Bloch's Cycle Complex of Zero Cycles with $\Mathbb Z/P^ N $ Coefficients

$Bloch's Cycle Complex of Zero Cycles with $\Mathbb Z/P^ N $ Coefficients$

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8.2. Etale´ descent 46 8.3. Birational geometry and rational singularities 47 8.4. Galois descent 48

Appendix 49 A. Semilinear algebra 49 References 51

Introduction

Let X be a separated scheme of finite type over k of dimension d with k being a perfect field of c n positive characteristic p. In this work, we show that Bloch’s cycle complex ZX of zero cycles mod p is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod pn coefficients for singular varieties. c As the first candidate for a motivic complex, Bloch introduced his cycle complex ZX in [Blo86] under i the framework of Beilinson-Lichtenbaum. Let i be an integer, and ∆ = Spec k[T0,...,Ti]/( Tj − c 1). Here Z := z0(−, −•) is a complex of sheaves in the Zariski or the ’etale topology. The global X P sections of its degree (−i)-term z0(X,i) is the free abelian group generated by dimension i-cycles in X × ∆i intersecting all faces properly and the differentials are the alternating sums of the cycle-theoretic intersection of the cycle with each face (cf. Section 2). In this article we define a complex Kn,X,log via Grothendieck’s duality theory of coherent sheaves following the idea in [Kat87] and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves νn,X (namely the Gersten complex of logarithmic de Rham-Witt sheaves, which is introduced and studied in [Kat86a, §1][Mos99, (1.3)-(1.5)]) to Kn,X,log for the étale topology and also for the Zariski topologye under the c extra assumption k = k. Combined with Zhong’s quasi-isomorphism from Bloch’s cycle complex ZX to νn,X [Zho14, 2.16], we deduce certain vanishing and finiteness properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with Z/pn-coefficients. The proofs in this article aree self-contained in respect to Kato’s work [Kat87]. Let us briefly recall Kato’s work in [Kat87] and introduce our main object of studies: Kn,X,log. Let π : X → Spec k be the structure morphism of X. Let WnX := (|X|, WnOX ), where |X| is the underlying topological space of X, and WnOX is the sheaf of length n truncated Witt vectors. Let Wnπ : WnX → Spec Wnk be the morphism induced from π via functoriality. According to Grothendieck’s duality theory, ! there exists an explicit Zariski complex Kn,X of quasi-coherent sheaves representing (Wnπ) Wnk (such a complex Kn,X is called a residual complex, cf. [Har66, VI.3.1]). There is a natural Cartier operator ′ d d C : Kn,X → Kn,X , which is compatible with the classical Cartier operator C : WnΩX → WnΩX in d the smooth case via Ekedahl’s quasi-isomorphism (see Theorem 1.7). Here WnΩX denotes the degree d := dim X part of the de Rham-Witt complex. We define the complex Kn,X,log to be the mapping cone of C′−1. What Kato did in [Kat87] is the FRP counterpart, where FRP is the ”flat and relatively perfect” topology (this is a topology with étale coverings and with the underlying category lying in between the small and the big étale site). Kato then showed that Kn,X,log in the topology FRP acts as a dualizing complex in a rather big triangulated subcategory of the derived category of Z/pn-sheaves, containing all coherent sheaves and sheaves like logarithmic de Rham-Witt sheaves [Kat87, 0.1]. Kato also showed that in the smooth setting, Kn,X,log is concentrated in one degree and this only nonzero cohomology sheaf is the top degree logarithmic de Rham-Witt sheaf [Kat87, 3.4]. For the latter, an analogy on the small étale site naturally holds. Rülling later observed that with a trick from p−1-linear algebra, [Kat87, 3.4] can be done on the Zariski site as well, as long as one assumes k = k (cf. Proposition 1.15). Comparing this with the Kato-Moser complex νn,X , which is precisely the Gersten resolution of the logarithmic de Rham-Witt sheaf in the smooth setting, one gets an identification in the smooth setting νn,X ≃ Kn,X,log on the Zariski topology. Similar ase in [Kat87, 4.2] (cf. Proposition 1.21), Rülling also built up the localization sequence for Kn,X,log on the Zariski site in his unpublished notes (cf. Propositione 1.22). c Compared with the localization sequence for ZX [Blo94, 1.1] and for νn,X (which trivially holds in the Zariski topology), it is reasonable to expect a chain map relating these objects in general. ≃ The aim of this article is to build a quasi-isomorphism ζlog : νn,X −→e Kn,X,log in the singular setting, c such that when pre-composed with Zhong’s quasi-isomorphism ψ : ZX → νn,X [Zho14, 2.16], it gives e e CYCLE COMPLEX VIA COHERENT DUALITY 3

another perspective of Bloch’s cycle complex with Z/pn-coefficients in terms of Grothendieck’s coherent duality theory. More precisely, we prove the following result. Theorem 0.1 (Theorem 5.10, Theorem 6.1). Let X be a separated scheme of finite type over k with k being a perfect field of positive characteristic p. Then there exists a chain map ≃ ζlog,ét : νn,X,ét −→ Kn,X,log,ét, which is a quasi-isomorphism. If moreover k = k, the chain map e ≃ ζlog,Zar : νn,X,Zar −→ Kn,X,log,Zar is also a quasi-isomorphism. Composed with Zhong’s quasi-isomorphisme ψ, we have the following composition of chain maps

c n ≃ ζlog,ét ◦ ψét : ZX,ét/p −→ Kn,X,log,ét which is a quasi-isomorphism. If moreover k = k, the composition

c n ≃ ζlog,Zar ◦ ψZar : ZX,Zar/p −→ Kn,X,log,Zar is also a quasi-isomorphism.

We explain more on the motivation behind the definition of Kn,X,log. In the smooth setting, the d logarithmic de Rham-Witt sheaves can be defined in two ways: either as the subsheaves of WnΩX generated by log forms, or as the invariant part under the Cartier operator C. In the singular case, these two perspectives give two different (complexes of) sheaves. The first definition can also be done in the singular case, and this was studied by Morrow [Mor15]. For the second definition one has to replace d WnΩX by a dualizing complex on WnX: for this Grothendieck’s duality theory yields a canonical and explicit choice, and this is what we have denoted by Kn,X . And then this method leads naturally to c n Kato’s and also our construction of Kn,X,log. Now with our main theorem one knows that ZX /p sits in a distinguished triangle ′ c n C −1 +1 ZX /p → Kn,X −−−→ Kn,X −−→ in the derived category Db(X, Z/pn), in either the étale topology, or the Zariski topology with an extra k = k assumption. In particular, when X is Cohen-Macaulay of pure dimension d, then the triangle above becomes ′ c n C −1 +1 ZX /p → WnωX [d] −−−→ WnωX [d] −−→ .

where WnωX is the only non-vanishing cohomology sheaf of Kn,X (when n = 1, W1ωX = ωX is the c n usual dualizing sheaf on X), and ZX /p is concentrated at degree −d (cf. Proposition 8.1). This is a generalization of the top degree case of [GL00, 8.3], which in particular implies the above triangle in the smooth case. As corollaries, we arrive at some properties of the higher Chow groups of 0-cycles with p-primary torsion coefficients. (We have specialized several statements here in the introduction part. Please see the main text for more general statements,) Corollary 0.2 (Proposition 8.2, Corollary 8.6, Proposition 8.3, Corollary 8.4, Corollary 8.9, Corol- lary 8.12, Corollary 8.14). Let X be a separated scheme of finite type over a perfect field k of characteristic p> 0. (1) (Cartier invariance) Assume k = k. Then

n −q C′ −1 CH0(X, q; Z/p )= H (WnX,Kn,X,Zar) Zar . (2) (Semisimplicity) Assume k = k. Let X be proper over k. Then for any q, −q n H (WnX,Kn,X )ss = CH0(X, q; Z/p ) ⊗Z/pn Wnk. (We refer to Definition A.4 and Remark A.5(2) for the definition of the semisimplicity and the notation (−)ss in this context.) b n (3) (Relation with p-torsion Poincaréduality) There is an isomorphism in D ((WnX)ét, Z/p ) ! n Kn,X,log,ét ≃ R(Wnπ) (Z/p ), ! where R(Wnπ) is the extraordinary inverse image functor defined in [SGA4-3, ExposéXVIII, Thm 3.1.4]. 4 FEI REN

(4) (Affine vanishing) Assume k = k. Suppose X is affine and Cohen-Macaulay of pure dimension d. Then n CH0(X, q, Z/p )=0 for q 6= d. (5) (Etale´ descent) Assume k = k. Suppose X is Cohen-Macaulay of pure relative dimension d. Then i c n i R ǫ∗(ZX,ét/p )= R ǫ∗νn,X,ét =0, i 6= −d. (6) (Invariance under rational resolution) Assume k = k. For a rational resolution of singularities f : X → X (cf. Definition 8.10) of an integrale k-scheme X of pure dimension, the trace map induces an isomorphism e n ≃ n CH0(X, q; Z/p ) −→ CH0(X, q; Z/p ) for each q. (7) (Galois descent) Assume k = ke. Let f : X → Y be a finite étale Galois map with Galois group G. Then n n G CH0(Y, d; Z/p )=CH0(X, d; Z/p ) . Now we give a more detailed description of the structure of this article. The general setting is that X is a separated scheme of finite type over a perfect field k of positive characteristic p (we sometimes just say X is a k-scheme for short). In Part 1, we review the basic properties of the chain complexes to appear. Section 1 is devoted to the properties of the complex Kn,X,log, the most important object of our studies. We study the Zariski version in Section 1.1-Section 1.4. Fol- ′ lowing the idea in [Kat87], we define the Cartier operator C for the residual complex Kn,X , and then ′ ′ define the complex Kn,X,log to be the mapping cone of C − 1 in Section 1.1. We compare our C with the classical definition of the Cartier operator C for top degree de Rham-Witt sheaves in Section 1.2. To avoid interruption of a smooth reading we collect the calculation in the next two subsections (Sec- tion 1.2.2-Section 1.2.3). The localization sequence appears in Section 1.3. In these subsections, the most important ingredients are a surjectivity result of C′ − 1 (cf. Proposition 1.15. See also Section A for a short discussion on semilinear algebra), the trace map of a nilpotent thickening (cf. Proposition 1.21), and the localization sequence (cf. Proposition 1.22). They are observed already by Rülling and are only re-presented here by the author. After a short discussion on functoriality in Section 1.4, we move to the étale case in Section 1.5. Most of the properties hold true in a similar manner, except that the surjectivity d d of Cét − 1 : WnΩX,ét → WnΩX,ét over a smooth k-scheme X holds true without any extra assumption of the base field (except perfectness, which is already needed in defining the Cartier operator). This enables us to build the quasi-isomorphism ζlog,ét without assuming k being algebraically closed in the c next part. The rest of the sections in Part 1 are introductory treatments of Bloch’s cycle complex ZX , M Kato’s complex of Milnor K-theory CX,t and the Kato-Moser complex of logarithmic de Rham Witt sheaves νn,X,t, respectively. There are no new results in these three sections. ≃ In Part 2 we construct the quasi-isomorphism ζlog : νn,X −→ Kn,X,log and study its properties in M Section 5e. We first build a chain map ζ : CX → Kn,X and then we show that it induces a chain map M ζlog : CX → Kn,X,log. This map actually factors throughe a chain map ζlog : νn,X → Kn,X,log via the Bloch-Gabber-Kato isomorphism [BK86, 2.8]. We prove that ζlog is a quasi-isomorphism for t = ét, and also for t = Zar with an extra k = k assumption. In Section 6, we review the maine results of [Zho14, §2] c n and compose Zhong’s quasi-isomorphism ψ : ZX /p → νn,X with our ζlog. This composite map enables us to use tools from the coherent duality theory in calculation of certain higher Chow groups of 0-cycles. In Part 3 we discuss the applications. Section 7 mainlye serves as a preparation section for Section 8. In Section 8 we arrive at several results for higher Chow groups of 0-cycles with p-primary torsion coefficients: affine vanishing, finiteness (reproof of a theorem of Geisser), étale descent, and invariance under rational resolutions. Prerequisites. The reader will be assumed to be familiar with Grothendieck duality theory in the sense of [Har66] and [Con00]. One can find a short survey of the properties of the residual complexes and the trace map in [CR12, §1.7]. Acknowledgments. This paper is adapted from my PhD thesis. I’d like to expresses the deepest gratitude to my advisor, Kay Rülling, for suggesting me this topic, for providing a lecture on the de Rham-Witt theory, for sharing his private manuscript and observations, and for the numerous discussions and guidance during the whole time of my PhD. In particular, Proposition 1.15, Proposition 1.21, and Proposition 1.22 were already contained in his unpublished manuscript. Alexander Schmitt has read the CYCLE COMPLEX VIA COHERENT DUALITY 5

preliminary version of this paper with great care and provided detailed comments on the mathematical contents, grammatical errors and typos. Thomas Geisser has also read the preliminary version of this work and pointed out that Proposition 8.8 is in fact a corollary of [GL00, 8.4]. Yun Hao kindly provided me his notes on an elementary proof of Proposition A.3. I am indebted to them all. I thank also the Berlin mathematical school for providing ﬁnancial support during my studies in Berlin.

Part 1. The complexes

1. Kato’s complex Kn,X,log,t

1.1. Definition of Kn,X,log. Let k be a perfect field of characteristic p. Let Wnk be the ring of Witt vectors of length n of k. Notice that Wnk is an injective Wnk-module by Baer’s criterion. So Spec Wnk is a Gorenstein scheme by [Har66, V. 9.1(ii)], and its structure sheaf placed at degree 0 is a residual complex • (with codimension function being the zero function and the associated filtration being Z (Wnk) = 0 0 {Z (Wnk)}, where Z (Wnk) is the set of the unique point in Spec Wnk) by [Har66, p299 1.] and the • categorical equivalence [Con00, 3.2.1] (note that in this case the Cousin functor EZ (Wnk) applied to △ Wnk is still Wnk). This justifies the symbol (WnFk) to appear. To avoid possible confusion we will distinguish the source and target of the absolute Frobenius by using the symbols k1 = k2 = k. Absolute Frobenius is then written as Fk : (Spec k1, k1) → (Spec k2, k2), and the n-th Witt lift is written as WnFk : (Spec Wnk1, Wnk1) → (Spec Wnk2, Wnk2). There is a natural isomorphism of Wnk1-modules (the last isomorphism is given by [Har66, VI.3.1])

≃ ∗ △ (1.1.1) Wnk1 −→ WnFk HomWnk2 ((WnFk)∗(Wnk1), Wnk2) ≃ (WnFk) (Wnk2),

a 7→ a ⊗ [(WnFk)∗1 7→ 1] (= [(WnFk)∗a 7→ 1]), where WnFk : (Spec Wnk1, Wnk1) → (Spec Wnk2, (WnFk)∗(Wnk1)) is the natural map of ringed spaces, and the Hom set is given the (WnFk)∗(Wnk1)-module structure via the first place. In fact, it is clearly a bijection: identify the target with Wnk2 via the evaluate-at-1 map, then one can see that the map −1 (1.1.1) is identified with a 7→ (WnFk) (a). Let X be a separated scheme of finite type over k with structure map π : X → k. Since Wnk is a Gorenstein scheme as we recalled in the last paragraph, △ Kn,X := (Wnπ) Wnk

is a residual complex on WnX, associated to the codimension function dKn,X with

dKn,X (x)= − dim {x}, • p and the ﬁltration Z (Kn,X )= {Z (Kn,X )} with

p Z (Kn,X )= {x ∈ X | dim {x}≤−p}.

In particular, Kn,X is a bounded complex of injective quasi-coherent WnOX -modules with coherent cohomologies sitting in degrees [−d, 0]. ′ When n = 1, we write KX := K1,X . Now we turn to the deﬁnition of C . Denote the level n Witt lift

of the absolute Frobenius FX by WnFX : (WnX1, WnOX1 ) → (WnX2, WnOX2 ). The structure maps of WnX1, WnX2 are Wnπ1, Wnπ2 respectively. These schemes ﬁt into a commutative diagram

WnFX WnX1 / WnX2

Wnπ1 Wnπ2

WnFk Spec Wnk1 / Spec Wnk2. Denote △ Kn,Xi := (Wnπi) (Wnki), i =1, 2.

Via functoriality, one has a WnOX1 -linear map

△ △ (Wnπ1) (1.1.1) △ △ (1.1.2) Kn,X = (Wnπ1) (Wnk1) −−−−−−−−−−−→ (Wnπ1) (WnFk) (Wnk2) 1 ≃ △ △ △ ≃ (WnFX ) (Wnπ2) (Wnk2) ≃ (WnFX ) Kn,X2 . 6 FEI REN

Here the isomorphism at the beginning of the second line is given by [Con00, (3.2.3)]. Then via the

adjunction with respect to the morphism WnFX , one has a WnOX2 -linear map Tr ′ ′ (WnFX )∗(1.1.2) △ WnFX (1.1.3) C := C : (WnFX )∗Kn,X −−−−−−−−−−−→ (WnFX )∗(WnFX ) Kn,X −−−−−→ Kn,X , n 1 ≃ 2 2 where the last map is the trace map of WnFX for residual complexes, cf. [Con00, §3.4]. We call it the (level n) Cartier operator for residual complexes. We sometimes omit the (WnFX )∗-module structure of ′ the source and write simply as C : Kn,X → Kn,X . Now we come to the construction of Kn,X,log (cf. [Kat87, §3]). Deﬁne C′−1 (1.1.4) Kn,X,log := Cone(Kn,X −−−→ Kn,X )[−1]. This is a complex of abelian sheaves sitting in degrees [−d, 1].

When n = 1, we set KX,log := K1,X,log. Writing more explicitly, Kn,X,log is the following complex −d −d+1 −d 0 −1 0 (Kn,X ⊕ 0) → (Kn,X ⊕ Kn,X ) → ... → (Kn,X ⊕ Kn,X ) → (0 ⊕ Kn,X ).

The diﬀerential of Kn,X,log at degree i is given by i i+1 dlog = dn,log : Kn,X,log → Kn,X,log i i−1 i+1 i (Kn,X ⊕ Kn,X ) → (Kn,X ⊕ Kn,X ) (a,b) 7→ (d(a), −(C′ − 1)(a) − d(b)), where d is the diﬀerential in Kn,X . The sign conventions we adopt here for shifted complexes and the cone construction are the same as in [Con00, p6, p8]. And naturally, one has a distinguished triangle

C′−1 +1 (1.1.5) Kn,X,log −→ Kn,X −−−→ Kn,X −−→ Kn,X,log[1]. Explicitly, the ﬁrst map is in degree i given by i i i−1 i Kn,X,log = Kn,X ⊕ Kn,X −→ Kn,X , (a,b) 7→ a. The ”+1” map is given by i i i+1 i+1 i Kn,X −→ (Kn,X,log[1]) = Kn,X,log = (Kn,X ⊕ Kn,X ), b 7→ (0,b). Both maps are indeed maps of chain complexes.

d 1.2. Comparison of WnΩX,log with Kn,X,log. Recall the following result from classical Grothendieck duality theory [Har66, IV, 3.4][Con00, 3.1.3] and Ekedahl [Eke84, §1] (see also [CR12, proof of 1.10.3 and Rmk. 1.10.4]). Proposition 1.1 (Ekedahl). When X is smooth and of pure dimension d over k, then there is a canonical quasi-isomorphism d ≃ WnΩX [d] −→ Kn,X . Remark 1.2. Suppose X is a separated scheme of finite type over k of dimension d. Denote by U the smooth locus of X, and suppose that the complement Z of U is of dimension e. Suppose moreover that U is non-empty and equidimensional (it is satisfied for example, when X is integral). Then Ekedahl’s quasi-isomorphism Proposition 1.1 gives a quasi-isomorphism of dualizing complexes d ≃ (1.2.1) WnΩU [d] −→ Kn,U . Note that by the very definition, the associated filtrations of quasi-isomorphic dualizing complexes are the same (cf. [Har66, 3.4]). As explained above, the associated filtration of Kn,U is its dimension filtration. Let Z• be the codimension filtration of U (cf. [Con00, p.105]). Since U is of pure dimension d, we know that its dimension filtration is just a shift of the codimension filtration, i.e., Z•[d]. Apply the Cousin functor associated to the shifted codimension filtration Z•[d] (cf. [Har66, IV, §1, Motif G]) to the quasi-isomorphism (1.2.1) between dualizing complexes, we have an isomorphism of residual complexes d ≃ EZ•[d](WnΩU [d]) −→ Kn,U • with the same filtration Z [d] (cf. [Con00, 3.2.1]). Since Wnj is an open immersion, we can canonically ∗ identify the residual complexes (Wnj) Kn,X ≃ Kn,U by [Har66, VI, 3.1 and 5.3]. Since Kn,X is a CYCLE COMPLEX VIA COHERENT DUALITY 7

residual complex and in particular is a Cousin complex (cf. [Con00, p. 105]), the adjunction map ∗ Kn,X → (Wnj)∗(Wnj) Kn,X ≃ (Wnj)∗Kn,U is an isomorphism at degrees [−d, −e − 1]. Thus the induced chain map d Kn,X → (Wnj)∗EZ•[d](WnΩU [d]) is an isomorphism at degrees [−d, −e − 1]. 1.2.1. Compatibility of C′ with the classical Cartier operator C. We review the absolute Cartier operator in classical literature (see e.g. [BK05, Chapter 1 §3], [Ill79, §0.2], [Katz70, 7.2], [IR83, III §1]). Let X be a k-scheme. The (absolute) inverse Cartier operator γX of degree i on a scheme X is affine locally, say, on Spec A ⊂ X, given additively by the following expression (Hi(−) denotes the cohomology sheaf of the complex) i i • (1.2.2) γA : ΩA/k −→ H (FA,∗ΩA/k) p p−1 p−1 ada1 ...dai 7→ a a1 da1 ...ai dai, i • i • where a,a1,...ai ∈ A. Here H (FA,∗ΩA/k) denotes the A-module structure on H (ΩA/k) via the absolute p • Frobenius FA : A → A, a 7→ a (note that FA,∗ΩA/k is a complex of A-modules in positive characteristic). For each degree i, γA thus defined is an A-linear map. These local maps patch together and give rise to a map of sheaves i i • (1.2.3) γX :ΩX −→H (FX,∗ΩX ) which is OX -linear. When X is smooth of dimension d, γX is a isomorphism of OX -modules, which is called the (absolute) Cartier isomorphism. See [BK05, 1.3.4] for a proof (note that although the authors there assumed the base field to be algebraically closed, the proof of this theorem works for any perfect field k of positive characteristic). This can be generalized to the de Rham-Witt case.

′i i Lemma 1.3 (cf. [Kat86b, 4.1.3]). Denote by WnΩX the abelian sheaf F (Wn+1ΩX ) regarded as a WnOX - i submodule of (WnFX )∗WnΩX . When X is smooth of dimension d, the map i ′i n−1 i−1 F : WnΩX → WnΩX /dV ΩX i i induced by Frobenius F : Wn+1ΩX → R∗(WnFX )∗WnΩX is an isomorphism of WnOX -modules. In particular, when i = d, d d n−1 d−1 F : WnΩX → (WnFX )∗WnΩX /dV ΩX is an isomorphism of WnOX -modules. Proof. Since i i n i n i−1 Ker(R : Wn+1Ω → WnΩ )= V Ω + dV Ω , n i n i−1 n−1 i−1 i i F V Ω = 0 and F dV Ω = dV Ω , F : Wn+1Ω → WnΩ reduces to i i n−1 F : WnΩ → WnΩ /dV . i i−1 n−1 Surjectivity is clear. We show injectivity. Suppose x ∈ Wn+1Ω , y ∈ Ω , such that F (x) = dV y. Then F (x − dV ny) = 0, which implies by [Ill79, I (3.21.1.2)] that x − dV ny ∈ V nΩi. d d The second claim follows from the fact that F : Wn+1Ω → R∗(WnFX )∗WnΩ is surjective on top ′d d degree d [Ill79, I (3.21.1.1)], and therefore WnΩ = (WnFX )∗WnΩ as WnOX -modules.

Deﬁnition 1.4 ((absolute) Cartier operator). Let X be a smooth scheme of dimension d over k. (1) The composition

−1 i • i • (γX ) i (1.2.4) C := CX :Z (FX,∗ΩX ) → H (FX,∗ΩX ) −−−−−→ ΩX i • i d i+1 with Z (FX,∗ΩX ) := Ker(FX,∗ΩX −→ FX,∗ΩX )

is called the (absolute) Cartier operator of degree i, denoted by C or CX . (2) (cf. [Kat86b, 4.1.2, 4.1.4]) More generally, for n ≥ 1, deﬁne the (absolute) Cartier operator Cn := Cn,X of level n to be the composite

−1 ′i ։ ′i n−1 i−1 F i (1.2.5) Cn : WnΩX WnΩX /dV Ω −−−→ WnΩX , X ≃ 8 FEI REN

i ≃ ′i n−1 i−1 where F : WnΩX −→ WnΩX /dV ΩX is the map in Lemma 1.3. When i = d is the top degree we obtain the WnOX -linear map

−1 d ։ d n−1 d−1 F d (1.2.6) Cn : (WnFX )∗WnΩX (WnFX )∗WnΩX /dV ΩX −−−→ WnΩX .

Remark 1.5. (1) According to the explicit formula for F , we have C = C1 [Ill79, I 3.3]. For this reason we will simply write C for Cn sometimes. (2) Cn (for all n) are compatible with ´etale pullbacks. Actually any de Rham-Witt system (e.g. • (WnΩX ,F,V,R,p, d)) is compatible with ´etale base change [CR12, 1.3.2]. (3) The n-th power of Frobenius F induces a map

n i ≃ i n • F : WnΩX −→H ((WnFX )∗ WnΩX ), which is the same as [IR83, III (1.4.1)]. −1 (4) Notice that on Spec Wnk, Cn : Wnk → Wnk is simply the map (WnFk) , because F : Wn+1k → Wnk equals R ◦ Wn+1Fk in characteristic p. (5) We sometimes omit ”(WnFX )∗” in the source. But one should always keep that in mind and be careful with the module structure.

Remark 1.6. Before we move on, we state a remark on ´etale schemes over WnX.

(1) Notice that every ´etale WnX-scheme is of the form Wng : WnU → WnX, where g : U → X is an ´etale X-scheme. In fact, there are two functors

F : {´etale WnX-schemes} ⇆ {´etale X-schemes} : G

V 7→ V ×WnX X

WnU ←[ U The functor F is a categorical equivalence according to [EGAIV-4, Ch. IV, 18.1.2]. The functor G is well-deﬁned (i.e. produces ´etale WnX-schemes) and is a right inverse of F by [Hes15, Thm. 1.25]. We want to show that there is a natural isomorphism GF ≃ id, and this is the consequence of the following purely categorical statement: If F : A→B and G : B → A are two functors satisfying both F being a categorical equivalence and F G ≃ id, then G is a quasi-inverse of F , i.e., there exists a canonical natural isomorphism GF ≃ id. We leave this as an easy exercise for the reader. (2) The square

WnFU WnU / WnU

Wng Wng

WnFX WnX / WnX.

is a cartesian square. This is because for any étale map g : U → X, the relative Frobenius FU/X is an isomorphism by [Fu15, 10.3.1]. Thus WnFU/X is an also isomorphism and the claim follows. We shall now state the main result in this subsection, which seems to be an old folklore (cf. proof of [Kat87, 3.4]). To eliminate possible sign inconsistency of the Cartier operator with the Grothendieck trace map calculated via residue symbols [Con00, Appendix A], we reproduce the proof by explicit calculations (see Section 1.2.2-Section 1.2.3). And at the same time, this result justifies our notation for C′: The classical Cartier operator C is simply the (−d)-th cohomology of our C′ in the smooth case. Theorem 1.7 (Compatibility of C′ with C). Suppose that X is a smooth scheme of dimension d over a perfect field k of characteristic p> 0. Then the top degree classical Cartier operator d d C : (WnFX )∗WnΩX/k → WnΩX/k as defined in Definition 1.4, agrees with the (−d)-th cohomology of the Cartier operator for residual complexes ′ d d C : (WnFX )∗WnΩX/k → WnΩX/k as defined in (1.1.3) via Ekedahl’s quasi-isomorphism Proposition 1.1.

Proof. The Cartier operator is stable under ´etale base change, i.e., for any ´etale morphism Wng : WnX → WnY (which must be of this form according to Remark 1.6(1)), we have ∗ d d CX ≃ (Wng) CY : (WnFX )∗WnΩX → WnΩX . CYCLE COMPLEX VIA COHERENT DUALITY 9

We claim that the map C′ defined in (1.1.3) is also compatible with étale base change. That is, whenever we have an étale morphism Wng : WnX → WnY , there is a canonical isomorphism ′ ∗ ′ CX ≃ (Wng) CY : (WnFX )∗Kn,X → Kn,X .

First of all, the Grothendieck trace map TrWnFX for residual complexes is compatible with étale base change by [Har66, VI.5.6], i.e., ∗ △ TrWnFX ≃ g TrWnFY : (WnFX )∗(WnFX ) Kn,X → Kn,X . Secondly, because of the cartesian square in Remark 1.6(2) and the flat base change theorem ∗ ∗ (Wng) (WnFX )∗ ≃ (WnFX ) (Wng)∗, we are reduced to show that (1.1.2) is compatible with étale base change. And this is true, because we have ∗ △ (Wng) ≃ (Wng) by [Har66, VI, 3.1 and 5.3], and the compatibility of (−)△ with composition by [Con00, (3.2.3)]. This finishes the claim. Note that the question is local on WnX. Thus to prove the statement for smooth k-schemes X, using ′ d the compatibility of C and C with respect to étale base change, it suffices to prove for X = Ak. That is, we need to check that the expression given in Lemma 1.14 for C′ agrees with the expression for C given in Lemma 1.10. This is apparent. 1.2.2. Proof of Theorem 1.7: C for the top Witt differentials on the affine space. Let k be a perfect field of positive characteristic p. The aim of this subsection is to provide the formula for the Cartier operator on the top degree de Rham-Witt sheaf over the affine space (Lemma 1.10). But before this, we first show a lemma which will be used in the calculation of Lemma 1.10. Lemma 1.8 (cf. [Kat86b, 4.1.2]). Let X be a smooth k-scheme. Then ′i i V = p ◦ Cn : R∗WnΩX → Wn+1ΩX , ′q i i where WnΩX denotes the abelian sheaf F (Wn+1ΩX ) regarded as a WnOX -submodule of (WnFX )∗WnΩX . Proof. Consider the following diagram p

F V * W Ωi / / W Ω′i / W Ωi . n+1 X n X❏ n+1 X ❏❏ ✉✉: ❏❏Cn ✉✉ R pr ❏ ✉ ❏❏ ✉✉p ❏❏ ✉✉ ′i −1 $ , ✉ i F WnΩX F i W Ω / i−1 / W Ω n X ≃ dV n−1Ω ≃ n X X 7 id i ′i n−1 i−1 Notice that F : WnΩ → WnΩ /dV Ω is an isomorphism by Lemma 1.3, and therefore we can take the inverse. All the small parts commute by definition (among these one notices that the top part commutes because X is of characteristic p), except the triangle on the right. Moreover one has the outer d ′d diagram commutes, due to the definition of p [Ill79, I 3.4]. Since F : Wn+1ΩX → WnΩX is surjective, commutativity of the right triangle follows from the known commutativities. m Notation 1.9. When we write an element in WnΩX in terms of a product with respect to an totally ordered index set, we make the following assumptions: when an index set is empty, the respective factor of the product does not occur; when an index set is non-empty, the factors of the product are ordered such that the indices are increasing. With these assumptions we avoid any confusion concerning signs. d d Lemma 1.10 (Cn on A ). Let X = Ak. Then the Cartier operator (cf. Definition 1.4) d d C := Cn : WnΩX → WnΩX is given by the following formula:

ji−1 ji−1 C α [Xi ]nd[Xi]n · [Xi ]nd[Xi]n

i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0

si ji si ji · dV ([Xi ]n−si ) · dV ([Xi ]n−si ) ! i/∈I,sYi6=n−1 i/∈I,sYi=n−1 10 FEI REN

−1 ji/p−1 1 ji = (WnFk) (α) [Xi ]nd[Xi]n · dV ([Xi ]n−1) ji i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 si+1 ji · dV ([Xi ]n−si−1) · 0 , i/∈I,sYi6=n−1 i/∈I,sYi=n−1 where α ∈ Wnk. As for the sign of the product we follow Notation 1.9.

Proof. Write X = {X1,...,Xd−1} (empty when d = 1) and S = Xd. According to [HM04, (4.2.1)], any d element in WnΩk[X,S] is uniquely written as n−1 (n) j−1 s (n−s) j (1.2.7) b0,j [S ]nd[S]n + dV (bs,j [S ]n−s), s=1 Xj≥1 X Xp∤j (n) d−1 (n−s) d−1 d where b0,j ∈ WnΩk[X], and for s ∈ [1,n − 1], bs,j ∈ Wn−sΩk[X]. (Here we have used WnΩk[X] = 0 d and Wn−sΩk[X] = 0.) Now compute d d F : Wn+1Ωk[X,S] → WnΩk[X,S], (n+1) j−1 (n+1) pj−1 (1.2.8) b0,j [S ]n+1d[S]n+1 7→ F (b0,j )[S ]nd[S]n; s (n+1−s) j s−1 (n+1−s) j (1.2.9) dV (bs,j [S ]n+1−s) 7→ dV (bs,j [S ]n+1−s); and d d V : WnΩk[X,S] → Wn+1Ωk[X,S], p (1.2.10) b(n)[Sj−1] d[S] 7→ (−1)d−1 dV (b(n)[Sj ] ) when v (j) = 0. 0,j n n j 0,j n p (n) d In the last equation we used db0,j ∈ WnΩk[X] = 0. Therefore, according to (1) p = p ◦ R where p is injective, by [Ill79, I 3.4], (2) V = p ◦ Cn, by Lemma 1.8, and (3) Cn ◦ F = R (because of (1)(2)), d d (4) F : Wn+1Ωk[X,S] → WnΩk[X,S] is surjective, one gets d d (1.2.11) Cn : WnΩk[X,S] → WnΩk[X,S]

C (b(n))[Sj/p−1] d[S] , v (j) 6= 0; (by (1.2.8) (n) j−1 n 0,j n n p b [S ]nd[S]n 7→ d−1 0,j (−1) dV (R(b(n))[Sj ] ), v (j) = 0. (by (1.2.10)) (j ≥ 1) ( j 0,j n−1 p

s+1 (n−s) j s (n−s) j dV (R(bs,j )[S ]n−s−1), 1 ≤ s ≤ n − 2; (by (1.2.9)) dV (bs,j [S ]n−s) 7→ −1 (s ∈ [1,n − 1],p ∤ j) ( 0, s = n − 1. (by Cn = F ◦ pr) (n) Note that Cn(b0,j ) is computed via the induction on d: when d = 1, (n) −1 (n) Cn(b0,j ) = (WnFk) (b0,j ) ∈ Wnk because F = R ◦ WnFk : Wnk → Wn−1k (note that chark = p). (n) d−1 Since b0,j ∈ WnΩk[X] could also be written in expression (1.2.7), we could further write (1.2.11) out. d That is to say, every element in WnΩAd is uniquely written as a sum of expressions of the form k

ji−1 si ji (1.2.12) α [Xi ]nd[Xi]n dV ([Xi ]n−si ) , Yi∈I i∈[1Y,d]\I ji−1 where α ∈ Wnk, I ⊂ [1, d] an index subset (I is the set of indices taken the form [Xi ]d[Xi] and the si ji rest indices takes the form dV ([Xi ]n−si )), and

{ji}i∈[1,d], {si}i∈[1,d]\I some integers, satisfying

• ji ≥ 1, when i ∈ I, and CYCLE COMPLEX VIA COHERENT DUALITY 11

• vp(ji) = 0 and si ∈ [1,n − 1] when i ∈ [1, d] \ I.

Cn maps each of the factors of (1.2.12) in the following way: −1 α 7→ Wn(Fk) (α), α ∈ Wnk,

ji/p−1 ji−1 [Xi ]nd[Xi]n, vp(ji) ≥ 1; [Xi ]nd[Xi]n 7→ 1 ji dV ([X ]n−1), vp(ji) = 0. ( ji i

s +1 ji j dV i ([X ] ), s 6= n − 1; dV si ([X i ] ) 7→ i n−si−1 i i n−si 0, s = n − 1. i

1.2.3. Proof of Theorem 1.7: C′ for the top Witt differentials on the affine space. The aim of Section 1.2.3 is to calculate C′ for top de Rham-Witt sheaves on the affine space (Lemma 1.14). To do this, one needs to first calculate the trace map of the canonical lift of the absolute Frobenius.

1.2.3.1. Trace map of the canonical lift F of absolute Frobenius FX . Let k be a perfect ﬁeld of positive Xe d characteristic p. Let X = Ak, and let X := Spec Wn(k)[X1,...,Xd] be the canonical smooth lift of X over Wn(k). To make explicit the modulee structures, we distinguish the source and the target of the absolute Frobenius of Spec k and write ite as

Fk : Spec k1 → Spec k2 Similarly, write the absolute Frobenius on X as

FX : X = Spec k1[X1,...,Xd] → Y = Spec k2[X1,...,Xd].

There is a canonical lift F of FX over X, and we write it as Xe

F : X = Spec Wn(k1)[X1,...,Xd] → Y := Spec Wn(k2)[Y1,...,Yd]. Xe e e F is given by Xe e e e ∗ F : Γ(Y, OY )= Wn(k2)[Y1,...,Yd] → Wn(k1)[X1,...,Xd]=Γ(X, OX ), e Xe e e Wnk2 ∋ α 7→ Wn(Fk)(α), e e p e Yi 7→ Xi .

on the level of global sections. Clearly F restricts to FX on X. Let Xe

πX : X → Spec k1, πY : Y → Spec k2, π : X → Wnk1, π : Y → Wnk2 e Xe Ye be the structure maps. The composition F ◦ π : X → Spec Wnk2 gives X a Wnk2-scheme structure, Xe Ye e e and the map FX is then a map of Wnk2-schemes. Therefore the trace map e e e e Tr : F F △K → K e FX X,∗ X Y Y ef e e e e makes sense. Consider the following map of complexese e F π△(1.1.1) eX,∗ X △ f f △ △ FX,∗KX ≃ FX,∗π Wnk1 −−−−−−−−−−→FX,∗π WnFk Wnk2 ≃ e e e Xe ∼ e Xe

TrF eX e e e △ △ △ f FX,∗F π Wnk2 ≃ FX,∗F KY −−−→ KY . e Xe Ye e Xe e e Taking the (−d)-th cohomology, it induces a map e e e e d d (1.2.13) FX,∗Ω → Ω X/We nk1 Ye /Wnk2 In the following lemma we will computee this map. Lemma 1.11. The notations are the same as above. The map (1.2.13) has the following expression:

(1.2.13) (1.2.14) Ωd −−−−−−→ Ωd X/We nk1 Ye /Wnk2 (W F )−1(α)Y µdY , when λ = p − 1 for all i; αXλ+pµdX 7→ n k i 0, when λ 6= p − 1 for some i. i 12 FEI REN

Proof. Construct a regular immersion of X into P = Ad associated to the following homomorphism of Y rings: e e e Γ(P, O )= Wn(k2)[Y1,...,Yd,T1,...,Td] → Wn(k1)[X1,...,Xd]=Γ(X, O ), Pe Xe α 7→ (WnFk)(α), α ∈ Wn(k2), e p e Yi 7→ Xi , i =1, . . . , d, Ti 7→ Xi, i =1, . . . , d. Its kernel is p p I = (T1 − Y1,...,Td − Yd). Denote p ti = Ti − Yi,i =1, . . . , d.

Obviously the ti’s form a regular sequence in Γ(P, O ). Denote by i the associated closed immersion. Pe Then one has a factorization of FX : e e i (1.2.15) X = Spec Wn(ke1)[X1,...,Xd] / P = Spec Wn(k2)[Y1,...,Yd,T1,...,Td] ❳❳❳❳❳ ❳❳❳❳❳ ❳❳❳❳❳ π e e ❳❳❳❳❳ FX ❳❳ ef + Y = Spec Wn(k2)[Y1,...,Yd].

Regarding X as a Wnk2-scheme via the composite map F ◦eπ , the diagram (1.2.15) is then a diagram Xe Ye in the category of Wnk2-schemes. A generale element in Γ(X, Ωd ) is a sum of expressionse of the form X/We nk1 λ+pµ d d (1.2.16) eαX dX, α ∈ Wnk1, λ ∈ [0,p − 1] , µ ∈ N .

λ λ1 λd µ Here λ = {λ1,...,λd}, µ = {µ1,...,µd} are multi-indices, and X := X1 ...Xd (similar for Y , λ+pµ d X , etc.), dX := dX1 ...dXd (similar for dT , etc.). The element (1.2.16) in Γ(X, Ω ) corre- X/We nk1 sponds to −1 λ+pµ d d e (1.2.17) (WnFk) (α)X dX, α ∈ Wnk2, λ ∈ [0,p − 1] , µ ∈ N , in Γ(X, Ωd ) under (−d)-th cohomology of the map F π△(1.1.1), and X/W k X,∗ e n 2 e Xe −1 λ µ d d e (WnFk) (α)T Y dT , α ∈ Wnk2e, λ ∈ [0,p − 1] , µ ∈ N is a lift of (1.2.17) to Γ(P, Ωd ). Write Pe /Wnk2 −1 λ µ eβ := dtd ∧···∧ dt1 ∧ (WnFk) (α)T Y dT d −1 λ µ = (−1) dYd ∧···∧ dY1 ∧ (WnFk) (α)T Y dT

in Γ(P,ω ) (ω denotes the dualizing sheaf with respect to the smooth morphism P → Pe /Wnk2 Pe /Wnk2 Wnk2). One can write out the image of β under map [Con00, p.30 (a)], i.e., e e ω ≃ ω ⊗ π∗ω , P /W k P /Y OP Y /W k e n 2 e e e e n 2 d(3d+1) −1 λ ∗ µ β 7→ (−1) 2 (WnFk) (α)T dT ⊗ π Y dY .

where ω and ω denote the dualizing sheaf with respect to the smooth morphisms π : P → Y P/e Ye Ye /Wnk2 and Y → Wnk2. It’s easily seen that FX is a finite flat morphism between smooth Wnk2-schemes. Applying [CR11, Lemma A.3.3], one has e e e e e λ −1 λ+pµ −1 T dT µ TrF ((WnFk) (α)X dX) = (WnFk) (α)ResP/Y Y dY , eX e e t ,...,t f 1 d T λdT where ResP/Y ∈ Γ(Y, OY ) is the residue symbol defined in [Con00, (A.1.4)], and TrF is e e t ,...,t e eX 1 d f the trace map on top differentialse of the Wnk2-morphism FX [Con00, (2.7.36)]. We consider the following cases (in the following (RN) withe N ∈ [1, 10] being a positive integer means the corresponding property from [Con00, §A]): e CYCLE COMPLEX VIA COHERENT DUALITY 13

λ d−1 • When (λ1,...,λn) 6= (p − 1,...,p − 1), T dT = dη for some η ∈ Ω . Suppose without loss of P/e Ye generality that λ1 6= p − 1. Then we can take

1 λ1+1 λ2 λd η = T1 T2 ...Td dT2 ...dTd. λ1 +1 Noticing that p p−1 dti = d(Ti − Yi)= pTi dTi in Ω , and that λ1 + mp + 1 (m ∈ Z>0) is not divisible by p when λ1 +1 is so. Now we P/e Ye calculate

λ λ1+1 λ2 λd T dT 1 d(T1 T2 ...Td dT2 ...dTd) ResP/Y = ResP/Y e e t1,...,tn λ +1 e e t ,t ,...,t 1 1 2 n λ1+p λ2 λd p T1 T2 ...Td dT1dT2 ...dTd = Res 2 (by (R9)) λ +1 P/e Ye t ,t ,...,t 1 1 2 n λ1+p+1 λ2 λd p d(T1 T2 ...Td dT2 ...dTd) = Res 2 (λ + 1)(λ + p + 1) P/e Ye t ,t ,...,t 1 1 1 2 n 2 λ1+2p λ2 λd 2p T1 T2 ...Td dT1dT2 ...dTd = Res 3 (by (R9)) (λ + 1)(λ + p + 1) P/e Ye t ,t ,...,t 1 1 1 2 n 2 λ1+2p+1 λ2 λd 2p d(T1 T2 ...Td dT2 ...dTd) = Res 3 2 P/e Ye t ,t ,...,t i=0(λ1 + ip + 1) 1 2 n 6p3 λ1+3p λ2 λd Q T1 T2 ...Td dT1dT2 ...dTd = ResP 4 (by (R9)) 2 e/Ye t ,t ,...,t i=0(λ1 + ip + 1) 1 2 n = ... Q n ( i) · pn λ1+np λ2 λd i=1 T1 T2 ...Td dT1dT2 ...dTd = ResP n+1 (by (R9)) n−1 e/Ye t ,t ,...,t i=0Q(λ1 + ip + 1) 1 2 n =0. Q The last step is because pn = 0 in Γ(Y, O ). Ye • When (λ1,...,λn) = (p − 1,...,p − 1), consider

′ ′ ′ ′ e ′ Z[Y1 ,...,Yd,T1,...,Td] ′ ′ ′ ′ ′ (1.2.18) X := Spec ′ p ′ ′ p ′ / Spec Z[Y1 ,...,Yd,T1,...,Td] =: P (T1 −Y1 ,...,Td −Yd ) ❲❲❲❲❲ ❲❲❲❲❲ ❲❲❲❲❲ f ❲❲❲❲❲ + ′ ′ ′ Spec Z[Y1 ,...,Yd] =: Y .

∗ ′ ′ ′p ′ d f is given by f (Yi )= Yi = Ti in Γ(X , OX′ ). This is a ﬁnite locally free morphism of rank p . Consider the map ′ ′ ′ h : Γ(Y , OY ′ )= Z[Y1 ,...,Yd] → Wn(k2)[Y1,...,Yd]=Γ(Y, O ), Ye ′ Yi 7→ Yi for all i, e ′ that relates the two diagrams (1.2.18) and (1.2.15). In Γ(Y , OY ′ ), we have ′p−1 ′p−1 ′ ′ ′p ′ ′p ′ d T1 ...Td dT1 ...dTd d(T1 − Y1 ) ...d(Td − Yd) p · Res ′ ′ = Res ′ ′ P /Y T ′p − Y ′,...,T ′p − Y ′ P /Y T ′p − Y ′,...,T ′p − Y ′ 1 1 d d 1 1 d d (R6) = TrX′/Y ′ (1) = pd.

The notation TrX′/Y ′ denotes the classical trace map associated to the ﬁnite locally free ring ′ ′ d extension Γ(Y , OY ′ ) → Γ(X , OX′ ). As for the last equality, TrX′/Y ′ (1) = p because f is a d d ′ ﬁnite locally free map of rank p . Since p is a non-zerodivisor in Γ(Y , OY ′ ), one deduces ′p−1 ′p−1 ′ ′ T1 ...Td dT1 ...dTd Res ′ ′ =1. P /Y T ′p − Y ′,...,T ′p − Y ′ 1 1 d d Set p−1 p−1 p−1 T = T1 ...Td , 14 FEI REN which is the canonical lift of Xλ via the map i : X ֒→ P in our current case. Pulling back to

Γ(Y, OY ) via h, one has e e e e −1 p−1 p−1 p (R5) ′ ′ ′ ′ T dT ∗ T1 ...Td dT1 ...dTd (1.2.19) Res = h ResP ′/Y ′ ′p ′ ′p ′ =1. P/e Ye t ,...,t T − Y ,...,T − Y 1 d 1 1 d d

Altogether, we know that the map (1.2.13) takes the following expression

Ωd → Ωd X/We nk1 Ye /Wnk2 (W F )−1(α)Y µdY , when λ = p − 1 for all i; αXλ+pµdX 7→ n k i 0, when λ 6= p − 1 for some i. i

′ 1.2.3.2. C for top Witt differentials. Now we turn to the Wn-version. The aim of this subsection is to ′ d calculate C for top Witt differentials on Ak (Lemma 1.14). Let f : X → Y be a finite morphism between smooth, separated and equidimensional k-schemes of dimension d. Same as before, we denote by πX : X → k and πY : Y → k the respective structure maps. △ △ Kn,X := (WnπX ) Wnk,Kn,Y = (WnπY ) Wnk are residual complexes on X and Y . Then we define the trace map

d d (1.2.20) TrWnf : (Wnf)∗(WnΩX ) → WnΩY

to be the (−d)-th cohomology map of the composition

ev. at 1 (1.2.21) TrWnf : (Wnf)∗Kn,X ≃ HomWnOY ((Wnf)∗WnOX ,Kn,Y ) −−−−−→ Kn,Y

d −d via Ekedahl’s isomorphism WnΩX ≃ H (Kn,X ) Proposition 1.1. Computation of the trace map is a local problem on Y . Thus by possibly shrinking Y we could assume that Y and (therefore also X) is affine. In this case, there exist smooth affine Wnk-schemes X and Y which lift X and Y . Denote the structure morphisms of X, Y by πX and πY , respectively. Then there e e e e exists a finite Wnk-morphism f : X → Y lifting f : X → Y by the formal smoothness property of Y . Consider the map of abelian sheaves [Eke84, I (2.3)] e e e e e e

∗ ϑY 0 • , Y : WnOY −−→ H (Ω ) ֒→ OY̺ (1.2.22) ≃ Ye /Wnk e n−1 i pn pn−1 n−1 p V ([ai]) 7→ a0 + pa1 + ··· + p an−1, i=0 X e e e where ai ∈ OY , and ai ∈ OY being arbitrary liftings of ai. The map ϑY appearing above is the i = 0 case of the canonical isomorphisme deﬁned in [IR83, III. 1.5] e

i ≃ i • (1.2.23) ϑY : WnΩY −→H (Ω ). Ye /Wnk

∗ Note that the map ̺Y : WnOY → O is a morphism of sheaves of rings, and it induces a ﬁnite morphism Ye ̺Y : WnY → Y (cf. [Eke84, I, paragraph after (2.4)]). Altogether we have the following commutative

e CYCLE COMPLEX VIA COHERENT DUALITY 15

diagram of schemes (cf. [Eke84, I. (2.4)])

̺X X / WnX ④④= ③< iX ④ iX ③③ e ④④ ③③ ④④ e ③③ . ④④ . ③③ W f X fe X n

f f ̺Y Y / WnY ④④= ③< iY ④ iY ③③ e ④④ ③③ ④④ e ③③ . ④④ . ③③ πY W π Y e Y n Y

πY πY ≃ Wnk / Wnk. W (F n) ③③< n k ②②< ③③ ②② ③③ ②② ③③ ② . ③ ≃ - ②② k n / k Fk

△ △ Lemma 1.12. Set KX = π Wnk, and KY = π Wnk. The (−d)-th cohomology of the map Trf : e Xe e Ye e d d f∗KX → KY gives a map f∗Ω → Ω , which we again denote by Trf . Then by passing to quotients, e e Xe Ye e this map Trf induces a well-deﬁned map e e e

d • d • τf : H (f∗Ω ) → H (Ω ). e Xe Ye e Moreover, the map τ is compatible with TrWnf deﬁned in (1.2.21) : fe

Tr d Wnf d (Wnf)∗WnΩX / WnΩY

ϑY ≃ (Wnf)∗ϑX ≃

(̺Y )∗τ d • fe d • (̺Y f)∗H (Ω ) / (̺Y )∗H (Ω ) Xe Ye e d • d • Proof. We do it the other way around, namely we define the map τf : H (f∗Ω ) → H (Ω ) via e Xe Ye d d d d TrWnf : (Wnf)∗WnΩX → WnΩY , and then show that this is the reduction of Tr : f∗Ω → Ω . fe X Y d e d e ee First of all, via isomorphisms ϑX , ϑY , the map TrWnf : (Wnf)∗WnΩX → WnΩY defined in (1.2.21) d • d • e induces a well-defined map τ : H (f∗Ω ) → H (Ω ). To show compatibility with Tr , one needs an fe X Y fe observation of Ekedahl: Ekedahl observede that the compositee e

△ n (̺Y )∗π (1.1.1) ≃ Y d △ e tY : (̺Y )∗ΩY /W k[d] −→ (̺Y )∗KY ≃ (̺Y )∗π Wnk −−−−−−−−−−−→ e n e Ye ≃ Tr △ n △ △ △ ̺Y (̺Y )∗π (WnFk ) Wnk ≃ (̺Y )∗(̺Y ) (WnπY ) Wnk −−−→ Kn,Y Ye

d • factors through tY : (̺Y )∗H (Ω )[d] → Kn,Y (cf. [Eke84, §1 (2.6)]). Then he deﬁned the map Ye /Wnk d WnΩY [d] → Kn,Y to be the composite

d ϑY d • tY (1.2.24) sY : WnΩY [d] −−→ H (Ω )[d] −→ Kn,Y . ≃ Ye /Wnk 16 FEI REN

Now consider the following diagram of complexes of sheaves

TrWnf (Wnf)∗Kn,X / Kn,Y ♠6 E O ♥♥6 F O (W f) s ♠♠ ☞ ♥♥ ✌ n ∗ X♠♠ ☞ sY ♥♥ ✌ ♠♠♠ ☞ ♥♥♥ ✌ ♠♠♠ ☞☞ ♥♥♥ ✌✌ ♠ ☞ Tr ♥ ✌ d ☞ Wnf d ✌ (Wnf)∗WnΩ [d] ☞ / WnΩ [d] ✌ X ☞☞ Y ✌✌ ☞☞ ✌✌ ☞☞ ✌✌ ☞ (Wnf)∗tX ✌ tY (Wnf)∗tX ☞ tY ✌ ☞☞ ✌✌ (Wnf)∗ϑX ≃ ☞ ϑY ≃ ✌ ☞☞ ✌ ☞ ✌✌ ☞ (̺Y )∗ Tr ✌ ☞☞ d fe ✌ d ☞ (̺Y f)∗Ω [d] ✌✌ / (̺Y )∗Ω [d]. ☞ Xe ✌ Ye ☞☞ ♥♥♥ ✌ ♦♦♦ ☞ ♥♥♥ ✌✌ ♦♦♦ ☞ ♥♥♥ e ✌ ♦♦ ☞☞ ♥♥♥ ✌ ♦♦♦ w (̺Y )∗τ ✌ w♦ d • fe d • (̺Y f)∗H (Ω )[d] / (̺Y )∗H (Ω )[d] Xe Ye The unlabeled arrows are given by the natural quotient maps. The front commutes by the definition of τ . e fe d d The top commutes by the definition of TrWnf : (Wnf)∗WnΩX → WnΩY . The triangles in the right (resp. the left) side commute due to the definition of tY and sY (resp. tX and sX ). The back square commutes, because the trace map Tr is functorial with respect to maps residual complexes with the same associated fe filtration by [Con00, 3.4.1(1)]. We want to show that the bottom square commutes. To this end, it suffices d d d d to show (̺Y )∗ Tr : (̺Y f)∗Ω → (̺Y )∗Ω is compatible with TrWnf : (Wnf)∗WnΩX → WnΩY via ϑX fe X Y e e d d and ϑY . Because the map TrWnf : (Wnf)∗WnΩX → WnΩY is determined by the degree −d part of the e d map TrWnf : (Wnf)∗Kn,X → Kn,Y , we are reduced to show compatibility of (̺Y )∗ Trf : (̺Y f)∗Ω → e Xe (̺ ) Ωd with Tr : (W f) K → K via (W f) (s ◦ ϑ−1) and s ◦ ϑ−1. By commutativity Y ∗ Y Wnf n ∗ n,X n,Y n ∗ X X Y Y of the lefte and right squares, this is reduced to the commutativity of the square on the back,e which is known. Therefore the bottom square commutes as a result. τ is just a temporary notation for the lemma above. Later we will denote τ by Tr . fe fe fe Before we proceed to the main result of this subsection, we present the following lemma, which gives • the explicit formula for ϑY . We will use WnΩ to denote the relative de Rham-Witt complex defined Ye /Wnk by [LZ04]. Lemma 1.13 ([BER12, 8.4(ii)]). Notations are the same as above. There is a commutative diagram

F n W Ωq / Zq(Ω• ) n+1 Y /W k Ye /Wnk e n

q ϑY q • WnΩY / H (Ω ). Ye /Wnk In particular, n−1 n−1 i n i pn−1 pn−1−1 p−1 ϑY ( dV ([ai])) = F dV ([ai]) = a0 da0 + a1 da1 + ··· + an−1dan−1. i=0 i=0 X X n Proof. One just need to check for q = 0: fore highereq, botheϑY eand F aree generatede by thee q = 0 case as n−1 i n i morphisms of diﬀerential graded algebras. Take i=0 V ([ai]n−i) ∈ WnOY . Then i=0 V ([ai]n+1−i) ∈ n−i n n i n i n−i n i p Wn+1OY is a lift. F ( i=0 V ([ai]n+1−i)) = i=0 p F ([ai]n+1−i)= i=0 p ai . We are done with e n P P the relation p an = 0 in O . e P Ye P P d e e e Lemma 1.14. Let X = Ak. Let e ′ ′ d d C = Cn : WnΩX → WnΩX . be the map given by the −d-th cohomology of the level n Cartier operator for residual complexes (cf. (1.1.3)). Then C′ is given by the following formula:

′ ji−1 ji−1 C α [Xi ]nd[Xi]n [Xi ]nd[Xi]n

i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 CYCLE COMPLEX VIA COHERENT DUALITY 17

si ji si ji dV ([Xi ]n−si ) dV ([Xi ]n−si ) ! i/∈I,sYi6=n−1 i/∈I,sYi=n−1 −1 ji/p−1 1 ji = (WnFk) (α) [Xi ]nd[Xi]n dV ([Xi ]n−1) ji i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 si+1 ji dV ([Xi ]n−si−1) 0 , i/∈I,sYi6=n−1 i/∈I,sYi=n−1 where α ∈ Wnk. As for the sign of the product we follow Notation 1.9. d Proof. Consider the map WnFX : WnX → WnX with X := Ak. It is not a map of Wnk-schemes a priori, but after labeling the source by W X := W Ad and the target by W Y := W Ad , one can n n k1 n n k2 realize WnFX as a map of Wnk2-schemes (the Wnk2-scheme structure of WnX is given by WnFX ◦WnπY , where πY : Y → k2 denotes the structure morphism of the scheme Y ). Write Ad Ad X = Wnk1 = Spec Wnk1[X1,...,Xd] (resp. Y = Wnk2 = Spec Wnk2[X1,...,Xd]), and take thee canonical lift FX of FX as in Lemma 1.11.e Consider (1.2.25) e

(WenFX )∗(1.1.2) TrWnF (W F ) W Ωd / (W F ) W Ωd X / W Ωd n X ∗ n X/k1 ≃ n X ∗ n X/k2 n Y/k2

(WnFX )∗ϑX (WnFX )∗ϑX ϑY △ (̺Y F )∗π (1.1.1) (̺Y )∗(TrF ) eX X eX d • f f d • f d • (̺Y FX )∗H (Ω ) ≃ / (̺Y FX )∗H (Ω ) / (̺Y )∗H (Ω ). e X/We nk1 e X/We nk2 Ye /Wnk2 The compositee map of the top row is C′ (cf. (1.1.3e ) and Ekedahl’s quasi-isomorphism Proposition 1.1). The composite of the bottom row is induced from ̺Y,∗(1.2.13). The right side commutes due to Lemma 1.12. The left side commutes by naturality. Given an index set I ⊂ [1, d] and integers {ji}i∈[1,d], {si}i∈[1,d]\I satisfying

• ji ≥ 1, when i ∈ I, and • vp(ji) = 0 and si ∈ [1,n − 1] when i ∈ [1, d] \ I, recall that a general element in W Ωd is the sum of expressions of the following form (cf. (1.2.12)) n X/k1

ji−1 si ji (1.2.26) α [Xi ]nd[Xi]n dV ([Xi ]n−si ) , Yi∈I Yi/∈I where α ∈ Wnk1. One can also write this expression (1.2.26) in terms of ﬁner index sets:

ji−1 ji−1 α [Xi ]nd[Xi]n [Xi ]nd[Xi]n i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 si ji si ji dV ([Xi ]n−si ) dV ([Xi ]n−si ) . i/∈I,sYi6=n−1 i/∈I,sYi=n−1 where α ∈ Wnk1 (which might diﬀer from the α in (1.2.26) by a sign because we might have changed the order of the factors in the wedge product). As we explained after diagram (1.2.25), we can decompose C′ in the following way: ′ −1 d d C = ϑY ◦ (1.2.13) ◦ ϑX : WnΩX/k1 → WnΩY/k2 . According to the explicit formula (1.2.14) of the map (1.2.13), and the explicit formula for the maps ϑX , ϑY (Lemma 1.13), one could perform the following calculations:

′ ji−1 ji−1 C α [Xi ]nd[Xi]n [Xi ]nd[Xi]n

i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0

si ji si ji dV ([Xi ]n−si ) dV ([Xi ]n−si ) ! i/∈I,sYi6=n−1 i/∈I,sYi=n−1 n n −1 n p ji−1 p ji−1 = (ϑY ◦ (1.2.13)) Wn(Fk) (α) Xi dXi Xi dXi

i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 18 FEI REN

n−s n−s p i ji−1 p i ji−1 jiXi dXi jiXi dXi ! i/∈I,sYi6=n−1 i/∈I,sYi=n−1 n−1 n−1 −1 n−1 p ji−1 p ji−1 = ϑY Wn(Fk) (α) Xi dXi Xi dXi

i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0

n−s −1 n−s −1 p i ji−1 p i ji−1 jiXi dXi jiXi dXi ! i/∈I,sYi6=n−1 i/∈I,sYi=n−1 −1 ji/p−1 1 ji = (WnFk) (α) [Xi ]nd[Xi]n dV ([Xi ]n−1) ji i∈I,vYp(ji)≥1 i∈I,vYp(ji)=0 si+1 ji dV ([Xi ]n−si−1) 0 . i/∈I,sYi6=n−1 i/∈I,sYi=n−1

1.2.4. Criterion for surjectivity of C′ − 1. The following proposition is proven in the smooth case by Illusie-Raynaud-Suwa [Suw95, 2.1]. The proof presented here is due to Rülling. Proposition 1.15 (Raynaud-Illusie-Suwa). Let k = k be an algebraically closed field of characteristic p> 0. X is a separated scheme of finite type over k. Then for every i, C′ − 1 induces a surjective map on global cohomology groups

′ i i C −1 i H (WnX,Kn,X ) := R Γ(WnX,Kn,X ) −−−→ H (WnX,Kn,X ). In particular, i i C′−1 R Γ(WnX,Kn,X,log) ≃ H (WnX,Kn,X ) . Proof. Take a Nagata compactification of X, i.e., an open immersion j : X ֒→ X such that X is proper over k. The boundary X \ X is a closed subscheme in X. By blowing up in X one could assume X \ X is the closed subscheme associated to an effective Cartier divisor D on X. We could thus assume j is an affine morphism. Therefore

Wnj : WnX ֒→ WnX is also an affine morphism. ∗ For any quasi-coherent sheaf M on WnX, the difference between M and (Wnj)∗(Wnj) M are precisely those sections that have poles (of any order) at SuppD = WnX \ WnX. Suppose that the effective Cartier divisor D is represented by (Ui,fi)i, where {Ui}i is an affine cover of X, and fi ∈ Γ(Ui, OX ). Recall that OX (mD) denotes the line bundle on X which is the inverse (as line bundles) of the m-th power of the ideal sheaf of X \ X ֒→ X. Locally, one has an isomorphism 1 OX (mD) |Ui ≃ OUi · m fi for each i. Denote by WnOX (mD) the line bundle on WnX such that 1 WnOX (mD) |Ui ≃ WnOUi · m , [fi] where [−] = [−]n denotes the Teichmüller lift. Denote

M(mD) := M⊗WnOX WnOX (mD). The natural map ≃ ∗ ∗ (1.2.27) M(∗D) := colimm M(mD) −→ (Wnj)∗(Wnj) (M(mD)) = (Wnj)∗(Wnj) M is an isomorphism of sheaves. Here the inductive system on the left hand side is given by the natural map

M(mD) := M⊗WnOX WnOX (mD) →M⊗WnOX WnOX ((m + 1)D) induced from the inclusion WnOX (mD) ֒→ WnOX ((m + 1)D), i.e., locally on Ui, this inclusion is the map

WnOX (mD) |Ui ֒→ WnOX ((m + 1)D) |Ui CYCLE COMPLEX VIA COHERENT DUALITY 19

a a[fi] m 7→ m+1 . [fi] [fi]

where a ∈ WnOUi . As a result, i ∗ i ∗ (1.2.28) H (WnX, (Wnj) M)= H (RΓ(WnX, R(Wnj)∗(Wnj) M)) i ∗ = H (RΓ(WnX, (Wnj)∗(Wnj) M)) (Wnj is aﬃne) i = H (RΓ(WnX, colimm M(mD)) (1.2.27) i = colimm H (WnX, M(mD)).

Apply this to the bounded complex Kn,X of injective quasi-coherent WnOX -modules. Taking into ∗ account Kn,X ≃ (Wnj) Kn,X by [Har66, VI.5.3], (1.2.27) gives an isomorphism of complexes ≃ (1.2.29) Kn,X (∗D) := colimm Kn,X (mD) −→ (Wnj)∗Kn,X ,

and (1.2.28) gives an isomorphism of Wnk-modules i i colimm H (WnX,Kn,X (mD)) = H (WnX,Kn,X ). Via the projection formula [Har66, II.5.6] and tensoring ′ C : (WnFX )∗Kn,X → Kn,X

with WnOX (mD), one gets a map ∗ (WnFX )∗(Kn,X ⊗WnOX WnOX (pmD)) ≃ (WnFX )∗(Kn,X ⊗WnOX (WnFX ) WnOX (mD)) ′ C ⊗idW O (mD) n X ≃ ((WnFX )∗Kn,X ) ⊗WnOX WnOX (mD) −−−−−−−−−−−→ Kn,X ⊗WnOX WnOX (mD). Precomposing with the natural map

(WnFX )∗(Kn,X ⊗WnOX WnOX (mD)) → (WnFX )∗(Kn,X ⊗WnOX WnOX (pmD)), and taking global section cohomologies, one gets ′ i i C : H (WnX,Kn,X (mD)) → H (WnX,Kn,X (mD)). To show surjectivity of ′ i i C − 1 : H (WnX,Kn,X ) −→ H (WnX,Kn,X ), it suffices to show surjectivity for ′ i i C − 1 : H (WnX,Kn,X (mD)) → H (WnX,Kn,X (mD)). q q Because H (Kn,X ) are coherent sheaves on the proper scheme X for all q, H (Kn,X ⊗WnOX WnOX (mD)) = q H (Kn,X ) ⊗WnOX WnOX (mD) are also coherent, therefore the local-to-global spectral sequence implies that i M := H (WnX,Kn,X (mD)) ′ −1 is a finite Wnk-module. Now M is equipped with an endomorphism C which acts p -linearly (cf. Definition A.4). The proposition is then a direct consequence of Proposition A.6. The following proposition is a corollary of [Suw95, Lemma 2.1]. We restate it here as a convenient reference. Proposition 1.16 (Raynaud-Illusie-Suwa). Assume k = k. When X is separated smooth over k of pure dimension d, d d C − 1 : WnΩX → WnΩX is surjective. −d d Proof. Apply affine locally the H -case of Proposition 1.15. Then Ekedahl’s quasi-isomorphism WnΩX [d] ≃ ′ Kn,X from Proposition 1.1 together with compatibility of C and C from Theorem 1.7 gives the claim.

Remark 1.17. When X is Cohen-Macaulay of pure dimension d, WnX is also Cohen-Macaulay by Serre’s criterion of the same pure dimension, and thus Kn,X,t is concentrated at degree −d for all n [Con00, 3.5.1]. Denote by WnωX the only nonzero cohomology sheaf of Kn,X in this case. Then the same reasoning as in Proposition 1.16 shows that when k = k and X is Cohen-Macaulay over k of pure dimension, the map ′ C − 1 : WnωX → WnωX is surjective. 20 FEI REN

d 1.2.5. Comparison between WnΩX,log and Kn,X,log. Let X be a k-scheme. Denote by d log the following map of abelian étale sheaves ∗ ⊗q q d log : (OX,ét) → WnΩX,ét,

a1 ⊗···⊗ aq 7→ d log[a1]n ...d log[aq]n,

∗ d[ai]n where a1,...,aq ∈ O ,[−]n : OX,ét → WnOX,ét denotes the Teichmüller lift, and d log[ai]n := . X,ét [ai]n q We will denote its sheaf theoretic image by WnΩX,log,ét and call it the étale sheaf of log forms. We q q q denote by WnΩX,log := WnΩX,log,Zar := ǫ∗WnΩX,log,ét, and call it the Zariski sheaf of log forms. Lemma 1.18 ([CSS83, lemme 2], [GS88a, 1.6(ii)]). Let X be a smooth k-scheme. Then we have the following left exact sequences

q q 1−F q n−1 (1.2.30) 0 → WnΩX,log → WnΩX −−−→ WnΩX /dV ,

q ′q C−1 q (1.2.31) 0 → WnΩX,log → WnΩX −−−→ WnΩX , ′q q where WnΩX := F (Wn+1ΩX ). The last maps are also surjective when t = ´et. The following proposition collects what we have done so far. Proposition 1.19 (cf. [Kat87, Prop. 4.2]). X is smooth of pure dimension d over a perfect ﬁeld k. Then −d d i (1) we have H (Kn,X,log)= WnΩX,log, and H (Kn,X,log)=0 for all i 6= −d, −d +1. (2) When k = k, the natural map d WnΩX,log[d] −→ Kn,X,log is a quasi-isomorphism of complexes of abelian sheaves.

′ d C−1 Proof. (1) Since C is compatible with C by Theorem 1.7, the natural map Cone(WnΩX [d] −−−→ d WnΩX [d])[−1] → Kn,X,log is a quasi-isomorphism by the ﬁve lemma and the Ekedahl quasi- isomorphism Proposition 1.1. The claim thus follows from the exact sequence (1.2.31). (2) Proposition 1.16+(1) above.

1.3. Localization triangle associated to Kn,X,log.

1.3.1. Deﬁnition of TrWnf,log. Proposition 1.20 (Proper pushforward, cf. [Kat87, (3.2.3)]). Let f : X → Y be a proper map between separated schemes of ﬁnite type over k. Then so is Wnf : WnX → WnY , and we have a map

TrWnf,log : (Wnf)∗Kn,X,log → Kn,Y,log of complexes that ﬁts into the following commutative diagram of complexes, where the two rows are b n distinguished triangles in D (WnX, Z/p )

C′−1 (Wnf)∗Kn,X,log / (Wnf)∗Kn,X / (Wnf)∗Kn,X /

TrWnf,log TrWnf TrWnf C′−1 Kn,Y,log / Kn,Y / Kn,Y / .

Moreover TrWnf,log is compatible with composition and open restriction.

This is the covariant functoriality of Kn,X,log with respect to proper morphisms. Thus we also denote

TrWnf,log by f∗. Proof. It suﬃces to show the following diagrams commute.

(W F ) (W f) ( ) n Y ∗ n ∗ 1.1.2 △ (WnFY )∗(Wnf)∗Kn,X ≃ / (WnFY )∗(Wnf)∗(WnFX ) Kn,X

(WnFY )∗ TrWnf (WnFY )∗ TrWnf (W F ) ( ) n Y ∗ 1.1.2 △ (WnFY )∗Kn,Y ≃ / (WnFY )∗(WnFY ) Kn,Y , CYCLE COMPLEX VIA COHERENT DUALITY 21

(W f) Tr △ ≃ △ n ∗ WnFX (WnFY )∗(Wnf)∗(WnFX ) Kn,X / (Wnf)∗(WnFX )∗(WnFX ) Kn,X / (Wnf)∗Kn,X

(WnFY )∗ TrWnf TrWnf Tr △ WnFY (WnFY )∗(WnFY ) Kn,Y / Kn,Y ,

where TrWnf on the right of the first diagram and the left of the second diagram denotes the trace map △ of residual complex (WnFY ) Kn,Y : △ △ △ △ TrWnf : (Wnf)∗(WnFX ) Kn,X ≃ (Wnf)∗(Wnf) (WnFY ) Kn,Y → (WnFY ) Kn,Y . Commutativity of the first diagram is due to functoriality of the trace map with respect to residual complexes with the same associated filtration [Con00, 3.4.1(1)]. Commutativity of the second is because of compatibility of the trace map with composition of morphisms [Con00, 3.4.1(2)].

1.3.2. TrWnf,log in the case of a nilpotent immersion.

Proposition 1.21 (R¨ulling. Cf. [Kat87, 4.2]). Let i : X0 ֒→ X be a nilpotent immersion (thus so is Wni : Wn(X0) → WnX). Then the natural map

TrWni,log : (Wni)∗Kn,X0,log → Kn,X,log is a quasi-isomorphism.

Proof. Put In := Ker(WnOX → (Wni)∗WnOX0 ). Apply HomWnOX (−,Kn,X ) to the sequence of WnOX - modules

(1.3.1) 0 → In → WnOX → (Wni)∗WnOX0 → 0, we get again a short exact sequence of complexes of WnOX -modules

TrWni 0 → (Wni)∗Kn,X0 −−−−→ Kn,X → Qn := HomWnOX (In,Kn,X ) → 0. ∗ The ﬁrst map is clearly TrWni by duality. The restriction of the map (WnFX ) : WnOX → (WnFX )∗WnOX to In gives a map ∗ (WnFX ) |In : In → (WnFX )∗In, n−1 n−1 p V ([ai]) 7→ V ([ai ]). i=0 i=0 X X Deﬁne ′ (1.3.2) CIn : (WnFX )∗Qn = (WnFX )∗HomWnOX (In,Kn,X )

→ HomWnOX ((WnFX )∗In, (WnFX )∗Kn,X )

(WnFX )∗(1.1.2)◦ △ −−−−−−−−−−−−→ HomW O ((WnFX )∗In, (WnFX )∗(WnFX ) Kn,X ) ≃ n X

TrWnFX ◦ −−−−−−→HomWnOX ((WnFX )∗In,Kn,X ) ∗ ∨ ((WnFX )|In ) −−−−−−−−−→HomWnOX (In,Kn,X )= Qn. ′ ′ ′ According to the deﬁnition of C in (1.1.3), C is compatible with CIn . Thus one has the following commutative diagram

(WnFX )∗ TrWni 0 / (WnFX )∗(Wni)∗Kn,X0 / (WnFX )∗Kn,X / (WnFX )∗Qn / 0

′ ′ C′ C C In

TrWni 0 / (Wni)∗Kn,X0 / Kn,X / Qn / 0.

′ ′ ′ ′ Replace C by C − 1, and CIn by CIn − 1, we arrive at the two lower rows of the following diagram. Denote C′ −1 In Qn,log := Cone(Qn −−−−→ Qn)[−1]. ′ ′ Taking into account the shifted cones of C − 1 and CIn − 1, we get the ﬁrst row of the following diagram which is naturally a short exact sequence. Now we have the whole commutative diagram of complexes, 22 FEI REN where all the three rows are exact, and all the three columns are distinguished triangles in the derived category:

TrWni,log 0 / (Wni)∗Kn,X0,log / Kn,X,log / Qn,log / 0

TrWni 0 / (Wni)∗Kn,X0 / Kn,X / Qn / 0

′ ′ C′ −1 C −1 C −1 In

TrWni 0 / (Wni)∗Kn,X0 / Kn,X / Qn / 0.

+1 +1 +1

We want to show that TrWni,log is a quasi-isomorphism. By the exactness of the first row, it suffices to show Qn,log is an acyclic complex. Because the right column is a distinguished triangle, it suffices to ′ show CIn − 1 : Qn → Qn is a quasi-isomorphism. Actually it’s even an isomorphism of complexes: since ∗ (WnFX ) |In : In → (WnFX )∗In is nilpotent (because I1 = Ker(OX → i∗OX0 ) is a finitely generated ′ nilpotent ideal of OX ), CIn : Qn → Qn is therefore nilpotent (because one can alter the order of the three ′ labeled maps in (1.3.2) in the obvious sense), and CIn − 1 is therefore an isomorphism of complexes.

Localization triangles associated to Kn,X,log. Let i : Z ֒→ X be a closed immersion with j : U ֒→ X .1.3.3 its open complement. Recall −1 (1.3.3) ΓZ (F) := Ker(F → j∗j F) i for any abelian sheaf F. Denote its i-th derived functor by HZ (F). Notice that ′ ′ • ΓZ′ (F)=ΓZ (F) for any nilpotent thickening Z of Z (e.g. Z = WnZ), −1 • F → j∗j F is surjective whenever F is flasque, and • flasque sheaves are ΓZ -acyclic ([Har67, 1.10]) and f∗-acyclic for any morphism f. • n Therefore, for any complex of flasque sheaves F of Z/p -modules on WnX, • • • 0 → ΓZ (F ) →F → (Wnj)∗(F |WnU ) → 0 is a short exact sequence of complexes. Thus the induced triangle

• • • +1 (1.3.4) ΓZ (F ) →F → (Wnj)∗(F |WnU ) −−→ b n • is a distinguished triangle in D (WnX, Z/p ), whenever F is a flasque complex with bounded cohomolo- • gies. In particular, since Kn,X,log is a bounded complex of flasque sheaves, this is true for F = Kn,X,log. The following proposition is proven in the smooth case by Gros-Milne-Suwa [Suw95, 2.6]. The proof presented here comes from unpublished manuscript of Rülling. .Proposition 1.22 (Rülling). Let i : Z ֒→ X be a closed immersion with j : U ֒→ X its open complement Then (1) (Purity) The map

TrWni,log (Wni)∗Kn,Z,log =ΓZ ((Wni)∗Kn,Z,log) −−−−−−→ ΓZ (Kn,X,log) is a quasi-isomorphism of complexes of sheaves. (2) (Localization triangle) The following

TrWni,log +1 (1.3.5) (Wni)∗Kn,Z,log −−−−−−→ Kn,X,log → (Wnj)∗Kn,U,log −−→ b n is a distinguished triangle in D (WnX, Z/p ).

Note that we are working on the Zariski site and abelian sheaves on WnX can be identiﬁed with abelian sheaves on X canonically. Thus we can replace (Wni)∗Kn,Z,log by i∗Kn,Z,log, and (Wnj)∗Kn,U,log by j∗Kn,U,log freely.

Proof. (1) Let In be the ideal sheaf associated to the closed immersion Wni : WnZ ֒→ WnX, and m let Zn,m be the closed subscheme of WnX determined by m-th power ideal In . In particular, Zn,1 = WnZ. Denote by in,m : Zn,m ֒→ WnX and by jn,m : WnZ ֒→ Zn,m the associated closed CYCLE COMPLEX VIA COHERENT DUALITY 23

immersions. In this way one has a series of decomposition of Wni as maps of Wnk-schemes indexed by m:

jn,m in,m WnZ ❳❳ / Zn,m / WnX ❳❳❳❳❳ ◗◗ ❳❳❳❳❳ ◗◗◗ ❳❳❳ ◗◗ π ❳❳❳❳ ◗◗ Zn,m WnπX WnπZ ❳❳❳❳❳ ◗◗◗ ❳❳❳❳❳◗◗ ❳❳(+ Wnk. △ Denote KZn,m := (πZn,m ) (Wnk), where πZn,m : Zn,m → Wnk is the structure morphism. We have a canonical isomorphism i i (1.3.6) in,m,∗H (KZn,m ) ≃ExtWnOX (in,m,∗OZn,m ,Kn,X )

by [Con00, (3.2.3)] and [Har66, VI.3.1] associated to the closed immersion in,m. The trace maps associated to the closed immersions

Zn,m ֒→ Zn,m+1 for diﬀerent m make the left hand side of (1.3.6) an inductive system. The right hand side also lies in an inductive system when m varies: the canonical surjections

in,m+1,∗OZn,m+1 → in,m,∗OZn,m induce the maps

(1.3.7) HomWnOX (in,m,∗OZn,m ,Kn,X ) → HomWnOX (in,m+1,∗OZn,m+1 ,Kn,X ) whose i-th cohomologies are the connecting homomorphisms of the inductive system. By duality, the map (1.3.7) is the trace map associated to the closed immersion Zn,m ֒→ Zn,m+1, and thus is compatible with the inductive system on the left hand side of (1.3.6). Consider the trace map associated to the closed immersion in,m : Zn,m ֒→ WnX, i.e., the evaluation-at-1 map

HomWnOX (in,m,∗OZn,m ,Kn,X ) → Kn,X .

Its image naturally lies in ΓWnZ (Kn,X ). It induces an isomorphism on cohomology sheaves after taking colimit on m

i ev1 i colimm Ext (in,m,∗OZ ,Kn,X ) −−→ H (Kn,X ) WnOX n,m ≃ Z by [Har66, V.4.3]. Now we consider i i (1.3.8) colimm in,m,∗H (KZn,m ) ≃ colimm ExtWnOX (in,m,∗OZn,m ,Kn,X ) ev1 i −−→ HZ (Kn,X ). ≃

The composite map of (1.3.8) is colimm Trin,m . On the other hand, consider the log trace associated to the closed immersion in,m (cf. Proposition 1.20) i i (1.3.9) Trin,m,log : H (in,m,∗KZn,m,log)= H (ΓZ (in,m,∗KZn,m,log)) i i −→H (ΓZ (Kn,X,log)) = HZ (Kn,X,log). The maps (1.3.8), (1.3.9) give the vertical maps in the following diagram (due to formatting reason we omit in,m,∗ from every term of the ﬁrst row) which are automatically compatible by Proposition 1.20:

′ ′ i−1 C −1 i−1 i i C −1 i H (KZn,m ) / H (KZn,m ) / H (KZn,m,log) / H (KZn,m ) / H (KZn,m )

Trin,m Trin,m Trin,m,log Trin,m Trin,m ′ ′ i−1 C −1 i−1 i i C −1 i HZ (Kn,X ) / HZ (Kn,X ) / HZ (Kn,X,log) / HZ (Kn,X ) / HZ (Kn,X ).

Taking the colimit with respect to m, the ﬁve lemma immediately gives that colimm Trin,m,log is

an isomorphism. Then TrWni,log, which is the composition of colim Tr colim Tr i m jn,m,log i m in,m,log i (Wni)∗H (Kn,Z,log) −−−−−−−−−−−−→ colimm in,m,∗H (KZn,m,log) −−−−−−−−−−−→ HZ (Kn,X,log), Proposition 1.21,≃ ≃ is an isomorphism. This proves the statement. 24 FEI REN

+1 (2) Since ΓZ (Kn,X,log) → Kn,X,log → (Wnj)∗Kn,U,log −−→ is a distinguished triangle, the second part follows from the ﬁrst part.

1.4. Functoriality. The push-forward functoriality of Kn,X,log has been done in Proposition 1.20 for proper f. Now we define the pullback map for an étale morphism f. Since Wnf is then also étale, we ∗ △ have an isomorphism of functors (Wnf) ≃ (Wnf) by [Har66, VI, 3.1 and 5.3]. Define a chain map of complexes of WnOY -modules

∗ adj ∗ △ (1.4.1) f : Kn,Y −−→ (Wnf)∗(Wnf) Kn,Y ≃ (Wnf)∗(Wnf) Kn,Y ≃ (Wnf)∗Kn,X . ∗ Here adj stands for the adjunction map of the identity map of (Wnf) Kn,Y . Proposition 1.23 (Etale´ pullback). Suppose f : X → Y is an étale morphism. Then ∗ f : Kn,Y,log → (Wnf)∗Kn,X,log, defined by termwise applying (1.4.1), is a chain map between complexes of abelian sheaves. Proof. It suffices to prove that C′ is compatible with f ∗ defined above. Consider the following diagram in the category of complexes of WnOY -modules

( ) TrW F 1.1.2 △ n Y (WnFY )∗Kn,Y ≃ / (WnFY )∗(WnFY ) Kn,Y / Kn,Y

adj a) adj d) adj

( ) TrW F ∗ 1.1.2 ∗ △ n Y ∗ (Wnf)∗(Wnf) (WnFY )∗Kn,Y ≃ / (Wnf)∗(Wnf) (WnFY )∗(WnFY ) Kn,Y / (Wnf)∗(Wnf) Kn,Y

α ≃ b) β ≃ (1.1.2) ∗ ∗ △ ≃ (Wnf)∗(WnFX )∗(Wnf) Kn,Y ≃ / (Wnf)∗(WnFX )∗(Wnf) (WnFY ) Kn,Y e)

≃ c) ≃

( ) TrW F 1.1.2 △ n X (Wnf)∗(WnFX )∗Kn,X ≃ / (Wnf)∗(WnFX )∗(WnFX ) Kn,X / (Wnf)∗Kn,X . In this diagram we use shortened notations for the maps due to formatting reasons, e.g. we write (1.1.2) instead of (Wnf)∗(WnFX )∗(1.1.2), etc.. The maps labelled α and β are base change maps, and they are isomorphisms because Wnf is flat (actually Wnf is étale because f is étale) [Har66, II.5.12]. The ∗ ∗ ∗ composite of the maps on the very left and very right are (WnFY )∗(f ) and f (where f is as defined in ′ ′ (1.4.1)). The composite of the maps on the very top and very bottom are CY and (Wnf)∗CX . Diagrams a),b),c), d) commute due to naturality. Diagram e) commutes, because we have a cartesian square

WnFX WnX / WnX

Wnf Wnf

WnFY WnY / WnY by Remark 1.6(2), and then the base change formula of the Grothendieck trace map as given in [Har66, VI.5.6] gives the result. Lemma 1.24. Consider the following cartesian diagram

f ′ W / Z

g′ g f X / Y with g being proper, and f being ´etale. Then we have a commutative diagram of residual complexes

′∗ f ≃ (Wng)∗Kn,Z / (Wng)∗(Wnf)∗Kn,W / (Wnf)∗(Wng)∗Kn,W

TrWng TrWng′ f ∗ Kn,Y / (Wnf)∗Kn,X . CYCLE COMPLEX VIA COHERENT DUALITY 25

Proof. We decompose the diagram into the following two diagrams and show their commutativity one by one.

adj ′ ′ ∗ (Wng)∗Kn,Z / (Wng)∗(Wnf )∗(Wnf ) Kn,Z

≃ ′ ′ ∗ △ (Wnf)∗(Wng )∗(Wnf ) (Wng) Kn,Y O

TrWng ≃ α

∗ △ (Wnf)∗(Wnf) (Wng)∗(Wng) Kn,Y

TrWng

adj ∗ Kn,Y / (Wnf)∗(Wnf) Kn,Y .

Here α denotes the base change map, it is an isomorphism because Wnf is ﬂat [Har66, II.5.12]. This diagram commutes by naturality. Next consider

′ ′ ∗ ≃ ′ (Wng)∗(Wnf )∗(Wnf ) Kn,Z / (Wng)∗(Wnf )∗Kn,W

≃ ≃ ′ ′ ∗ △ ≃ ′ (Wnf)∗(Wng )∗(Wnf ) (Wng) Kn,Y / (Wnf)∗(Wng )∗Kn,W O ≃ α

∗ △ Tr ′ (Wnf)∗(Wnf) (Wng)∗(Wng) Kn,Y Wng

TrWng

∗ ≃ (Wnf)∗(Wnf) Kn,Y / (Wnf)∗Kn,X . The top part commutes by naturality. The bottom part commutes by the base change formula of the Grothendieck trace maps with respect to ´etale morphisms [Har66, VI.5.6].

∗ Since both f for log complexes in Proposition 1.23 and g∗ := TrWng,log are deﬁned termwise, we arrive immediately the following compatibility as a consequence of Lemma 1.24. Proposition 1.25. Notations are the same as Lemma 1.24. One has a commutative diagram of complexes ′∗ f ≃ (Wng)∗Kn,Z,log / (Wng)∗(Wnf)∗Kn,W,log / (Wnf)∗(Wng)∗Kn,W,log

g ′ ∗ g∗ f ∗ Kn,Y,log / (Wnf)∗Kn,X,log. ´ 1.5. Etale counterpart Kn,X,log,ét. Let X be a separated scheme of finite type over k of dimension d. In this subsection we will use t = Zar, ét to distinguish objects, morphisms on different sites. When t is omitted, it means t = Zar unless otherwise stated. Denote the structure sheaf on the small étale site (WnX)ét by WnOX,ét. Denote ∗ (ǫ∗,ǫ ) : ((WnX)ét, WnOX,ét) → ((WnX)Zar, WnOX ) the module-theoretic functors. Recall that every étale WnX-scheme is of the form Wng : WnU → WnX, where g : U → X is an étale X-scheme by Remark 1.6(1). Now let F be a WnOX,ét-module on (WnX)ét. Consider the following map (cf. [Kat87, p. 264])

(1.5.1) τ : (WnFX )∗F −→F, which is deﬁned to be

Wng pr1 ((WnFX )∗F)(WnU −−−→ WnX)= F(WnX ×WnFX ,WnX WnU −−→ WnX) W F ∗ n U/X Wng −−−−−−→ F(WnU −−−→ WnX) ≃ 26 FEI REN for any étale map Wng : WnU → WnX (here we use pr1 to denote the first projection map of the fiber product). This is an automorphism of F as an abelian étale sheaf, but it changes the WnOX,ét-module structure of F. Define ∗ Kn,X,ét := ǫ Kn,X

to be the complex of étale WnOX,ét-modules associated to the Zariski complex Kn,X of WnOX -modules. This is still a complex of quasi-coherent sheaves with coherent cohomologies. For a proper map f : X → Y of k-schemes, define

∗ ∗ ǫ TrWnf TrWnf,ét : (Wnf)∗Kn,X,ét = ǫ ((Wnf)∗Kn,X ) −−−−−−→ Kn,Y,ét to be the étale map of WnOY,ét-modules associated to the Zariski map TrWnf : Kn,X → Kn,X of ′ WnOX -modules. Define the Cartier operator Cét for étale complexes to be the composite

−1 ∗ ′ τ ∗ ǫ (1.1.3) Cét : Kn,X,ét −−→ (WnFX )∗Kn,X,ét = ǫ ((WnFX )∗Kn,X ) −−−−−−→ Kn,X,ét. ≃ Define ′ Cét−1 Kn,X,log,ét := Cone(Kn,X,ét −−−−→ Kn,X,ét)[−1]. We also have the sheaf-level Cartier operator. Let X be a smooth k-scheme. Recall that by definition, Cét is the composition of the inverse of (1.5.1) with the module-theoretic etalization of the WnOX -linear map (1.2.6) (it has appeared in Lemma 1.18 before):

−1 ∗ d τ d ∗ d ǫ (1.2.6) d Cét : WnΩX,ét −−→ (WnFX )∗WnΩX,ét = ǫ ((WnFX )∗WnΩX ) −−−−−−→ WnΩX,ét. ≃ ′ ′ Proposition 1.26 (cf. Theorem 1.7). Cét is the natural extension of C to the small étale site, i.e., ′ ′ ǫ∗Cét = C : Kn,X → Kn,X .

When X is smooth, Cét is the natural extension of C to the small étale site d d ǫ∗Cét = C : WnΩX → WnΩX . And one has compatibility −d ′ Cét = H (Cét). Proof. The first two claims are clear. The last claim follows from the compatibility of C and C′ in the Zariski case Theorem 1.7. Proposition 1.27 (cf. Proposition 1.15). Let X be a separated scheme of finite type over k with k = k. Then ′ i i Cét−1 i H (WnX,Kn,X,ét) := R Γ((WnX)t,Kn,X,ét) −−−−→ H (WnX,Kn,X,ét) is surjective for every i. In particular, i i C′ −1 R Γ(WnX,Kn,X,log,ét) ≃ H (WnX,Kn,X,ét) ét . Proof. The quasi-coherent descent from the étale site to the Zariski site gives

RΓ((WnX)ét,Kn,X,ét)= RΓ((WnX)Zar,Kn,X,Zar). Taking the i-th cohomology groups, the desired surjectivity then follows from the compatibility of C′ ′ and Cét Proposition 1.26 and the Zariski case Proposition 1.15. In the étale topology and for any perfect field k, surjectivity of d d Cét − 1 : WnΩX,ét → WnΩX,ét is known without the need of Proposition 1.27 (cf. Lemma 1.18). For the same reasoning as in Proposi- tion 1.19, we have Proposition 1.28 (cf. Proposition 1.19). Assume X is smooth of pure dimension d over a perfect field k. Then the natural map d WnΩX,log,ét[d] −→ Kn,X,log,ét is a quasi-isomorphism of complexes of abelian complexes. We go back to the general non-smooth case. The proper pushforward property in the étale setting is very similar to the Zariski case. CYCLE COMPLEX VIA COHERENT DUALITY 27

Proposition 1.29 (Proper pushforward, cf. Proposition 1.20). For f : X → Y proper, we have a well-deﬁned map of complexes of ´etale sheaves

(1.5.2) TrWnf,log,ét : (Wnf)∗Kn,X,log,ét → Kn,X,log,ét

given by applying TrWnf,´et termwise. Proof. The map τ −1 is clearly functorial with respect to any map of abelian sheaves. The rest of the proof goes exactly as in Proposition 1.20.

Proposition 1.30 (cf. Proposition 1.21). Let i : X0 ֒→ X be a nilpotent immersion. Then the natural map

TrWni,log,ét : (Wni)∗Kn,X0,log,ét → Kn,X,log,ét is a quasi-isomorphism. Proof. This is is a direct consequence of the functoriality of the map τ −1 and Proposition 1.21.

Let i : Z ֒→ X be a closed immersion with j : U ֒→ X being the open complement as before. Deﬁne

−1 ΓZ (F) := Ker(F → j∗j F) for any étale abelian sheaf F on X, just as in the Zariski case (cf. (1.3.3)). Replacing Z (resp. X) by a nilpotent thickening will define the same functor as ΓZ (−), because the étale site of any scheme is the same as the étale site of its reduced scheme [EGAIV-4, Ch. IV, 18.1.2]. Recall that when F = I is an injective Z/pn-sheaf, −1 0 → ΓZ (I) → I → j∗j I → 0 n n n is exact. In fact, because j!Z/p is a subsheaf of the constant sheaf Z/p on X, the map HomX (Z/p , I) → n n n −1 n −1 HomX (j!Z/p , I) is surjective. Since HomX (j!Z/p , I) = HomX (Z/p , j I) = HomX (Z/p , j∗j I), n n −1 HomX (Z/p , I) → HomX (Z/p , j∗j I) is surjective, hence the claim. This implies that for any complex F • of étale Z/pn-sheaves with bounded cohomologies,

• • −1 • +1 (1.5.3) RΓZ (F ) →F → j∗j F −−→ is a distinguished triangle in Db(X, Z/pn) (cf. (1.3.4)). Proposition 1.31 (cf. Proposition 1.22). Let i : Z ֒→ X be a closed immersion with j : U ֒→ X being the open complement as before. Then (1) (Purity) We can identify canonically the functors

b n b n (Wni)∗ = RΓZ ◦ (Wni)∗ : D ((WnZ)ét, Z/p ) → D ((WnX)ét, Z/p ). The composition of this canonical identification with the trace map

TrWni,log,ét (Wni)∗Kn,Z,log,ét = RΓZ ((Wni)∗Kn,Z,log,ét) −−−−−−−−→ RΓZ (Kn,X,log,ét) is a quasi-isomorphism of complexes of étale Z/pn-sheaves. (2) (Localization triangle)

TrWni,log,ét +1 (Wni)∗Kn,Z,log,ét −−−−−−−−→ Kn,X,log,ét −→ (Wnj)∗Kn,U,log,ét −−→

b n is a distinguished triangle in D ((WnX)´et, Z/p ).

Proof. (1) One only needs to show that (Wni)∗ = RΓZ ◦ (Wni)∗, and then the rest of the proof n is the same as in Proposition 1.22(1). Let I be an injective étale Z/p -sheaf on WnZ. Since −1 −1 HomWnX (−, (Wni)∗I) = HomWnZ ((Wni) (−), I) and (Wni) is exact, we know (Wni)∗I is an injective abelian sheaf on (WnX)ét. This implies that R(ΓZ ◦ (Wni)∗)= RΓZ ◦ (Wni)∗ by the Leray spectral sequence, and thus (Wni)∗ = R(Wni)∗ = R(ΓZ ◦ (Wni)∗)= RΓZ ◦ (Wni)∗. (2) One only need to note that (Wnj)∗Kn,U,log,ét = R(Wnj)∗Kn,U,log,ét. In fact, the terms of Kn,U,log,ét are quasi-coherent WnOX,ét-modules which are (Wnj)∗-acyclic in the étale topology i ∗ ∗ i (because R f∗(ǫ F)= ǫ (R f∗F) for any quasi-coherent Zariski sheaf F and any quasi-compact quasi-separated morphism f [Stacks, Tag 071N].). Now the first part and the distinguished triangle (1.5.3) imply the claim. 28 FEI REN

Bloch’s cycle complex c 2. ZX,t Let X be a separated scheme of ﬁnite type over k of dimension d. Let i ∆ = Spec k[T0,...,Ti]/( Tj − 1). i Deﬁne z0(X,i) to be the free abelian group generated by closedX integral subschemes Z ⊂ X × ∆ that intersect all faces properly and dim Z = i. We say two closed subschemes Z1,Z2 of a scheme Y intersect properly if for every irreducible component W of the schematic intersection Z1 ∩ Z2 := Z1 ×Y Z2, one has

(2.0.1) dim W ≤ dim Z1 + dim Z2 − dim Y i (cf. [Gei05, A.1]). A subvariety of X × ∆ is called a face if it is determined by some Tj1 = Tj2 = ··· =

Tjs =0 (0 ≤ j1 < ··· < js ≤ i). Note that a face is Zariski locally determined by a regular sequence of i X × ∆ . Therefore the given inequality condition (2.0.1) in the definition of z0(X,i) is equivalent to the equality condition [Gei05, (53)]. The above definition defines a sheaf z0(−,i) in both the Zariski and the étale topology on X ([Blo86, p.270]. See also [Gei04, Lemma 3.1]). Define the complex of sheaves d → z0(−,i) −→ z0(−,i − 1) → ...z0(−, 0) → 0 with differential map j d(Z)= (−1) [Z ∩ V (Tj )]. j X Here we mean by V (Tj) the closed integral subscheme determined by Tj and by [Z ∩ V (Tj )] the linear combination of the reduced irreducible components of the scheme theoretic intersection Z ∩ V (Tj ) with coefficients being intersection multiplicities. z0(X, •) is then a homological complex concentrated in degree [0, ∞). Labeling cohomologically, we set c i (ZX ) = z0(−, −i). This complex is nonzero in degrees (−∞, 0]. Define the higher Chow group −i c CH0(X,i) := Hi(z0(X, •)) = H (ZX (X)) for any i. The higher Chow groups with coefficients in an abelian group A will be denoted −i c CH0(X,i; A) := H (ZX (X) ⊗Z A). c The complex ZX,t, with either t = Zar or t = ét, is covariant for proper morphisms, and contravariant for quasi-finite flat morphisms [Blo86, Prop. 1.3].

Kato’s complex of Milnor -theory M 3. K CX,t M Recall that given a field L, the q-th Milnor K-group Kq (L) of L is defined to be the q-th graded piece of the non-commutative graded ring (L∗)⊗q KM (L)= q≥0 , q (a ⊗ (1 − a) | a, 1 − a ∈ L∗) q≥0 L M where (a ⊗ (1 − a) | a, 1 − a ∈ L∗) denotes the two-sided ideal of the non-commutative graded ring ∗ ⊗q ∗ q≥0(L ) generated by elements of the form a ⊗ (1 − a) with a, 1 − a ∈ L . The image of an element ∗ ⊗q M a1 ⊗···⊗ aq ∈ (L ) in Kq (L) is denoted by {a1,...,aq}. L M Recall the definition of a Milnor K-sheaf on a point X = Spec L, where L is any field. KSpec L,q,Zar is M the constant sheaf associated to the abelian group Kq (L) (without the assumption that L is an infinite M field, cf. [Ker10, Prop. 10(4)]), and KSpec L,q,ét is the étale sheaf associated to the presheaf ′ M ′ ′ L 7→ Kq (L ); L /L finite separable. Choose a separable closure Lsep of L. Then the geometric stalk at the geometric point Spec Lsep over M ′ M sep Spec L is colimL⊂L′⊂Lsep Kq (L ), which is equal to Kq (L ) because the filtered colimit commutes with the tensor product and the quotient. Now by Galois descent of the étale sheaf condition, the sheaf M KSpec L,q,ét is precisely ′ M sep Gal(Lsep/L′) ′ L 7→ Kq (L ) ; L /L finite separable. CYCLE COMPLEX VIA COHERENT DUALITY 29

Here the Galois action is given on each factor, according to the very definition of the étale presheaf M,pre KX,q,ét. Let X be a separated scheme of finite type over k of dimension d. Now with the topology t = Zar or M t = ét, we have the corresponding Gersten complex of Milnor K-theory, denote by CX,t (the differentials dM will be introduced below):

M M M M d d M d M (3.0.1) ιx∗Kx,d,t −−→ ... −−→ ιx∗Kx,1,t −−→ ιx∗Kx,0,t, x∈MX(d) x∈MX(1) x∈MX(0) ,where ιx : Spec k(x) ֒→ X the natural inclusion map. As part of the convention M i M (CX,t) = ιx,∗Kx,−i,t. x∈MX(−i) In other words, (3.0.1) sits in degrees [−d, 0]. It remains to introduce the differential maps. M When t = Zar, the differential map d in (3.0.1) is defined in the following way. Let x ∈ X(q) be a dimension q point, and ρ : X′ → {x} be the normalization of {x} with generic point x′. Define

x′ P ∂ P Nm ′ M x M M ′ y′ M ′ y /y M (d )y : Kq (x)= Kq (x ) −−−−→ Kq−1(y ) −−−−−−→ Kq−1(y). ′ My |y M M ′ ′ ′(1) Here we have used the shortened symbol Kq (x) := Kq (k(x)). The notation y |y means that y ∈ X is in the fiber of y. x′ M ′ M ′ (3.0.2) ∂y′ : Kq (x ) → Kq−1(y ) is the Milnor tame symbol defined by y′. And M ′ M (3.0.3) Nmy′/y : Kq−1(y ) → Kq−1(y) is the Milnor norm map of the finite field extension k(y) ⊂ k(y′). The differential dM of this complex is given by M M x M M d := (d )y : Kq (x) → Kq−1(y). x∈X x∈X y∈X X(q) y∈X(qX−1) ∩{x} M(q) M(q−1) There are different sign conventions concerning the tame symbol in the literature and we clarify the Spec L M one that we adopt. Following [Ros96, p.328], we define the tame symbol ∂Spec k(v) := ∂v : Kn (L) → M Kn−1(k(v)) for a field L, a normalized discrete valuation v on L and k(v) the residue field with respect to v, via

(3.0.4) ∂v({πv,u1,...,un−1})= {u1,..., un−1}.

Here πv is a local parameter with respect to v, u1,...un−1 are units in the valuation ring of v, and u1,..., un−1 are the images of u1,...un−1 in the residue field k(v). ′ When t = ét, set x ∈ X(q), y ∈ X(q−1) ∩ {x}. Denote by ρ : X → {x} the normalization map and ′ ′ M denote by x the generic point of X . One can canonically identify the étale abelian sheaves Kx,q,ét and M M M ′ ′(1) ′ ρ∗Kx′,q,ét on {x}, and thus identify ιx,∗Kx,q,ét and ιx,∗ρ∗Kx′,q,ét on X. Let y ∈ X such that ρ(y )= y. Then the componentwise differential map M x M M (d )y : ιx,∗Kx,q,ét → ιy,∗Ky,q−1,ét is defined to be the composition M x (d )y = ιy,∗(Nm) ◦ ρ∗(∂). x′ Here ∂ := y′∈X′(1)∩ρ−1(y) ∂y′ , where x′ M M (3.0.5) P ∂y′ : ιx′,∗Kx′,q,ét → ιy′,∗Ky′,q−1,ét on X′ is defined to be the sheafification of the tame symbol on the presheaf level. Indeed, the tame symbol is a map of étale presheaves by [Ros96, R3a]. And Nm := y′∈X′(1) ∩ρ−1(y) Nmy′/y, where M M (3.0.6) Nmy′/y : ρ∗Ky′,q−1,ét → Ky,q−1,Pét on y is defined to be the sheafification of the norm map on the presheaf level. The norm map is a map of étale presheaves by [Ros96, R1c]. 30 FEI REN

M The complex CX,t, either t = Zar or t = ét, is covariant for proper morphisms and contravariant for quasi-finite flat morphisms ([Ros96, (4.6)(1)(2)]). The pushforward map associated to a proper morphism is induced by the Milnor norm map, and the pullback map associated to a quasi-finite flat morphism is induced by the pullback map of the structure sheaves.

4. Kato-Moser’s complex of logarithmic de Rham-Witt sheaves νn,X,t Let X be a separated scheme of finite type over k of dimension d. Kato first defined the Gersten complex of the logarithmic de Rham-Witt sheaves in [Kat86a, §1]. Moser in [Mos99, (1.3)-(1.5)]e sheafified Kato’s construction on the étale site and studied its dualizing properties. We will adopt here the sign conventions in [Ros96]. q Let Y be a k-scheme. Let q ∈ N be an integer. Recall that in Section 1.2.5, we have defined WnΩY,log,t, q with either t = Zar or t = ét, to be the abelian subsheaf of WnΩY,t étale locally generated by log forms. q q We will freely use WnΩL,log,t for WnΩSpec L,log,t below. Now let X be a separated scheme of finite type over k of dimension d. Define the Gersten complex M n νn,X,t, in the topology t = Zar or t = ét, to be the complex of t-sheaves isomorphic to CX,t/p via the Bloch-Gabber-Kato isomorphism [BK86, 2.8]: e d 1 0 (4.0.1) 0 → ιx,∗WnΩk(x),log,t → ... → ιx∗WnΩk(x),log → ιx,∗WnΩk(x),log,t → 0. x∈MX(d) x∈MX(1) x∈MX(0) Here ιx : Spec k(x) → X is the natural map. We will still denote by ∂ the reduction of the tame symbol ∂ mod pn (cf. (3.0.2)(3.0.5)), but denote by tr the reduction of Milnor’s norm Nm mod pn (cf. (3.0.3)(3.0.6)). The reason for the later notation will be clear from Lemma 5.3. As part of the convention,

i −i νn,X,t = ιx∗WnΩk(x),log,t, x∈MX(−i) i.e. νn,X is concentrated in degrees e [−d, 0]. Propositione 4.1. Let i : Z ֒→ X be a closed immersion with j : U ֒→ X its open complement. We have the following short exact sequence for t = Zar:

0 / i∗νn,Z,Zar / νn,X,Zar / j∗νn,U,Zar / 0. For t = ét, one has the localization triangle e e e +1 i∗νn,Z,ét → νn,X,ét → Rj∗νn,U,ét −−→ .

Proof. νn,X,Zar is a complex of flasque sheaves (therefore Rj∗(νn,X,Zar) = j∗νn,X,Zar), and one has the sequence being short exact in thise case. Whene t = ét, thee purity theorem holds [Mos99, Corollary on ≃ p.130],e i.e., i∗νn,Z,ét = ΓZ (νn,X,ét) −→ RΓZ (νn,X,ét). We aree done with thee help of the distinguished triangle (1.5.3) in the étale topology.

e e e M Functoriality of νn,X,t is the same as that of CX,t via d log. We omit the statement.

Part 2. The mapse Construction of the chain map M 5. ζn,X,log,t : CX,t −→ Kn,X,log,t M 5.1. Construction of the chain map ζn,X,t : CX,t −→ Kn,X,t. Let x ∈ X(q) be a dimension q point. ιx : Spec k(x) → X is the canonical map and ix : {x} ֒→ X the closed immersion. At degree i = −q, and over a point x, we deﬁne the degree i map to be ζi := ζi , with n,X,t x∈X(q) n,x,t

i M d log qP q (5.1.1) ζn,x,t : (Wnιx)∗Kx,q,t −−−→ (Wnιx)∗WnΩk(x),log,t ⊂ (Wnιx)∗WnΩk(x),t (−1)i Tr = (W i ) Ki −−−−−−−−−→Wnix Ki . n x ∗ n,{x},t n,X,t i We will use freely the notation ζn,X,t with some of its subscript or superscript dropped. It’s worth noticing that all the maps of étale sheaves involved here are given by the sheafification of the respective Zariski maps on the étale presheaf level. So to check commutativity of a composition of such maps between étale sheaves, it suffices to check on the t = Zar level. Keeping the convention as before, we usually omit the subscript Zar when we are working over the Zariski topology. CYCLE COMPLEX VIA COHERENT DUALITY 31

Proposition 5.1. Let X be a separated scheme of finite type over k with k being a perfect field of characteristic p> 0. For t = Zar and t = ét, the map

M ζn,X,t : CX,t −→ Kn,X,t,

as deﬁned termwise in (5.1.1), is a chain map of complexes of sheaves on the site (WnX)t.

Note that we have a canonical identiﬁcation (WnX)t = Xt for both t = Zar and t = ´et. We use (WnX)t just for the convenience of describing the WnOX -structure of residual complexes appearing later.

Proof. To check ζn,X,t is a map of complexes, it suﬃces to check that the diagram

dM M i X M i+1 (CX,t) / (CX,t)

i i+1 ζn,X,t ζn,X,t i dX i+1 i+1 (Kn,X,t) / (Kn,X,t)

commutes for t = Zar. To this end, it suﬃces to show: for each x ∈ X(q), and y ∈ X(q−1) which is a specialization of x, the diagram

(dM )x M X y M (5.1.2) (Wnιx)∗Kx,q / (Wnιy)∗Ky,q−1

ζn,y −q+1 ζn,x (Wniy,x)∗K n,{y}

− TrWniy,x

d{x} K−q / K−q+1 n,{x} n,{x}

.(commutes (iy,x : {y} ֒→ {x} denotes the canonical closed immersion M Since the deﬁnition of the diﬀerential maps in CX involves normalization, consider the normalization ρ : X′ → {x} of {x}, and form the cartesian square

′ ′ ′ {y}×{x} X / X = {x }

ρ iy,x {y} / {x}.

Denote the generic point of X′ by x′. Suppose y′ is one of the generic points of the irreducible components ′ ′ ′ ′ of {y}×{x} X , and denote by Y the irreducible component corresponding to y . In particular, y is a codimension 1 point in the normal scheme X′, thus is regular. Because the base ﬁeld k is perfect, y′ is ′ also a smooth point in X . According to Remark 1.2, the degree [−q, −q +1] terms of Kn,X′ are of the form

0 q δ 1 q (Wnιx′ )∗Hx′ (WnΩX′ ) −→ (Wnιy′ )∗Hy′ (WnΩX′ ) −→ ..., y′∈X′ M(q−1)

′ ′(1) where δ denotes the diﬀerential map of the Cousin complex Kn,X . After localizing at a single y ∈ X in the Zariski sense, one gets

0 q δy′ 1 q (Wnιx′ )∗Hx′ (WnΩX′ ) −−→ (Wnιy′ )∗Hy′ (WnΩX′ ) −→ .... 32 FEI REN

′ ′ ′ ′ ,Consider the following diagrams. Write ιx′ : Spec k(x ) ֒→ X , ιy′ : Spec k(y ) ֒→ X the inclusions ′ ′ ′ iy′,x′ : Y = {y } ֒→ X the closed immersion, we have a diagram

x′ ∂ ′ M y M (5.1.3) (Wnιx′ )∗Kx′,q / (Wnιy′ )∗Ky′,q−1

d log d log q q−1 (Wnιx′ )∗WnΩk(x′) (Wnιy′ )∗WnΩk(y′) ❙❙❙ ❙❙❙ δy′ ❙❙ − TrW (i ) ❙❙ n y′,x′ ❙❙❙❙ ❙) 1 q (Wnιy′ )∗Hy′ (WnΩX′ ).

For any y′ ∈ ρ−1(y) ⊂ X′(1), we have a diagram

Nm ′ M y /y M (5.1.4) (Wnρ)∗Ky′,q−1 / Ky,q−1

d log d log q−1 TrWnρ q−1 (Wnρ)∗WnΩk(y′) / WnΩk(y).

′ ′ ′ Write iy′,x′ : Y = {y } ֒→ X , iy,x : {y} ֒→ {x}, we have a diagram

q−1 TrWnρ q−1 (5.1.5) (Wnρ)∗(Wnιy′ )∗WnΩk(y′) / (Wnιy)∗WnΩk(y)

TrW (i ) Tr n y′ ,x′ Wn(iy,x)

Tr 1 q Wnρ −(q−1) (Wnρ)∗(Wnιy′ )∗H ′ (WnΩ ′ ) / K , y X n,{x}

and a diagram

d ′ =P δ ′ q X y 1 q−1 (5.1.6) (Wnιx′ )∗WnΩk(x′) / y′∈ρ−1(y)(Wnιy′ )∗Hy′ (WnΩX′ ) L TrWnρ ≃ TrWnρ d{x} K−q / K−(q−1). n,{x} n,{x}

All the trace maps above are trace maps of residual complexes at a certain degree. (5.1.5) is the degree q − 1 part of the diagram

TrWnρ ′ ′ ′ (Wnρ)∗(Wniy ,x )∗Kn,Y / (Wniy,x)∗Kn,{y}

TrW (i ) Tr n y′ ,x′ Wn(iy,x )

TrWnρ (Wnρ)∗Kn,X′ / Kn,{x}

′ (the trace map on top is the trace map of the restriction of Wnρ to WnY ), and thus is commutative by the functoriality of the Grothendieck trace map with respect to composition of morphisms [Con00, 3.4.1(2)].

′ (5.1.6) is simply the degree −q to −q + 1 terms of the trace map TrWnρ : (Wnρ)∗Kn,X → Kn,{x}, thus is also commutative. It remains to check the commutativity of (5.1.3) and (5.1.4). And these are Lemma 5.2 and Lemma 5.3. CYCLE COMPLEX VIA COHERENT DUALITY 33

One notices that diagram (5.1.2) decomposes into the four diagrams (5.1.3)-(5.1.6):

x′ L ′ ∂ ′ P ′ Nm ′ M y |y y M y /y y /y M (Wnρ)∗(Wnιx′ )∗Kx′,q / y′|y(Wnρ)∗(Wnιy′ )∗Ky′,q−1 / (Wnιy)∗Ky,q−1 (5.1.4) d log (5.1.3) L d log d log q q−1 TrWnρ q−1 (Wnρ)∗(Wnιx′ )∗WnΩk(x′) y′|y(Wnρ)∗(Wnιy′ )∗WnΩk(y′) / (Wnιy)∗WnΩk(y) ❱❱ ❱❱❱ L ′ δ ′ ❱❱❱ y |y y (5.1.5) ❱❱ L L ′ − TrW (i ) − Tr ❱❱❱ y |y n y′,x′ Wn(iy,x) ❱❱❱❱ ❱❱❱❱ + Tr 1 q Wnρ −(q−1) TrWnρ ′ (Wnρ)∗(Wnιy′ )∗H ′ (WnΩ ′ ) / K y |y y X n,{x}

L (5.1.6)

d{x} K−q / K−(q−1) n,{x} n,{x}

Here by symbol y′|y we mean that y′ ∈ ρ−1(y). Notice that we have added a minus sign to both vertical arrows of (5.1.5) in the corresponding square above, but this does not aﬀect its commutativity. Since one can canonically identify

M M (Wnρ)∗(Wnιx′ )∗Kx′,q with (Wnιx)∗Kx,q, to show the commutativity of the diagram (5.1.2), it only remains to show Lemma 5.2 and Lemma 5.3.

Lemma 5.2. For an integral normal scheme X′, with x′ ∈ X′ being the generic point and y′ ∈ X′(1) being a codimension 1 point, the diagram (5.1.3) is commutative.

′ ′(1) M ′ Proof. Given a y ∈ X lying over y, Kq (x ) is generated by ′ {π ,u1,...,uq−1} and {v1,...,vq−1, vq}

∗ ′ as an abelian group, where u1,...,uq−1, v1,...,vq−1, vq ∈ OX′,y′ , and π is a chosen uniformizer of the discrete valuation ring OX′,y′ . It suﬃces to check the commutativity for these generators. In the ﬁrst case, the left-bottom composition gives

′ ′ (δy′ ◦ d log)({π ,u1,...,uq−1})= δy′ (d log[π ]nd log[u1]n ...d log[uq−1]n)

′ = d[π ]nd log[u1]n ...d log[uq−1]n . ′ " [π ]n # The last equality above is given by [CR11, A.1.2]. Here we have used the fact that [π′] is a regular ′ ′ ′ element in WnX , since π is regular in X . The top-right composition gives

x′ ′ ′ (− TrWn(iy′ ,x′ ) ◦d log ◦∂y )({π ,u1,...,uq−1})

= (− TrWn(iy′,x′ ) ◦d log){u1,..., uq−1}

= − TrWn(iy′ ,x′ )(d log[u1]n ...d log[uq−1]n)

′ = d[π ]nd log[u1]n ...d log[uq−1]n . ′ " [π ]n # The last equality is given by [CR11, A.2.12]. So the diagram (5.1.3) is commutative in this case. x′ In the second case, since ∂y′ ({v1,...,vq}) = 0, we need to check the left-bottom composite also gives zero. In fact,

(δy′ ◦ d log)({v1,...,vq})= δy′ (d log[v1]n ...d log[vq]n)

′ = [π ]n · d log[v1]n ...d log[vq]n ′ " [π ]n # =0. The second equality is due to [CR11, A.1.2]. The last equality is because, in a small neighborhood V of ′ ′ q ′ q y , the element [π ]n · d log[v1]n ...d log[vq]n ∈ WnΩV lies in the WnOV -submodule [π ]n · WnΩV . 34 FEI REN

Lemma 5.3 (Compatibility of Milnor norm and Grothendieck trace). Let F/E be a finite field extension with both fields E and F being of transcendence degree q−1 over k. Suppose there exists a finite morphism g between integral separated finite type k-schemes, such that F is the function field of the source of g and E is the function field of the target of g, and the field extension F/E is induced via the map g. Then the following diagram commutes Nm M F/E M Kq−1(F ) / Kq−1(E)

d log d log q−1 TrWng q−1 WnΩF / WnΩE .

Here the norm map NmF/E denotes the norm map from Milnor K-theory, and TrWng denotes the Grothendieck trace map associated to the finite morphism g. Remark 5.4. The compatibility of the trace map with the norm and the pushforward of cycles in various settings has been a folklore, and many definitions and properties of the trace map in the literature reflect this viewpoint. But since we have not found a proof of the compatibility of the Milnor norm with the trace map defined via the Grothendieck duality theory, we include a proof here.

Proof. We start the proof by some reductions. Since both NmF/E and TrF/E are independent of the choice of towers of simple field extensions, without loss of generality, one could suppose F is a finite simple E[T ] field extension over E. Now F = E(a)= f(T ) for some monic irreducible polynomial f(T ) ∈ E[T ] with 1 a ∈ F being one of its roots. This realizes Spec F as an F -valued point P of PE, namely,

iP Spec F = P / P1 ▼▼▼ E ▼▼▼ ▼▼ π g ▼▼▼ ▼& Spec E. All the three morphisms on above are morphisms of finite type (although not between schemes of finite type over k), so it makes sense to talk about the associated trace maps for residual complexes. But for the particular residual complexes we are interested in, we need to enlarge the schemes involved to schemes of finite type over k, while preserving the morphism classes (e.g., closed immersion, smooth morphism, etc) of the morphisms between them. To this end, take Y to be any separated smooth connected scheme of finite type over k with E 1 1 being the function field. Since PE is the generic fiber of Y ×k Pk, by possibly shrinking Y to an affine 1 neighborhood Spec B of pr1(P ) (here pr1 : Y ×k Pk → Y is the first projection map) one can extend the above diagram to the following:

iW 1 Spec F ∈ W ❙❙ / PY ❙❙❙❙ ❙❙❙❙ ❙❙❙ π g ❙❙❙❙ ❙❙❙❙ ) Y = Spec B ∋ Spec E.

P1 Y 1 Here W := {P } is the closure of the point P in PY . This is a commutative diagram of ﬁnite type k-schemes. In particular, it makes sense to talk about the residual complexes Kn,Y ,Kn,W and K P1 . n, Y Now it remains to show the commutativity of the following diagram

Nm M E(a)/E M (5.1.7) Kq−1(E(a)) / Kq−1(E)

d log d log q−1 TrWng q−1 WnΩE(a) / WnΩE ,

where TrWng denotes the trace map for residual complexes TrWng : (Wng)∗Kn,W → Kn,Y at degree −(q − 1).

We do induction on [E(a): E]. When [E(a): E] = 1, then both the Grothendieck trace TrWng : q−1 q−1 M M WnΩE(a)/k → WnΩE/k and the norm map NmE(a)/E : Kq−1(E(a)) → Kq−1(E) are the identity, therefore the claim holds. Now the induction step. Suppose the diagram (5.1.7) commutes for [E(a): E] ≤ r − 1. We will need to prove the commutativity for [E(a): E]= r. CYCLE COMPLEX VIA COHERENT DUALITY 35

First note that TrWng : (Wng)∗Kn,W → Kn,Y naturally decomposes into

(Wnπ)∗ TrWniP W TrWnπ (5.1.8) (Wng)∗Kn,W −−−−−−−−−−−−→ (Wnπ)∗K P1 −−−−→ Kn,Y . n, Y 1 q by [Con00, 3.4.1(2)]. HP (WnΩP1 ) is a direct summand of the degree −(q − 1) part of Kn,P1 . One can Y Y canonically identify 1 q 1 q (5.1.9) HP (WnΩP1 )= HP (WnΩP1 ), Y E 1 1 via pulling back along the natural map PE ֒→ PY . Thus on degree −(q − 1) and at the point P , the map (5.1.8) is canonically identiﬁed with Tr q−1 WniP W 1 q TrWnπ q−1 WnΩ −−−−−−→ HP (WnΩP1 ) −−−−→ WnΩE . E(a) E Consider Nm M ∂P M E(a)/E M Kq (E(T )) / Kq−1(E(a)) / Kq−1(E)

d log q−1 d log (−1) WnΩE(a) d log PPP PPTrWng TrW i W PP n P PP PPP Tr ' q δP 1 q Wnπ q−1 WnΩ / HP (WnΩP1 ) / WnΩE . E(T ) E We have used the identification (5.1.9) in this diagram. We have seen that the left square is commutative ′ 1 ′ up to sign −1, as a special case of Lemma 5.2 (i.e. take normal scheme X = PE and y := P = Spec F ). Since ∂P is surjective, to show the commutativity of the trapezoid on the right, it suffices to show that the composite square is commutative up to −1. For any element M s := {s1,...,sq−1} ∈ Kq−1(E(a)), one can always find a lift M s := {f, s1,..., sq−1} ∈ Kq (E(T )), such that each of the si = si(T ) is a polynomial of degree ≤ r − 1 (e.g. decompose E(a) as a r- r−1 j r−1 j dimensional E-vector space E(a)e = j=0eEa ande suppose si = j=0 bi,j a with bi,j ∈ E, then si = r−1 j si(T )= j=0 bi,j T satisfies the condition),L and ∂P (s)= s. DenoteP by e P yi,1,...,yi,ai (1 ≤ i ≤ q − 1) e 1 e the closed points of PE corresponding to the irreducible factors of the polynomials s1,..., sq−1. Note that the local section si,l cutting out yi,l is by definition an irreducible factor of si, and therefore deg si,l < r for all i and all l. e e We claim thate e e q q−1 (5.1.10) (TrWnπ)y ◦ δy = 0 : WnΩE(T )/k → WnΩE/k. y∈(P1 ) XE (0) In fact, q q 1 q (5.1.11) 0 → WnΩP1 → WnΩ → (Wnιy)∗Hy (WnΩP1 ) → 0 E E(T ) E y∈(P1 ) ME (0) 1 is an exact sequence [CR12, 1.5.9], where ιy : y ֒→ PE is the natural inclusion of the point y. Taking the long exact sequence with respect to the global section functor, one arrives at the following diagram with the row being a complex

q δ 1 q 1 1 q W Ω / P1 H (W Ω 1 ) / H (P , W Ω 1 ) n E(T ) y∈( )(0) y n P E n P E ❙❙ E E ❙❙❙❙ L ❙❙ TrW π ❙❙❙ n Py (TrWnπ)y ❙❙ ❙❙ ) q−1 WnΩE/k.

The trace maps on left of the above are induced from the degree 0 part of TrW π : (Wnπ)∗K P1 → Kn,Y . n n, Y The trace map on the right of the above is induced also by TrW π : (Wnπ)∗K P1 → Kn,Y , while the n n, Y global cohomology group is calculated via (5.1.11), i.e., one uses the last two terms of (5.1.11) as an 36 FEI REN

q 1 injective resolution of the sheaf WnΩP1 , and then TrWnπ : (Wnπ)∗Kn,P → Kn,Y induces the map of E Y complexes on global sections (sitting in degrees [−1, 0]), and then the map of cohomologies on degree 1 1 q q−1 0 gives our trace map H (PE , WnΩE) → WnΩE on the right. From the construction of these trace maps, the diagram on above is by deﬁnition commutative. Therefore (5.1.10) holds.

One notices that δy ◦ d log(s) = 0 unless y ∈{p,y1,1,...,yq−1,aq−1 , ∞}. Now calculate (Tr ◦ d log)(s) Wng e = (TrWng ◦d log ◦∂P )(s)

= −((TrWnπ)P ◦ δP ◦ d log)(s) (Lemma 5.2) e = ((TrWnπ)y ◦ δy ◦ d log)(s) (5.1.10) y∈{y ,...,y ,∞} e 1,1 Xq−1,aq−1 e = − (d log ◦ NmE(k(y))/E ◦∂y)(s) y∈{y ,...,y ,∞} 1,1 Xq−1,aq−1 (inductione hypothesis)

= (d log ◦ NmE(a)/E ◦∂P )(s) ([Ros96, 2.2 (RC)])

= (d log ◦ NmE(a)/E)(s). e This ﬁnishes the induction.

M 5.2. Functoriality of ζn,X,t : CX,t → Kn,X,t. Let k denote a perfect ﬁeld of positive characteristic p. Proposition 5.5 (Proper pushforward). ζ is compatible with proper pushforward. I.e., for f : X → Y a proper map, the following diagram is commutative

ζ M n,X,t (Wnf)∗CX,t / (Wnf)∗Kn,X,t

f∗ f∗

ζ M n,Y,t CY,t / Kn,Y,t.

Here f∗ on the left denotes the pushforward map for Kato’s complex of Milnor K-theory (cf. Section 3),

and f∗ on the right denotes the Grothendieck trace map TrWnf,t for residual complexes. Proof. We only need to prove the proposition for t = Zar and for degree i ∈ [−d, 0]. Then by the very deﬁnition of the ζ map and the compatibility of the trace map with morphism compositions [Con00, 3.4.1(2)], it suﬃces to check the commutativity at points x ∈ X(q), y ∈ Y(q), where q = −i:

M d log q Kq (x) / WnΩk(x)

f∗ f∗ M d log q Kq (y) / WnΩk(y).

(1) When y 6= f(x), both pushforward maps are zero maps, therefore we have the desired commutativity. (2) When y = f(x), by definition of ζ and the pushforward maps, we need to show commutativity of the following diagram for finite field extension k(y) ⊂ k(x)

M d log q Kq (x) / WnΩk(x)

Nmk(x)/k(y) TrWnf M d log q Kq (y) / WnΩk(y).

This is precisely Lemma 5.3. CYCLE COMPLEX VIA COHERENT DUALITY 37

Proposition 5.6 (Etale´ pullback). ζ is compatible with ´etale pullbacks. I.e., for f : X → Y an ´etale morphism, the following diagram is commutative

ζ M n,Y,t CY,t / Kn,Y,t

f ∗ f ∗

ζ M n,X,t (Wnf)∗CX,t / (Wnf)∗Kn,X,t. Here f ∗ on the left denotes the pullback map for Kato’s complex of Milnor K-theory (cf. Section 3), and f ∗ on the right denotes the pullback map for residual complexes (1.4.1).

Proof. It suﬃces to prove the proposition for t = Zar. Take y ∈ Y(q). Consider the cartesian diagram

f|W X × {y} =: W / {y} Y _ _

iW iy f X / Y. Then the desired diagram at point y decomposes in the following way at degree −q:

Tr M d log q −q Wniy −q K (y) / WnΩ = K / K q k(y) n,{y} n,Y

∗ ∗ ∗ f (f|W ) f

d log TrWni KM (x) / W Ωq = K−q W / K−q . x∈W(q) q x∈W(q) n k(x) n,W n,X

∗ ∗ ∗ The left square commutesL because both f andL (f |W ) are induced by the natural map f : OY → f∗OX . The right square commutes due to Lemma 1.24.

′ Ct−1 5.3. Extend to Kn,X,log,t. Recall the complex Kn,X,log,t := Cone(Kn,X,t −−−→ Kn,X,t)[−1], i.e., i i i−1 Kn,X,log,t = Kn,X,t ⊕ Kn,X,t. Notice that

(5.3.1) Kn,X,t → Kn,X,log,t, a 7→ (a, 0) is not a chain map. Nevertheless,

M Proposition 5.7. We keep the same assumptions as in Proposition 5.1. The chain map ζn,X,t : CX,t −→ Kn,X,t composed with (5.3.1) gives a chain map M ζn,X,log,t := (5.3.1) ◦ ζn,X,t : CX,t −→ Kn,X,log,t

of complexes of abelian sheaves on (WnX)t.

We will also use the shortened notation ζlog,t for ζn,X,log,t. When t = Zar, the subscript Zar will also be omitted.

Proof. Given x ∈ X(q), we prove commutativity of the following diagram

Tr C′ −1 M d log q Wnix,t −q X,t −q ιx∗Kx,q,t / ιx∗WnΩk(x),log,t / Kn,X,t / Kn,X,t _

C′ −1 {x},t Tr M d log q −q Wnix,t −q ιx∗Kx,q,t / ιx∗WnΩk(x),t / (ix,∗Kn,{x},t) / Kn,X,t. The left square naturally commutes. The right square also commutes, because C′ is compatible with the

Grothendieck trace map TrWnix (the proofs of Proposition 1.20 and Proposition 1.29 give the case for t = Zar and t = ét, respectively). Now because C′ − 1 : W Ωq → W Ωq , which is identified {x},t n k(x),t n k(x),t q with C{x},t − 1 as a result of Theorem 1.7 and Proposition 1.26, annihilates WnΩk(x),log,t, the composite of the second row is zero. Thus the composite of the first row is zero. This yields a unique chain map M ζn,X,log,t : CX,t → Kn,X,log,t, 38 FEI REN i.e., on degree i = −q, we have ζ = ζ with n,X,log,t x∈X(q) n,x,log,t i M P i i i−1 ζn,x,log,t : Kx,q,t → Kn,X,log,t = Kn,X,t ⊕ Kn,x,t, i s = {s1 ...,sq} 7→ (ζn,X,t(s), 0).

As a direct corollary of Proposition 5.5 and Proposition 5.6, one has the following proposition.

Proposition 5.8 (Functoriality). (1) ζlog,t is compatible with proper pushforward. I.e., for f : X → Y a proper map, the following diagram of complexes is commutative

ζ M n,X,log,t (Wnf)∗CX,t / (Wnf)∗Kn,X,log,t

f∗ f∗

ζ M n,Y,log,t CY,t / Kn,Y,log,t.

Here f∗ on the left denotes the pushforward map for Kato’s complex of Milnor K-theory (cf.

Section 3), and f∗ on the right denotes TrWnf,log,t as defined in Proposition 1.20 and Proposi- tion 1.29. (2) ζlog,t is compatible with étale pullbacks. I.e., for f : X → Y an étale morphism, the following diagram of complexes is commutative

ζ M n,Y,log,t CY,t / Kn,Y,log,t

f ∗ f ∗

ζ M n,X,log,t (Wnf)∗CX,t / (Wnf)∗Kn,X,log,t.

Here f ∗ on the left denotes the pullback map for Kato’s complex of Milnor K-theory (cf. Sec- tion 3), and f ∗ on the right denotes the pullback map deﬁned in Proposition 1.23.

M n 5.4. ζn,X,log,t : CX,t/p ≃ νn,X,t −→ Kn,X,log,t is a quasi-isomorphism. Since ζn,X,t is termwise deﬁned n M n M via the d log map, it annihilates p CX,t. Therefore ζn,X,log,t annihilates p CX,t as well, and induces a chain map e M n ζn,X,log,t : CX,t/p → Kn,X,log,t. M n Since the d log map induces an isomorphism of complexes CX,t/p ≃ νn,X,t, to show ζn,X,log,t is a quasi-isomorphism, it is equivalent to show e ζn,X,log,t : νn,X,t → Kn,X,log,t is a quasi-isomorphism. e Lemma 5.9. Suppose X is separated smooth over the perfect ﬁeld k of characteristic p > 0. Then for any level n,

ζn,X,log,ét : νn,X,ét −→ Kn,X,log,ét; is a quasi-isomorphism. If we moreover have k = k, then e ζn,X,log,Zar : νn,X,Zar −→ Kn,X,log,Zar is also a quasi-isomorphism. e Proof. This is a local problem, thus it suffices to prove the statement for each connected component of X. Therefore we assume X is of pure dimension d over k. Then for any level n, we have a quasi-isomorphism ([GS88b, Cor 1.6]) d ≃ WnΩX,log,t[d] −→ νn,X,t. We also have d ≃ e WnΩX,log,ét[d] −→ Kn,X,log,ét (by Proposition 1.28), and d ≃ WnΩX,log,Zar[d] −→ Kn,X,log,Zar when k = k (by Proposition 1.19). CYCLE COMPLEX VIA COHERENT DUALITY 39

On degree −d, we have a diagram

−d ζ −d d n,x,log,t −d 0 d νn,X,t = (Wnιx)∗WnΩk(x),log,t / Kn,X,log,t = (Wnιx)∗Hx(WnΩX,t) x∈X(0) x∈X(0) M O M O e (−1)d d d WnΩX,log,t / WnΩX,log,t

which is naturally commutative, due to the deﬁnition of ζn,X,log,t. It induces quasi-isomorphisms as stated in the lemma.

Theorem 5.10. Let X be a separated scheme of ﬁnite type over k with k being a perfect ﬁeld of characteristic p> 0. Then the chain map

ζn,X,log,ét : νn,X,ét −→ Kn,X,log,ét is a quasi-isomorphism. Moreover if k = k, e

ζn,X,log,Zar : νn,X,Zar −→ Kn,X,log,Zar is also a quasi-isomorphism. e Proof. One can assume that X is reduced. In fact, the complex νn,X,t is deﬁned to be the same complex ≃ as νn,Xred,t (see (4.0.1)), and we have a quasi-isomorphism Kn,Xred,log,t −→ Kn,X,log,t given by the trace

map, according to Proposition 1.21 and Proposition 1.30. One noticese that ζn,Xred,log,t is compatible with ζn,X,log,te because of the functoriality of the map ζlog,t with respect to proper maps Proposition 5.8(1). As long as we have a quasi-isomorphism

ζn,Xred ,log,t : νn,Xred,t −→ Kn,Xred,log, we get automatically that e ζ n,Xred,log,t ≃ ζn,X,log,t : νn,Xred,t = νn,X,t −−−−−−−−→ Kn,Xred,log −→ Kn,X,log,t is a quasi-isomorphism. Now we do induction on thee dimensione of the reduced scheme X. Suppose X is of dimension d, and suppose ζn,Y,log,t is a quasi-isomorphism for schemes of dimension ≤ d − 1. Now decompose X into the singular part Z and the smooth part U j i .U ֒−→ X ←−֓ Z Then Z has dimension ≤ d − 1. Consider the following diagram in the derived category of complexes of Z/pn-modules

+1 (5.4.1) i∗νn,Z,t / νn,X,t / Rj∗νn,U,t / i∗νn,Z,t[1]

i∗ζ ζ Rj∗ζ i∗ζ [1] e n,Z,log,t e n,X,log,t e n,U,log,t e n,Z,log,t TrWni,log +1 i∗Kn,Z,log,t / Kn,X,log,t / Rj∗Kn,U,log,t / i∗Kn,Z,log,t[1],

where the two rows are distinguished triangles coming from Proposition 1.22, Proposition 1.31 and Proposition 4.1. We show that the three squares in (5.4.1) are commutative in the derived category. The left square is commutative because of Proposition 5.8(1). The middle square of (5.4.1) is induced from the diagram

(5.4.2) νn,X,t / j∗νn,U,t

ζ j∗ζ e n,X,log,t e n,U,log,t Kn,X,log,t / j∗Kn,U,log,t

of chain complexes. Let x ∈ X(q). When x ∈ X(q) ∩ U, both νn,X,t → j∗νn,U,t and Kn,X,log,t → j∗Kn,U,log,t give identity maps at x, therefore the square (5.4.2) commutes in this case. When x ∈ X(q)∩Z, e e 40 FEI REN both of these give the zero map at x, therefore the square (5.4.2) is also commutative. The right square of (5.4.1) can be decomposed in the following way (cf. (1.3.4) and (1.5.3)):

+1 i∗ Rj∗νn,U,t / RΓZ (νn,X,t)[1] o ≃ i∗νn,Z,t[1]

Rj∗ζ RΓ (ζ )[1] i∗ζ [1] e n,U,log,t e Z n,X,log,t e n,Z,log,t +1 i∗ Rj∗Kn,U,log,t / RΓZ (Kn,X,log,t)[1] o ≃ i∗Kn,Z,log,t[1].

The map i∗ on the first row is induced by the norm map of Milnor K-theory Section 3. It is clearly an isomorphism of complexes when t = Zar. It is a quasi-isomorphism when t = ét due to the purity theorem [Mos99, p.130 Cor.]. The map i∗ on the second row is induced from TrWni,log,t as defined in Proposition 1.20 and Proposition 1.29, and it is an isomorphism due to Proposition 1.22(1) when t = Zar, and Proposition 1.31 when t = ét. The first square commutes by naturality of the +1 map. The second commutes because of the compatibility of ζlog,t with the proper pushforward Proposition 5.8(1). We thus deduce that the right square of (5.4.1) commutes. Now consider over any perfect field k for either of the two cases: (1) t = ét and k is a perfect field, or (2) t = Zar and k = k. The left vertical arrow of (5.4.1) is a quasi-isomorphism because of the induction hypothesis. The third one counting from the left is also a quasi-isomorphism because of Lemma 5.9. Thus so is the second one.

Combine c M with M 6. ψX,t : ZX,t → CX,t ζn,X,log,t : CX,t −→ Kn,X,log,t c M 6.1. The map ψX,t : ZX,t → CX,t. In [Zho14, 2.14], Zhong constructed a map of abelian groups c M ψX,t(X): ZX (X) → CX,Zar(X) based on the Nesterenko-Suslin-Totaro isomorphism [NS89, Thm. 4.9][Tot92]. It is straightforward to check that Zhong’s construction induces a well-defined map of c M complexes ψ := ψX,t : ZX,t → CX,t of sheaves for both t = Zar and t = ét. Zhong in [Zho14, 2.15] proved that ψ is covariant with respect to proper morphisms, and contravariant with respect to quasi-finite flat morphisms.

c n ≃ 6.2. ζn,X,log,t ◦ ψX,t : ZX,t/p −→ Kn,X,log,t is a quasi-isomorphism. In [Zho14, 2.16] Zhong proved that ψX,ét combined with the Bloch-Gabber-Kato isomorphism [BK86, 2.8], gives a quasi-isomorphism c n ≃ ψX,ét : ZX,ét/p −→ νn,X,ét. In the proof, Zhong actually showed that these two complexes of sheaves on each section of the big Zariski site over X are quasi-isomorphic. Therefore by restrictione to the small Zariski site, we have c n ≃ ψX,Zar : ZX,Zar/p −→ νn,X,Zar. Combining Zhong’s quasi-isomorphism with Theorem 5.10: e Theorem 6.1. Let X be a separated scheme of finite type over k with k being a perfect field of positive characteristic p. Then the following composition c n ≃ ζn,X,log,ét ◦ ψX,ét : ZX,ét/p −→ Kn,X,log,ét, is a quasi-isomorphism. If moreover k = k, then the following composition c n ≃ ζn,X,log,Zar ◦ ψX,Zar : ZX,Zar/p −→ Kn,X,log,Zar, is also a quasi-isomorphism.

Remark 6.2. From the construction of the maps ζn,X,log,t and ψX,t, we can describe explicitly their composite map. We write here only the Zariski case, and the étale case is just given by the Zariski version on the small étale site and then doing the étale sheafification. c i Let U be a Zariski open subset of X. Let Z ∈ (ZX,Zar) (U)= z0(U, −i) be a prime cycle. • When i ∈ [−d, 0] and dim pU (Z) = −i, set q = −i. Then Z as a cycle of dimension q in q q U × ∆ , is dominant over some u = u(Z) ∈ U(q) under projection pU : U × ∆ → U. By slight abuse of notation, we denote by T0,...,Tq ∈ k(Z) the pullbacks of the corresponding q ∗ coordinates via Z ֒→ U × ∆ . Since Z intersects all faces properly, T0,...,Tq ∈ k(Z) . Thus { −T0 ,..., −Tq−1 } ∈ KM (k(Z)) is well-defined. Take the Zariski closure of Spec k(Z) in U × ∆q, Tq Tq q CYCLE COMPLEX VIA COHERENT DUALITY 41

′ ′ U X X and denote it by Z . Then pU maps Z to {u} = {u} ∩ U. Denote by iu : {u} ֒→ X the closed immersion, and denote the composition

p U X iu Z′ −−→U {u} ֒→ {u} ֒−→ X by h. h is clearly generically finite, then there exists an open neighborhood V of u in X such −1 −1 that the restriction h : h (V ) → V is finite. Then Wnh : Wn(h (V )) → WnV is also finite. ′ Therefore it makes sense to consider the trace map TrWnh near the generic point of Z . Similarly, ′ it makes sense to consider the trace map TrWnpU near the generic point of Z . Then we calculate

i −T0 −Tq−1 ζlog(ψ(Z)) = (−1) TrWniu (d log(Nmk(Z)/k(u(Z)){ ,..., })) Tq Tq

i −T0 −Tq−1 = (−1) TrWniu (TrWnpU d log{ ,..., }) (Lemma 5.3) Tq Tq

i TqdT0 − T0dTq TqdTq−1 − Tq−1dTq = (−1) TrWnh( ··· ) T0Tq Tq−1Tq Here in the last step we have used the functoriality of the trace map with respect to composition of morphisms [Con00, 3.4.1(2)]. • When i∈ / [−d, 0] or dim pU (Z) 6= −i, we have ζlog(ψ(Z)) = 0. Combining the functoriality of Zhong’s map ψ with Proposition 5.8, one arrives at the following proposition.

c n ≃ Proposition 6.3 (Functoriality). The composition ζn,X,log,t ◦ ψX,t : ZX,t/p −→ Kn,X,log,t is covariant with respect to proper morphisms, and contravariant with respect to ´etale morphisms for both t = Zar and t = ´et.

Part 3. Applications

7. De Rham-Witt analysis of νn,X,t and Kn,X,log,t Let X be a separated scheme of finite type over k of dimension d. In this section we will use termi- nologies as defined in [CR12, §1], such as Witt residual complexes,e etc. Recall that Ekedahl defined a map of complexes of WnOX -modules (cf. [CR12, Def. 1.8.3])

p := p : R∗Kn−1,X,t → Kn,X,t. {Kn,X }n

By abuse of notations, we denote by R : Wn−1X ֒→ WnX the closed immersion induced by the restriction map on the structure sheaves R : WnOX → Wn−1OX .

Lemma 7.1. The map p : R∗Kn−1,X,t → Kn,X,t induces a map of complexes of abelian sheaves

(7.0.1) p : Kn−1,X,log,t → Kn,X,log,t by applying p on each summand.

′ Proof. It suffices to show that Ct : Kn,X,t → Kn,X,t commutes with p for both t = ét and t = Zar. For ′ −1 ∗ ′ t = ét, Cét is the composition of τ : Kn,X,ét → (WnFX )∗Kn,X,ét and ǫ (CZar) : (WnFX )∗Kn,X,ét → −1 Kn,X,ét. Since τ is functorial with respect to any map of abelian sheaves, we know that

τ −1 R∗Kn−1,X,´et / (WnFX )∗R∗Kn−1,X,´et

p p

τ −1 Kn,X,ét / (WnFX )∗Kn,X,ét is commutative, thus it suffices to prove the proposition for t = Zar. That is, it suffices to show the diagrams (7.0.2) and (7.0.3) commute:

R ( ) ∗ 1.1.2 △ (7.0.2) R∗Kn−1,X ≃ / R∗(Wn−1FX ) Kn−1,X

p p △ {(WnFX ) Kn,X }n ( ) 1.1.2 △ Kn,X ≃ / (WnFX ) Kn,X , 42 FEI REN

(7.0.3)

R∗ TrW F △ ≃ △ n−1 X (WnFX )∗R∗(Wn−1FX ) Kn−1,X / R∗(Wn−1FX )∗(Wn−1FX ) Kn−1,X / R∗Kn−1,X

(WnFX )∗p △ p {(WnFX ) Kn,X }n Tr △ WnFX (WnFX )∗(WnFX ) Kn,X / Kn,X . Here p := p is the lift-and-multiplication-by-p map associated to the Witt residual complex {Kn,X }n △ {Kn,X }n, while p △ denotes the one associated to Witt residual system {(WnFX ) Kn,X }n {(WnFX ) Kn,X }n (cf. [CR12, 1.8.7]). By deﬁnition, △ △ p △ : R∗(Wn−1FX ) Kn−1,X → (WnFX ) Kn,X {(WnFX ) Kn,X }n is given by the adjunction map of

(W F )△(ap) △ n−1 X △ △ △ △ (Wn−1FX ) Kn−1,X −−−−−−−−−−−→ (Wn−1FX ) R Kn,X ≃ R (WnFX ) Kn,X , ≃ where ap is the adjunction of p for residual complexes (cf. [CR12, Def. 1.8.3]). The second diagram

(7.0.3) commutes because the trace map TrWnFX induces a well-deﬁned map between Witt residual complexes [CR12, Lemma 1.8.9].

It remains to show the commutativity of (7.0.2). According to the deﬁnition of p △ in (WnFX ) Kn,X [CR12, 1.8.7], we are reduced to show the adjunction square commutes:

R△( ) △ 1.1.2 △ △ ≃ △ △ R Kn,X / R (WnFX ) Kn,X / (Wn−1FX ) R Kn,X O O a △ a p (Wn−1FX ) ( p) ( ) 1.1.2 △ Kn−1,X / (Wn−1FX ) Kn−1,X .

△ And this is (Wn−1π) applied to the following diagram

R△( ) △ 1.1.1 △ △ ≃ △ △ R Wnk / R (WnFk) Wnk / (Wn−1Fk) R Wnk O O a △ a p (Wn−1Fk) ( p) ( ) 1.1.1 △ Wn−1k / (Wn−1Fk) Wn−1k.

We are reduced to show its commutativity. Notice that this diagram is over Spec Wn−1k, where the only possible ﬁltration is the one-element set consisting of the unique point of Spec Wn−1k. This means that the Cousin functor associated to this ﬁltration sends any dualizing complex to itself, and the map ap in the sense of a map either between residual complexes [CR12, Def. 1.8.3] or between dualizing complexes [CR12, Def. 1.6.3] actually agree. Now we start the computation. Formulas for (1.1.1) and for ap (in the sense of a map between dualizing complexes) are explicitly given in Section 1.1 and [CR12, 1.6.4(1)], respectively. Label the source and the target of WnFk by Spec Wnk1 and Spec k2 respectively. Take a ∈ Wn−1k1. De- note WnFk : (Spec Wnk1, Wnk1) → (Spec Wnk2, (WnFk)∗(Wnk1)), and R : (Spec Wn−1ki, Wn−1ki) → (Spec Wnki, R∗Wn−1ki) (i = 1, 2) the natural maps of ringed spaces. Now the down-right composition △ a △ △ • ((Wn−1Fk) ( p)) ◦ (1.1.1) equals to the Cousin functor E(Wn−1Fk) R Z (Wnk) applied to the following composition

(1.1.1) ∗ Wn−1k1 −−−−−→ Wn−1Fk HomW k ((Wn−1Fk)∗(Wn−1k1), Wn−1k2) ≃ n−1 2 a p ∗ ∗ −→ Wn−1Fk HomW k ((Wn−1Fk)∗(Wn−1k1), R HomW k (R∗Wn−1k2, Wnk2)), ≃ n−1 2 n 2 −1 a 7→ [(Wn−1Fk)∗1 7→ (Wn−1Fk) (a)] −1 7→ [(Wn−1Fk)∗1 7→ [R∗1 7→ p(Wn−1Fk) (a)]]. △ a △ △ • And R (1.1.1) ◦ ( p) equals to the Cousin functor E(Wn−1Fk) R Z (Wnk) applied the following composition a p ∗ Wn−1k1 −→ R HomW k (R∗Wn−1k1, Wnk1) ≃ n 1 CYCLE COMPLEX VIA COHERENT DUALITY 43

(1.1.1) ∗ ∗ −−−−−→ R HomWnk1 (R∗Wn−1k1, WnFk HomWnk2 ((WnFk)∗(Wnk1), Wnk2)),

a 7→ [R∗1 7→ p(a)] −1 7→ [R∗1 7→ [(WnFk)∗1 7→ (WnFk) p(a)]]. −1 −1 It remains to identify p((Wn−1Fk) a) and (WnFk) p(a). And this is straightforward: write a = n−2 i i=0 V [ai] ∈ Wn−1k1, n−2 n−2 P −1 −1 i −1 i+1 p (7.0.4) (WnFk) p(a)= (WnFk) p(V [ai]) = (WnFk) (V [ai ]) i=0 i=0 X X n−2 n−2 i+1 i 1/p −1 = (V [ai]) = p (V [ai ]) = p((Wn−1Fk) a). i=0 i=0 X X Hence we ﬁnish the proof. However we don’t naturally have a restriction map R between residual complexes. Nevertheless, we ≃ could use the quasi-isomorphism ζn,X,log,t : νn,X,t −→ Kn,X,log,t to build up a map

(7.0.5) R : Kn,X,log,t → Kn−1,X,log,t e in the derived category Db(X, Z/pn). For this we will need to show that p and R induce chain maps for νn,X,t. This should be well-known to experts, we add here again due to a lack of reference. Lemma 7.2. e p : νn,X,t → νn+1,X,t, R : νn+1,X,t → νn,X,t given by p and R termwise, are well deﬁned maps of complexes for both t = Zar and t = ´et.

e e e e ′ Proof. It suﬃces to prove for t = Zar. Let x ∈ X(q) be a point of dimension q. Let ρ : X → {x} be the normalization of {x}. Let x′ be the generic point of X′ and y′ ∈ X′(1) be a codimension 1 point. Denote y := ρ(y′). It suﬃces to check the commutativity of the following diagrams in (1) and (2). (1) Firstly,

q ∂ q−1 q ∂ q−1 WnΩx′,log / WnΩy′,log Wn+1Ωx′,log / Wn+1Ωy′,log

p p R R q ∂ q−1 q ∂ q−1 Wn+1Ωx′,log / Wn+1Ωy′,log, WnΩx′,log / WnΩy′,log. ′ Notice that p = p◦R. Suppose π is a uniformizer of discrete valuation ring OX′,y′ and u1,...,uq are invertible elements in OX′,y′ . Calculate ′ p(∂(d log[π ]nd log[u2]n ...d log[uq]n))

= p(d log[u2]n ...d log[uq]n)

= p(d log[u2]n+1 ...d log[uq]n+1) ′ = p(∂(d log[π ]n+1d log[u2]n+1 ...d log[uq]n+1)) ′ = ∂(p(d log[π ]n+1d log[u2]n+1 ...d log[uq]n+1)) ′ = ∂(p(d log[π ]nd log[u2]n ...d log[uq]n)), and

p(∂(d log[u1]nd log[u2]n ...d log[uq]n)) =0

= p(∂(d log[u1]n+1d log[u2]n+1 ...d log[uq]n+1))

= ∂(p(d log[u1]n+1d log[u2]n+1 ...d log[uq]n+1))

= ∂(p(d log[u1]nd log[u2]n ...d log[uq]n)). This proves the ﬁrst diagram. Now the second. ′ R(∂(d log[π ]n+1d log[u2]n+1 ...d log[uq]n+1))

= R(d log[u2]n+1 ...d log[uq]n+1)

= d log[u2]n ...d log[uq]n 44 FEI REN

′ = ∂(d log[π ]nd log V [u2]n ...d log V [uq]n) ′ = ∂(R(d log[π ]n+1d log[u2]n+1 ...d log[uq]n+1)), and

R(∂(d log[u1]n+1d log[u2]n+1 ...d log[uq]n+1)) =0

= ∂(d log[u1]nd log[u2]n ...d log[uq]n)

= ∂(R(d log[u1]n+1d log[u2]n+1 ...d log[uq]n+1)). (2) Secondly,

q−1 tr q−1 q−1 tr q−1 WnΩy′,log / WnΩy,log Wn+1Ωy′,log / Wn+1Ωy,log

p p R R q−1 tr q−1 q−1 tr q−1 Wn+1Ωy′,log / Wn+1Ωy,log, WnΩy′,log / WnΩy,log.

X X′ X X Notice that ρ : X′ → {x} can be restricted to a map from {y′} to {y} ({x} denotes X′ X the closure of x in X, and similarly for {y′} , {y} ). Furthermore, y′ (resp. y) belongs to X′ X the smooth locus of {y′} (resp. {y} ), and there p and R come from the restriction of the p and R on the respective smooth locus. The map tr, induced by Milnor’s norm map, agrees

with the Grothendieck trace map TrWnρ due to Lemma 5.3. And according to compatibility of the Grothendieck trace map with the Witt system structure (i.e. de Rham-Witt system structure with zero diﬀerential) on canonical sheaves [CR12, 4.1.4(6)], we arrive at the desired commutativity.

Lemma 7.3. Assume either • t = Zar and k = k, or • t = ´et and k being a perfect ﬁeld of characteristic p> 0. Then we have the following short exact sequence

j p Ri (7.0.6) 0 → νi,X,t −→ νi+j,X,t −−→ νj,X,t → 0, in the category of complexes of sheaves over Xt, and a distinguished triangle e e e j p Ri +1 (7.0.7) Ki,X,log,t −→ Ki+j,X,log,t −−→ Kj,X,log,t −−→ b n in the derived category D (Xt, Z/p ). Proof. (1) Because of Lemma 7.2, it suﬃces to show

j q p q Ri q 0 → WiΩx,log,t −→ Wi+j Ωx,log,t −−→ Wj Ωx,log,t → 0

is short exact for any given point x ∈ X(q). And this is true for t = ét because of [CSS83, Lemme 1 q 3]. And for t = Zar, one further needs R ǫ∗WnΩx,log,ét = 0 for any x ∈ X(q) when k = k, which is proved in [Suw95, Cor. 2.3]. (2) Now it suffices to show that p and R for the system {Kn,X,log,t}n are compatible with p and R

of the system {νn,X,t}n, via the quasi-isomorphism ζn,X,log,t. The compatibility for R is clear by deﬁnition. It remains to check the compatibility for p. Because ζn,X,log,t = (5.3.1) ◦ ζn,X,t, it suﬃces to checke compatibility of p : νn−1,X,t → νn,X,t with p : Kn−1,X,t → Kn,X,t via ζn,X,t. At a given degree −q and given point x ∈ X(q), the map ζn,X,t : νn,X,t → Kn,X,t factors as

e e q q q q (−1) TrWnix,t q (Wnιx)∗WnΩ −→ (Wnix)∗WnΩ = (Wnix)∗K −−−−−−−−−−→ K . x,log,t x,t n,{x},t e n,X,t The ﬁrst arrow is the inclusion map and is naturally compatible with p. The compatibility of p via the trace map is given in [CR12, Lemma 1.8.9]. CYCLE COMPLEX VIA COHERENT DUALITY 45

8. Higher Chow groups of zero cycles Let k be a perfect field of characteristic p > 0 and X be a separated scheme of finite type over k of dimension d. 8.1. Vanishing and finiteness results. Proposition 8.1. There is a distinguished triangle ′ c n Cét−1 +1 ZX,ét/p → Kn,X,ét −−−−→ Kn,X,ét −−→ b n in the derived category D (Xét, Z/p ). When k = k, one also has the Zariski counterpart. Namely, we have a distinguished triangle ′ c n C −1 +1 (8.1.1) ZX /p → Kn,X −−−→ Kn,X −−→ in the derived category Db(X, Z/pn). c n In particular, when k = k and X is Cohen-Macaulay of pure dimension d, then ZX /p is concentrated at degree −d, and the triangle (8.1.1) becomes

′ c n C −1 +1 ZX /p → WnωX [d] −−−→ WnωX [d] −−→

in this case. Here WnωX is the only non-vanishing cohomology sheaf of Kn,X (when n =1, W1ωX = ωX is the usual dualizing sheaf on X). Proof. This is direct from the main result Theorem 6.1 and Remark 1.17. Proposition 8.2. Assume k = k. Then higher Chow groups of zero cycles equals the C′-invariant part of the cohomology groups of Grothendieck’s coherent dualizing complex, i.e., n −q C′−1 CH0(X, q; Z/p )= H (WnX,Kn,X ) . Proof. This follows directly from the Proposition 1.15 and the main result Theorem 6.1. b n Proposition 8.3. There is an isomorphism in D ((WnX)ét, Z/p ) ! n Kn,X,log,ét ≃ R(Wnπ) (Z/p ), ! where R(Wnπ) is the extraordinary inverse image functor defined in [SGA4-3, ExposéXVIII, Thm 3.1.4]. Proof. This follows directly from the main theorem Theorem 5.10 and [JSS14, Thm. 4.6.2]. Corollary 8.4 (Affine vanishing). Suppose X is affine and Cohen-Macaulay of pure dimension d. Then (1) When t = Zar and k = k, n CH0(X, q, Z/p )=0 for q 6= d. (2) When t = ét, −q c n R Γ(Xét, ZX /p )=0 for q 6= d, d − 1. If one further assumes k = k or smoothness, the possible non-vanishing occurs only in degree q = d.

Proof. When X is Cohen-Macaulay of pure dimension d, WnX is also Cohen-Macaulay of pure dimension d by Serre’s criterion, and Kn,X,t is concentrated at degree −d for all n [Con00, 3.5.1]. Now Serre’s affine −q −q vanishing theorem implies H (WnX,Kn,X,t)=0for q 6= d. This implies that R Γ(WnX,Kn,X,log,t)= n 0 unless q = d, d − 1. With the given assumptions, Theorem 6.1 implies that CH0(X, q, Z/p ) = −q c n R Γ(Xét, ZX /p ) = 0 unless q = d, d − 1. If one also assumes k = k, Proposition 8.2 gives the vanishing result for q = d − 1. d d When X is smooth, Cét − 1 : WnΩX,ét → WnΩX,ét is surjective by [GS88a, 1.6(ii)] (see (1.2.31)). By ′ ′ −d −d compatibility of Cét and Cét Proposition 1.26, one deduces that C − 1 : H (Kn,X,ét) → H (Kn,X,ét) is surjective. Generalizing Bass’s finiteness conjecture for K-groups (cf. [Wei13, IV.6.8]), the finiteness of higher Chow groups in various arithmetic settings has been a ”folklore conjecture” in literature (expression taken from [KS12, §9]). The following result was first proved by Geisser [Gei10, §5, eq. (12)] using the finiteness result from étale cohomology theory, and here we deduce it as a corollary of our main theorem, which essentially relies on the finiteness of coherent cohomologies on a proper scheme. We remark that Geisser’s result is more general than ours in that he allows arbitrary torsion coefficients. 46 FEI REN

Corollary 8.5 (Finiteness, Geisser). Assume k = k. Let X be proper over k. Then for any q, n CH0(X, q; Z/p ) is a ﬁnite Z/pn-module.

−q c n −q Proof. According to Theorem 6.1, R Γ(X, ZX /p )= R Γ(X,Kn,X,log). Thus it suffices to show that −q n −q for every q, R Γ(X,Kn,X,log) is a finite Z/p -module. First of all, since R Γ(X,Kn,X,log) is the ′ −q −q C -invariant part of R Γ(X,Kn,X ) by Proposition 1.15 and Proposition 1.27, R Γ(X,Kn,X,log) is a −1 1−WnF n −q module over the invariant ring (Wnk) X = Z/p . Because X is proper, R Γ(X,Kn,X ) is a finite Wnk-module by the local-to-global spectral sequence. Then Proposition A.7 gives us the result. −q Alternatively, we can also do induction on n. In the n = 1 case, because R Γ(X,KX,log) is the ′ −q C -invariant part of the finite dimensional k-vector space H (X,KX ) again by Proposition 1.15 and −1 Proposition 1.27, it is a finite Fp-module by p -linear algebra Proposition A.3. The desired result then follows from the long exact sequence associated to (7.0.7) by induction on n. We refer to Definition A.4 and Remark A.5(2) for the definition of the semisimplicity and the notation (−)ss in this context. Corollary 8.6 (Semisimplicity). Assume k = k. Let X be proper over k. Then for any q, −q n H (WnX,Kn,X )ss = CH0(X, q; Z/p ) ⊗Z/pn Wnk. −q Proof. Since X is proper, H (WnX,Kn,X ) is a finite Wnk-module for any q. Then according to Proposition A.8, −q −q C′−1 H (WnX,Kn,X )ss = H (WnX,Kn,X ) ⊗Z/pn Wnk. The claim now follows from Proposition 8.2. 8.2. Etale´ descent. The results Proposition 8.7, Proposition 8.8 in this subsection are well-known to experts. Proposition 8.7 (Gros-Suwa). Assume k = k. Then one has a canonical isomorphism ≃ νn,X,Zar = ǫ∗νn,X,ét −→ Rǫ∗νn,X,ét in the derived category Db(X, Z/pn). e e e Proof. When k = k, terms of the étale complex νn,X,ét are ǫ∗-acyclic according to [GS88a, 3.16]. The étale descent of Bloch’s cycle complex with Z-coefficients is shown in [Gei10, Thm 3.1], assuming the Beilinson-Lichtenbaum conjecture. Lookinge into the proof one sees that the mod pn version holds conjecture-free, and is a corollary of [GL00, 8.4] (we thank Geisser for pointing this out). But one could also deduce this as a corollary of Proposition 8.7 via Zhong’s quasi-isomorphism in Section 6.2 (which is again dependent on the main result of Geisser-Levine [GL00, 1.1]). Proposition 8.8 (Geisser-Levine). Assume k = k. Then one has a canonical isomorphism c n c n ≃ c n ZX,Zar/p = ǫ∗ZX,ét/p −→ Rǫ∗ZX,ét/p . in the derived category Db(X, Z/pn). As a result, n −q c n CH0(X, q; Z/p ) ≃ R Γ(Xét, ZX,ét/p ). Proof. Clearly, we have the compatibility

c n ≃ c n ZX,Zar/p / Rǫ∗ZX,´et/p

ψX,Zar Rǫ∗ψX,´et

≃ νn,X,Zar / Rǫ∗νn,X,´et.

ψ Rǫ ψ c n X,Zar Proposition 8.7 ∗ X,ét c n Thus Z /p −−−−→ νn,X,Zar = ǫ∗νen,X,ét −−−−−−−−−→eRǫ∗νn,X,ét −−−−−−→ Rǫ∗Z /p . X,Zar ≃ ≃ ≃ X,ét

Corollary 8.9. Assumeek = k. Supposee X is affine and Cohen-Macaulaye of pure dimension d. Then i c n i R ǫ∗(ZX,ét/p )= R ǫ∗νn,X,ét =0, i 6= −d. Proof. This is a direct consequence of Proposition 8.8, Proposition 8.7 and Corollary 8.4. e CYCLE COMPLEX VIA COHERENT DUALITY 47

8.3. Birational geometry and rational singularities. Recall the following definition of resolution- rational singularities, which are more often called rational singularities before in the literature, but here we follow the terminology from [Kov17] (see also Remark 8.11(1)). Definition 8.10 (cf. [Kov17, 9.1]). An integral k-scheme X is said to have resolution-rational singularities, if (1) there exists a birational proper morphism f : X → X with X smooth (such a f is called a resolution of singularities or simply a resolution of X), and i i (2) R f∗O = R f∗ω =0 for i ≥ 1. And f∗O = OeX . e Xe Xe Xe Such a map f : X → X is called a rational resolution of X. Note that the cohomological condition (2) is equivalent to the following condition e (2’) OX ≃ Rf∗O , f∗ω ≃ Rf∗ω in the derived category of abelian Zariski sheaves. Xe Xe Xe Remark 8.11. (1) According to [Kov17, 8.2], on integral k-schemes of pure dimension, our definitions for resolution-rational singularities and for rational resolutions are the same as the ones in [Kov17, 9.1]. (2) A necessary condition for an integral scheme to have such singularities is being Cohen-Macaulay. This is a standard result, cf. [Kov17, 8.3]. (3) According to [Kov17, 9.6], resolution-rational singularities are pseudo-rational. By definition [Kov17, 1.2], a k-scheme X is said to have pseudo-rational singularities, if it is normal Cohen- Macaulay, and for every normal scheme X′, every projective birational morphism f : X′ → X, Trf the composition f∗ωX′ → Rf∗ωX′ −−→ ωX is an isomorphism. Corollary 8.12. Let X and Y be integral k-schemes of pure dimensions which have pseudo-rational singularities and are properly birational, i.e., there are proper birational k-morphisms f : Z → X and g : Z → Y with Z being some integral scheme. Then we have

−q c n −q c n R Γ(Xét, Z /p )= R Γ(Yét, ZY,ét/p ) X,e ét for all q and all n ≥ 1. If we assume furthermore k = k, we also have

n n CH0(X, q, Z/p )=CH0(Y, q, Z/p ) for all q and all n ≥ 1.

Remark 8.13. (1) In particular, since for any rational resolution of singularities f : X → X, X and X are properly birational as k-schemes (i.e., take Z to be X), one can compute the higher Chow groups of zero cycles of X via those of X. e e (2) Deleting the pseudo-rational singularities assumption (in particular,e we relax the Cohen-Macaulay assumptions on X and Y ), the proof still passese through with the following assumption: X and Y are linked by a chain of proper birational maps and each of these maps has its trace map being a quasi-isomorphism between the residual complexes. Such a proper birational map is called a cohomological equivalence in [Kov17, 8.4]. Proof. Using Chow’s Lemma [Kov17, 4.1], we know that there exist projective birational morphisms ′ ′ ′ ′ f f g g f : Z1 → Z and g : Z2 → Z such that the compositions Z1 −→ Z −→ X and Z2 −→ Z −→ Y are also birational and projective. Let U ⊂ Z be an open dense subset such that f ′ and g′ restricted to the preimage of U are isomorphisms. Take Z′ be the Zariski closure of the image of the diagonal of U in ′ ′ Z1 ×Z Z2 with the reduced scheme structure. Then the two projections Z → Z1 and Z → Z2 are also projective and birational. This means that by replacing Z′ with Z, f with Z′ → X and g with Z′ → Y , we can assume our f : Z → X, g : Z → Y to be projective birational and our Z to be integral. Using Macaulayﬁcation [Kov17, 4.3, 4.4] we can additionally assume that Z is Cohen-Macaulay. This implies that f and g are pseudo-rational modiﬁcations by [Kov17, 9.7]. Suppose that X is of pure dimension d. Then so is Z. Now [Kov17, 8.6] implies that the trace map of f induces an isomorphism ≃ Trf : Rf∗KZ,t −→ KX,t b in D (Xt, Z/p). Thus ≃ Trf,log : Rf∗KZ,log,t −→ KX,log,t 48 FEI REN

b is also an isomorphism in D (Xt, Z/p). Consider the diagram

n−1 p R +1 (8.3.1) f∗KZ,log,t / f∗Kn,Z,log,t / f∗Kn−1,Z,log,t / f∗KZ,log,t[1]

Trf,log TrWnf,log TrWn−1f,log Trf,log [1] n−1 p R +1 KX,log,t / Kn,X,log,t / Kn−1,X,log,t / KX,log,t[1]

b in D (Xt, Z/p). The ﬁrst row is Rf∗ applied to the triangle (7.0.7) on Z. The second row is the triangle (7.0.7) on X. The left square commutes on the level of complexes by compatibility of the trace map with p [CR12, 1.8.9]. To prove commutativity of the middle square in the derived category, it suﬃces to show the square R f∗νn,Z,t / f∗νn−1,Z,t

f∗ f∗ e e R νn,X,t / νn−1,X,t commutes on the level of complexes. Since the vertical maps f∗ for Kato-Moser complexes are tr (cf. §4), which are by deﬁnition the reductione of the norme maps for Milnor K-theory, they agree with the

Grothendieck trace maps TrWnf , TrWn−1f by Lemma 5.3. And according to the compatibility of R with the Grothendieck trace maps [CR12, 4.1.4(6)], we arrive at the desired commutativity. The right square in (8.3.1) commutes by naturality of the ”+1” map. With all these commutativities we conclude that the vertical maps in (8.3.1) define a map of triangles. By induction on n we deduce that ≃ TrWnf,log : Rf∗Kn,Z,log,t −→ Kn,X,log,t b n is an isomorphism in D (Xt, Z/p ) for every n. The main result Theorem 6.1 thus implies −q c n −q c n R Γ(Zét, Z /p )= R Γ(Xét, ZX,ét/p ) X,e ét for all q and n. When k = k, the same theorem also implies that n n CH0(Z, q, Z/p )=CH0(X, q, Z/p ) for all q and n. Now replacing f with g everywhere in the above argument and we get the result.

8.4. Galois descent. Corollary 8.14. Let f : X → Y be a finite étale Galois map with Galois group G. Then −d c n −d c n G R Γ(Yét, ZY /p )= R Γ(Xét, ZX /p ) When k = k, we also have n n G CH0(Y, d; Z/p )=CH0(X, d; Z/p ) . Proof. The pullback f ∗ induces two canonical maps ∗ c c G ∗ G f : ZY,ét → (f∗ZX,ét) , f : Kn,Y,log,ét → (f∗Kn,X,log,ét) . Both of them are isomorphisms of complexes, because each term of these complexes are étale sheaves. Because of the contravariant functoriality with respect to étale morphisms (Proposition 6.3 and [Zho14,

2.15]), ζlog ◦ ψ is G-equivariant. In particular, the diagram

ζ ◦ψ c n log ZY,´et/p / Kn,Y,log,´et

f ∗ f ∗ ζ ◦ψ c n G log G (f∗ZX,ét/p ) / (f∗Kn,X,log,ét) commutes. −d ∗ G Apply R Γ(Yét, −) to the isomorphism f : Kn,Y,log,ét → (f∗Kn,X,log,ét) , one gets −d −d G R Γ(Yét,Kn,Y,log,ét)= R Γ(Yét, (f∗Kn,X,log,ét) ). CYCLE COMPLEX VIA COHERENT DUALITY 49

Consider the local-to-global spectral sequence associated to the right hand side of this equality, there’s only one non-zero term in the E∞-page with total degree −d (which is a term in the E2-page), thus we have −d G 0 −d G R Γ(Yét, (f∗Kn,X,log,ét) )= H (Yét, H ((f∗Kn,X,log,ét) )). Because (−)G commutes with taking kernels and with H0, we have 0 −d 0 −d G H (Yét, H (Kn,Y,log,ét)) = H (Yét, H (f∗Kn,X,log,ét)) .

Because f∗ preserves kernels, we have 0 −d G 0 −d G H (Yét, H (f∗Kn,X,log,ét)) = H (Xét, H (Kn,X,log,ét)) . Again by the observation from the spectral sequence, this means 0 −d G −d G H (Xét, H (Kn,X,log,ét)) = R Γ(Xét,Kn,X,log,ét) .

Since ζlog ◦ ψ is G-equivariant, the main theorem Theorem 6.1 implies −d c n −d c n G R Γ(Y´et, ZY /p )= R Γ(X´et, ZX /p ) . When k = k, Proposition 8.8 implies n n G CH0(Y, d; Z/p )=CH0(X, d; Z/p ) .

Appendix A. Semilinear algebra Definition A.1. Let k be a perfect field of positive characteristic p, and V be a finite dimensional k-vector space. A p-linear map (resp. p−1-linear map) on V is a map T : V → V , such that T (v + w)= T (v)+ T (w),T (cv)= cpT (v), v,w ∈ V,c ∈ k. (resp. T (v + w)= T (v)+ T (w),T (cv)= c−pT (v), v,w ∈ V,c ∈ k.) We say a map T : V → V is semilinear if it is either p-linear or p−1-linear. A semilinear map T : V → V is called semisimple, if ImT = V . Remark A.2. Let T be a semilinear map. (1) Note that p −p {c ∈ k | c = c} = Fp = {c ∈ k | c = c}. The fixed points of T V 1−T := {v ∈ V | T (v)= v}

is a Fp-vector space. (2) There is a descending chain of k-vector subspaces of V ImT ⊃ ImT 2 ⊃···⊃ ImT n ⊃ .... Since V is finite dimensional, it becomes stationary for some large N ∈ N. Define n N N+1 Vss := Im(T ) = Im(T ) = Im(T )= .... n\≥1 Obviously, (a) Vss is a k-vector subspace of V that is stable under T . T is semistable on Vss. 1−T (b) V ⊂ Vss. The proof of the following result is given in [SGA7-II] for p-linear maps, but an analogous proof also works for p−1-linear maps. Proposition A.3 ([SGA7-II, ExposéXXII, Cor. 1.1.10, Prop. 1.2]). Suppose k is a separably closed field of positive characteristic p. Then 1 − T : V → V is surjective. And 1−T Vss ≃ V ⊗Fp k, 1−T 1−T which in particular means V is a finite dimensional Fp-vector space with dimFp V = dimk Vss. We generalize the definition of a semilinear map. 50 FEI REN

Definition A.4. Let k be a perfect field of positive characteristic p, and let Wnk be the ring of the n-th truncated Witt vectors of k. Let M be a finitely generated Wnk-module. A p-linear map (resp. p−1-linear map) on M is a map T : M → M, such that

′ ′ ′ T (m + m )= T (m)+ T (m ),T (cm)= WnFk(c)T (m), m,m ∈ M,c ∈ Wnk. ′ ′ −1 ′ (resp. T (m + m )= T (m)+ T (m ),T (cm)= WnFk (c)T (m), m,m ∈ M,c ∈ Wnk.)

Here Fk denotes the p-th power Frobenius on the field k. We say that T is semilinear if it is either p-linear or p−1-linear in this sense. A semilinear map T : V → V is called semisimple, if ImT = V . Remark A.5. Let T be a semilinear map in the sense of Definition A.4. −1 (1) Write σ = WnFk (resp. σ = WnFk ). Then 1−σ n (Wnk) := {c ∈ Wnk | σ(c)= c} = Z/p for both cases. The fixed points of T M 1−T := {m ∈ M | T (m)= m} is a Z/pn-module. (2) As in the case of vector spaces, ImT ⊃ ImT 2 ⊃···⊃ ImT n ⊃ ...

is a descending chain of Wnk-submodules of M. It becomes stationary for some large N ∈ N, because M as a ﬁnitely generated Wnk-module is artinian. Deﬁne the Wnk-submodule of M

n N N+1 Mss := Im(T ) = Im(T ) = Im(T )= .... n\≥1 Then (a) Mss is a Wnk-submodule of M that is stable under T . T is semistable on Mss. 1−T (b) M ⊂ Mss. (c) (M/p)ss = Mss/p ⊂ M/p. Proposition A.6. Let k be a separably closed ﬁeld of positive characteristic p. Then 1 − T : M → M is surjective.

Proof. Take m ∈ M. Because M is finitely generated as a Wnk-module, M/pM is a finite dimensional k- vector space. Then Proposition A.3 implies that there exists a m′ ∈ M, such that (1−T )(m′)−m ∈ pM. That is, there exists a m1 ∈ M such that ′ (1 − T )(m )= m + pm1. ′ Do the same process with m1 instead of m, one gets a m1 ∈ M and a m2 ∈ M such that ′ (1 − T )(m1)= m1 + pm2. Thus ′ ′ 2 (1 − T )(m − pm1)= m − p m2. n Repeat this process. After finitely many times, because p = 0 in Wnk, ′ ′ n−1 n−1 ′ (1 − T )(m − pm1 + ··· + (−1) p mn−1)= m.

Proposition A.7. Let k be a separably closed ﬁeld of positive characteristic p. Then (1) M 1−T /(pM)1−T = (M/p)1−T . (2) M 1−T is a ﬁnite Z/pn-module.

n m Proof. Since Wnk is of p -torsion, we know that p M = 0 for some m ≤ n. Do induction on the smallest number m such that pmM = 0. When m = 1, the ﬁrst claim is trivial, and M = M/p is actually a ﬁnite dimensional k-vector space, thus the second claim follows from Proposition A.3. REFERENCES 51

Now we assume m> 1. Note that T induces a semilinear map on pM and pM is a ﬁnite Wnk-module, so by Proposition A.6 the map 1−T : pM → pM is surjective. Now we have the two rows on the bottom of the following diagram being exact: 0 00 O O O

1−T 0 / M 1−T /(pM)1−T / M/p / M/p / 0 O O O

1−T 0 / M 1−T / M / M / 0 O O O

1−T 0 / (pM)1−T / pM / pM / 0. O O O

0 0 0 The vertical maps between the last two rows are natural inclusions, and the first row is the cokernels of these inclusion maps. The snake lemma implies that the first row is exact, which means that M 1−T /(pM)1−T = (M/p)1−T . This is a finite Z/pn-module by the case m = 1. On the other hand, since pm−1 · pM = 0, the induction 1−T n hypothesis applied to the Wnk-module pM gives (pM) is a finite Z/p -module. Now the vertical exact sequence on the left gives that M 1−T is a finite Z/pn-module. Proposition A.8. Let k be a separably closed field of positive characteristic p. Then we have an identification of Wnk-modules 1−T Mss ≃ M ⊗Z/pn Wnk. Proof. For the finite dimensional k-vector space M/p, Proposition A.3 tells us that 1−T (M/p)ss ≃ (M/p) ⊗Fp k

In other words, there exists m1,...,md ∈ M (d = dimFp M/p), such that m1 + pM,...,md + pM ∈ 1−T (M/p) generate (M/p)ss asa k-vector space. Because of Proposition A.7(1), one can choose m1, ..., md ∈ 1−T M . Since M is a ﬁnite generated Wnk-module, Mss as a submodule is also ﬁnitely generated over 1−T Wnk. Note moreover that (M/p)ss = Mss/p. Apply Nakayama’s lemma, m1, ..., md ∈ M generate Mss as an Wnk-module.

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