomorphism ρ′ ∶ C X D → πabk X, D XSD ( S ) 1 ( ) with dense image such that the diagram ρ U~X / ab (9.14) CU X / π U ~ 1 ( ) ′ q′ pXSD XSD ′ ρXSD C X D / πabk X, D ( S ) 1 ( ) ′ is commutative. If X is normal and U is regular, then ρXSD induces an isomorphism of finite groups ′ ≅ abk ρ ∶ C X D 0 Ð→ π X, D 0. XSD ( S ) 1 ( ) ′ Proof. We first show the existence of ρXSD. Its continuity and density of its image will then follow by the corresponding assertions for ρU~X , shown in [15]. In view of (9.13), we abk ′ only have to show that if χ is a character of π1 X, D , then the composite χ○qXSD○ρU~X ′ ( ) ′ annihilates Ker pXSD . By (3.3), we only need to show that χ ○ qXSD ○ ρU~X annihilates M( h ) the image of K O ′ ID → CU X , where P is any maximal Parshin chain on U ⊂ X . ̂d ( X,P S ) ~ ( ) But the proof of this is completely identical to that of [15, Theorem 8.1], only difference being that we have to use part (1) of Theorem 9.2 instead of part (2). We now assume that X is normal and U is regular. In this case, we can replace C X D ′ ( S ) by CKS X D by Theorem 3.4. We show that ρ is injective on all of CKS X D . By ( S ) XSD ( S ) Lemmas 3.3 and 9.7, there is a diagram

ρ′∨ bk 1 XSD/ ∨ (9.15) fil H K / CKS X D D ( ) ( S ) α∨ β∨ ρ∨ 1 XSD/ ∨ filDH K / CKS X, D , ( ) ( ) 40 RAHULGUPTA,AMALENDUKRISHNA whose vertical arrows are injective. It is clear from the construction of the reciprocity maps that this diagram is commutative. ′ ′∨ To show that ρXSD is injective, it suffices to show that ρXSD is surjective (see [15, ∨ ∨ ∨ Lemma 7.10]). We fix a character χ ∈ CKS X D and let χ = β χ . Since ρXSD ( S ) ′̃ ( 1) is surjective by [16, Theorem 1.1], we can find a character χ ∈ filDH K such that ∨ ′ ′ bk 1 ( ) χ = ρXSD χ . We need to show that χ ∈ filD H K . ̃ ( ) ′ ( ) ′ 1 We fix a point x ∈ IrrC and let χx be the image of χ in H Kx . We need to show ′ bk 1 ( ) that χ ∈ fil H Kx , where nx is the multiplicity of D at x. By Corollary 9.4, it x nx−1 ( ) suffices to show that for some maximal Parshin chain P = p0,...,pd−2,x,η on U ⊂ X ′ 1 ( h ) ( ) and d-DV V ⊂ K dominating P , the image of χx in H Q V lies in the subgroup bk 1 h ( ( )) filnx−1H Q V . But this is proven by repeating the proof of [16, Theorem 1.1] mutatis mutandis( by( using)) only one modification. Namely, we need to use part (1) of Theorem 9.2 instead of part (2). ′ The surjectivity of ρXSD on the degree zero part follows because the top horizontal arrow in the commutative diagram ρ XSD adiv (9.16) CKS X, D 0 / π X, D 0 ( ) 1 ( ) β α ρ′ XSD/ abk CKS X D 0 / π X, D 0 ( S ) 1 ( ) is surjective by [16, Theorem 1.1]. The finiteness of CKS X D 0 follows because β is ( S ) surjective by Lemma 3.3 and CKS X, D 0 is finite by [16, Theorem 4.8]. This concludes the proof. ( ) Using Theorem 9.9 and [15, Lemma 8.4], we get the following. Corollary 9.10. Under the additional assumptions of Theorem 9.9, the map ′ abk ρ ∶ CKS X D m → π X, D m XSD ( S )~ 1 ( )~ is an isomorphism of finite groups for every integer m ≥ 1. 9.4. Filtration of Kerz-Zhao. Let k be a finite field of characteristic p and X an integral regular projective scheme over k of dimension d ≥ 1. Let D ⊂ X be an effective Cartier divisor with complement U such that Dred is a simple normal crossing divisor. Let K denote the function field of X. By [22, Theorems 1.1.5, 4.1.4], there is an isomorphism

1 m ≅ d M m ∨ (9.17) λm∶ H U, Z p Ð→ lim H X, K p . et´ ( ~ ) Ð→n et´ ( ̂d,XSnD~ ) Each group Hd X, KM pm is finite by [30, Theorem 3.3.1] and Corollary 9.10. It et´ ( ̂d,XSnD~ ) follows that (9.17) is an isomorphism of discrete torsion topological abelian groups. Since Hd X, KM pm ∨ ↪ lim Hd X, KM pm ∨, we can define et´ ( ̂d,XSD~ ) Ð→n et´ ( ̂d,XSnD~ ) bk 1 m −1 d M m ∨ fil H U, Z p ∶= λm H X, K p . D et´ ( ~ ) ( ) ( et´ ( ̂d,XSD~ ) ) bk 1 This filtration was defined by Kerz-Zhao [30, Definition 3.3.7]. We let filD Het´ U, Qp Zp = bk 1 m bk 1 1 ′ bk 1 ( ~ ) lim fil H U, Z p and fil H K = H K p fil H U, Qp Zp . Ð→m D et´ ( ~ ) D,et´ ( ) ( ){ }⊕ D et´ ( ~ ) Using [30, Theorem 3.3.1] and Corollary 9.10, we get the following logarithmic version of Theorem 8.11 (without assuming that Dred is regular). This identifies the filtration due to Kerz-Zhao to the one induced by the Brylinski-Kato filtration. Corollary 9.11. As subgroups of H1 K , one has ( ) filbk H1 K = filbkH1 K . D,et´ ( ) D ( ) RAMIFIED CLASS FIELD THEORY AND DUALITY OVER FINITE FIELDS 41

10. Counterexamples to Nisnevich descent We shall now prove Theorem 1.5. Let k be a finite field of characteristic p and X an integral regular projective scheme of dimension two over k. Let C ⊂ X be an integral regular curve with complement U. Let K = k X and f = k λ , where λ is the generic ( ) ( ) h point of C. Let Kλ be the Henselian discrete valuation field with ring of integers OX,λ r ′ ′ and residue field f. Fix a positive integer m0 = p m , where r ≥ 1 and p ∤ m . Assume that p ≠ 2. Then Matsuda has shown (see the proof of [37, Proposition 3.2.7]) that there is an isomorphism

ms 1 film0 H Kλ ≅ 1 (10.1) η∶ ( ) Ð→ BrΩ , bk 1 f fil H Kλ m0−1 ( ) 1 1 where B●Ωf is an increasing filtration of Ωf , recalled in the proof of Lemma 4.4. Suppose 1 1 1 that B1Ωf = d f ⊂ Ωf is not zero. Then BrΩf ≠ 0 for every r ≥ 1. Hence, the left ( ) 1 1 hand side of (10.1) is not zero. Since the restriction map δλ∶ H K → H Kλ is surjective, it follows that we can find a continuous character χ ∈(H1) K such( that) ms 1 bk 1 ( ) δλ χ ∈ film0 H Kλ ∖ film0−1H Kλ . (We) let U ′ ⊆ (U be) the largest( open) subscheme where χ is unramified and let C′ = X ∖ U ′ with the reduced closed subscheme structure. Since X is a surface, we can find a morphism f∶ X′ → X which is a composition of a sequence monoidal transformations such that the reduced closed subscheme f −1 C′ is a simple normal crossing divisor. In ( ) particular, E0 → C is an isomorphism if we let E0 be the strict transform of C. We let E = f −1 C′ with reduced structure. Note that there is a finite closed subset T ⊂ C′ such that f −1( X)∖ T → X ∖ T is an isomorphism. ( ) We let E = E0 + E1 +⋯+ Es, where each Ei is integral. We let λi be the generic 1 point of Ei so that λ0 = λ. Let arλ ∶ H K → Z be the Artin conductor (see [37, i ( ) Definition 3.2.5]). Theorem 9.1 implies that arλ0 χ = m0. We let mi = arλi χ for i ≥ 1 s ( ) ( ) ′ 1 bk 1 and define D =∑ miEi. It is then clear that χ ∈ filDH K ∖ filD H K . We have thus i=0 ( ) ( ) found a smooth projective integral surface X′ and an effective Cartier divisor D′ ⊂ X′ 1 ′ filD′ H (K) with the property that Dred is a simple normal crossing divisor and bk 1 ≠ 0 if filD′ H (K) ′ adiv ′ ′ abk ′ ′ K is the function field of X . It follows that Ker π1 X , D ↠ π1 X , D ≠ 0. adiv ′ ′ abk ′ ′ ( ( ) ( )) Equivalently, Ker π1 X , D 0 ↠ π1 X , D 0 ≠ 0. We now look at( the the( commutative) ( diagram) )

cyc ′ ′ ′ ′ X SD adiv ′ ′ (10.2) CH0 X D 0 / π X , D 0 ( S ) 1 ( )

′ cyc ′ ′ 4 ′ ′ X /SD abk ′ ′ H X D , Z 2 0 / π X , D 0. M( S ( )) 1 ( ) The top horizontal arrow is an isomorphism by [16, Theorems 1.2, 1.3] and the bottom horizontal arrow is an isomorphism by Corollary 1.4. We conclude that the left vertical arrow is surjective but not injective. To complete the construction of a counterexample, what remains is to ﬁnd a pair 1 X,C such that d f ⊂ Ωf is not zero. But this is an easy exercise. For instance, take ( )2 2( ) X = Pk and C ⊂ Pk a coordinate hyperplane. Then f ≅ k t is a purely transcendental extension of degree one and d t is a free generator of Ω( 1). We remark that we had ( ) f assumed above that p ≠ 2, but this condition can be removed using the proof of [15, Theorem 6.3]. 42 RAHULGUPTA,AMALENDUKRISHNA

Let X′, D′ be as above and let F = k t be a purely transcendental extension of degree( one. We) let X = X′ and D = D′ . Then( ) we have a commutative diagram ̃ F ̃ F ′ ′ CH0 X D / CH0 X D ( S ) ( ̃S ̃)

H4 X′ D′, Z 2 / H4 X D, Z 2 , M( S ( )) M( ̃S ̃ ( )) where the horizontal arrows are the flat pull-back maps. It easily follows from the proof of [32, Proposition 4.3] that the top horizontal arrow is injective. It follows that the right vertical arrow is not injective. This shows the failure of Nisnevich descent for the Chow groups with modulus over infinite fields too.

Acknowledgements. Gupta was supported by the SFB 1085 Higher Invariants (Univer- sit¨at Regensburg).

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Fakultat¨ fur¨ Mathematik, Universitat¨ Regensburg, 93040, Regensburg, Germany. Email address: [email protected]

Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India. Email address: [email protected]