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TROPICAL CYCLE CLASSES FOR NON-ARCHIMEDEAN SPACES ANDWEIGHT DECOMPOSITION OF DE RHAM COHOMOLOGY SHEAVES

YIFENG LIU

Abstract. This article has three major goals. First, we define tropical cycle class maps forsmooth varieties over non-Archimedean fields, valued in the Dolbeault cohomology definedin terms of real forms introduced by Chambert-Loir and Ducros. Second, we construct afunctorial decomposition of de Rham cohomology sheaves, called weight decomposition, forsmooth analytic spaces over certain non-Archimedean fields of characteristic zero, whichgeneralizes a construction of Berkovich and solves a question raised by himself. Third, wereveal a connection between the tropical theory and the algebraic de Rham theory. Asan application, we show that algebraic cycles that are trivial in the algebraic de Rhamcohomology are trivial as currents for Dolbeault cohomology as well.

Contents

1. Introduction 12. Chow cohomology revisited 73. Tropical cycle class map 104. Rigid cohomology and logarithmic differential forms 195. Weight decomposition of de Rham cohomology sheaves 246. Cohomological triviality before tropicalization 377. Cohomological triviality for tropical cycle classes 45References 52

1. Introduction

This article has three major goals. First, we define tropical cycle class maps for smoothvarieties over non-Archimedean fields, valued in the Dolbeault cohomology defined in termsof real forms introduced by Chambert-Loir and Ducros [CLD12]. Second, we construct afunctorial decomposition of de Rham cohomology sheaves, called weight decomposition, forsmooth analytic spaces over non-Archimedean fields embeddable into CF (see below), whichgeneralizes a construction of Berkovich and solves a question raised by himself in [Ber07].Third, we reveal a connection between the tropical theory and the algebraic de Rham theory.As an application, we show that algebraic cycles that are trivial in the algebraic de Rhamcohomology are trivial as currents for Dolbeault cohomology as well.

In this article, by a non-Archimedean field, we mean a complete topological field withrespect to a nontrivial non-Archimedean valuation of rank one. We fix a finite field Fthroughout the article. Denote by ZF the ring of Witt vectors in F and QF the field offractions of ZF. Then QF is naturally a non-Archimedean field, which is locally compact.

Date: July 7, 2017.2010 Mathematics Subject Classification. 14G22.

1

2 YIFENG LIU

Moreover, we fix a complete algebraic closure CF of QF, which is also a non-Archimedeanfield.

1.1. Tropical cycle class map. LetK be a non-Archimedean field. In [CLD12], Chambert-Loir and Ducros define, for every K-analytic (Berkovich) space1 X, a bicomplex (A •,•

X , d′, d′′)of sheaves of real vector spaces on X concentrated in the first quadrant. It is a non-Archimedean analogue of the bicomplex of (p, q)-forms on complex manifolds. In particular,we may define analogously the Dolbeault cohomology (Definition 3.1) of X to be

Hp,q(X) := ker(d′′ : A p,qX (X)→ A p,q+1

X (X))im(d′′ : A p,q−1

X (X)→ A p,qX (X))

.

Moreover, we have an integration map∫X

: A n,nX (X)c → R

for n = dim(X), where A n,nX (X)c is the space of (n, n)-forms on X whose support is compact

and disjoint from the boundary of X.By [Jel16] and [CLD12], we know that for every p ≥ 0, the complex (A p,•

X , d′′) is a fineresolution of the sheaf ker(d′′ : A p,0

X → A p,1X ). In §3, we will construct a canonical Q-subsheaf

T pX of ker(d′′ : A p,0

X → A p,1X ) such that the induced map

T pX ⊗Q R → ker(d′′ : A p,0

X → A p,1X )

is an isomorphism. In particular, we have a canonical isomorphismHq(X,T p

X)⊗Q R ∼= Hp,q(X)for every p, q ≥ 0.

Recall that in the complex world, for a smooth complex algebraic variety X , we havea cycle class map from CHp(X ) to the classical Dolbeault cohomology Hp,p

∂(X an) of the

associated complex manifold X an. Over a non-Archimedean field K, we may associate aseparated scheme X of finite type over K a K-analytic space X an [Ber93, §2.6]. We put

Hp,qtrop(X ) := Hq(X an,T p

Xan).The following theorem is an analogue of the above cycle class map in the non-Archimedeansetup.

Theorem 1.1 (Definition 3.6, Theorem 3.7, Corollary 3.10). Let K be a non-Archimedeanfield and X a separated smooth scheme of finite type over K of dimension n. Then there isa tropical cycle class map

cltrop : CHp(X )→ Hp,ptrop(X ),

functorial in X and K, such that for every algebraic cycle Z of X of codimension p,∫X an

cltrop(Z) ∧ ω =∫Zan

ω(1.1)

for every d′′-closed form ω ∈ A n−p,n−pX an (X an) with compact support.

In particular, if X is proper and Z is of dimension 0, then∫X an

cltrop(Z) = degZ.

1In this article, we assume that all K-analytic spaces are good, paracompact, Hausdorff and strictlyK-analytic. See §1.5.

TROPICAL CYCLE CLASSES AND WEIGHT DECOMPOSITION 3

The above theorem has the following corollary.

Corollary 1.2 (Corollary 3.11). Let K be a non-Archimedean field and X a proper smoothscheme over K. Let NSp(X ) be the quotient of CHp(X ) modulo numerical equivalence. Thenwe have

dim Hp,ptrop(X ) ≥ dim NSp(X )⊗Q

for every p ≥ 0.

Remark 1.3. Let the situation be as in Theorem 1.1.(1) The tropical cycle class respects products on both sides. More precisely, for Zi ∈

CHpi(X ) with i = 1, 2, we havecltrop(Z1 · Z2) = cltrop(Z1) ∧ cltrop(Z2),

where we have used the natural pairing∧ : Hp1,q1

trop (X )× Hp2,q2trop (X )→ Hp1+p2,q1+q2

trop (X ).(2) We may regard the formula (1.1) as a tropical version of the Cauchy formula in

multi-variable complex analysis.(3) Based on this theorem, we will give a counterexample of the Künneth decomposition

for the cohomology theory H•,•trop in Example 3.12.

1.2. Weight decomposition. Suppose that K is of characteristic zero. We have the fol-lowing complex of cX-modules in either analytic or étale topology:

Ω•X : OX = Ω0X

d−→ Ω1X

d−→ Ω2X

d−→ · · · ,

known as the de Rham complex, where cX := ker(d: OX → Ω1X) is the sheaf of constants.

It is not exact from the term Ω1X if dim(X) ≥ 1. The cohomology sheaves of the de Rham

complex Ωp,clX /dΩp−1

X are called de Rham cohomology sheaves.For p ≥ 0, denote by Υp

X the subsheaf of Ωp,clX /dΩp−1

X generated by sections of the form∑α

cαdfα1

fα1∧ · · · ∧ dfαp

fαp

where the sum is finite, cα are sections of cX , and fαi are sections of O∗X . In particular, wehave Υ0

X = cX , and that Υ1X is simply the sheaf ΥX defined in [Ber07, §4.3] in the case of

étale topology.

Theorem 1.4. Suppose that K is embeddable2 into CF. Let X be a smooth K-analyticspace. Then for every p ≥ 0, we have a decomposition

Ωp,clX /dΩp−1

X =⊕w∈Z

(Ωp,clX /dΩp−1

X )w

of cX-modules in either analytic or étale topology. It satisfies the following:(i) (Ωp,cl

X /dΩp−1X )w = 0 unless p ≤ w ≤ 2p;

(ii) ΥpX ⊂ (Ωp,cl

X /dΩp−1X )2p, and they are equal if p = 1;

(iii) (Ω1,clX /dOX)1 coincides with the sheaf ΨX defined in [Ber07, §4.5] in the case of étale

topology.Such decomposition is stable under base change, cup product, and functorial in X.

2However, we do not choose any such embedding.

4 YIFENG LIU

The proof of this theorem will be given at the end of Section 5. We call the decompositionin the above theorem the weight decomposition of de Rham cohomology sheaves.Corollary 1.5. Suppose that K is embeddable into CF. Then for every smooth K-analyticspace X, we have Ω1,cl

X /dOX = ΥX ⊕ ΨX in étale topology. This answers the question in[Ber07, Remark 4.5.5] for such K.

Remark 1.6. We expect that Theorem 1.4 and thus Corollary 1.5 hold by only requiringthat the residue field of K is algebraic over a finite field and K is of characteristic zero.In Theorem 1.4 (ii), it is probably not true that the inclusion Υp

X ⊂ (Ωp,clX /dΩp−1

X )2p is anisomorphism when p ≥ 2.1.3. Connection between tropical and de Rham theories. The following theoremreveals a connection between ker(d′′ : A p,0

X → A p,1X ) and the algebraic de Rham cohomology

sheaves of X (in the analytic topology).Theorem 1.7 (Proposition 6.2, Theorem 7.1). Let K be a non-Archimedean field embeddableinto CF, and X a smooth K-analytic space. Let L p

X be the subsheaf on X of Q-vector spacesof Ωp,cl

X /dΩp−1X generated by sections of the form df1

f1∧ · · ·∧ dfp

fpwhere fi’s are sections of O∗X .

Then(1) the canonical map L p

X ⊗Q cX → ΥpX is an isomorphism of sheaves on X;

(2) there is a canonical isomorphism L pX ' T p

X of sheaves on X.

For a proper smooth scheme X of dimension n over a general field K of characteristiczero, we have a cycle class map cldR : CHp(X ) → H2p

dR(X/K) with values in the algebraicde Rham cohomology. It is known that when K = C, the kernel of cldR coincides with thekernel of the cycle class map valued in Dolbeault cohomology. In particular, if cldR(Z) = 0,then

∫Zan ω = 0 for every ∂-closed (n − p, n − p)-form ω on X an. In the next theorem, we

prove that the same conclusion holds in the non-Archimedean setting as well, with mildrestriction on the field K.Theorem 1.8. Let k ⊂ CF be a finite extension of QF and X a proper smooth scheme overk of dimension n. Let Z be an algebraic cycle of X of codimension p such that cldR(Z) = 0.If we put Xa = X ⊗k CF and Za = Z ⊗k CF, then∫

Zana

ω = 0

for every d′′-closed form ω ∈ A n−p,n−pX an

a(X an

a ).

We remark that in the above theorem, we do not know whether cltrop(Z) = 0 or not. Ifwe know the Poincaré duality for H•,•(X an), then cltrop(Z) = 0. Nevertheless, we have thefollowing result for lower degree.Theorem 1.9. (Theorem 7.3) Let X be a proper smooth scheme over CF. Then

(1) H1,1trop(X ) is of finite dimension.

(2) For a line bundle L on X such that cldR(L) = 0, we have cltrop(L) = 0.To the best of our knowledge, the first conclusion in the above theorem is the only known

case of the finiteness of dim Hp,qtrop(X ) when both p, q are positive and X is of general dimen-

sion. Note that in the above theorem, we do not require that X can be defined over a finiteextension of QF.


Remark 1.10. We can interpret Theorem 1.8 in the following way. Let k be a number field.Let X be a proper smooth scheme over k of dimension n, and Z an algebraic cycle of X ofcodimension p. Suppose that there exists one embedding ι∞ : k → C such that∫

(Z⊗k,ι∞C)anω = 0

for every ∂-closed (n− p, n− p)-form ω on (X ⊗k,ι∞ C)an. Then for every finite field F andevery embedding ιF : k → CF, we have∫

(Z⊗k,ιF CF)anω = 0

for every d′′-closed (n− p, n− p)-form ω on (X ⊗k,ιF CF)an.

1.4. Structure of the article. We start in §2 by reviewing the relation between Chowcohomology and sheaves of Milnor K-theory, in order to define tropical cycle class mapslater.

In §3, we introduce the Dolbeault cohomology for non-Archimedean analytic spaces andconstruct the tropical cycle class map for smooth varieties over non-Archimedean fields.We show an integration formula for tropical cycle classes in Theorem 3.7, which can beregarded as a tropical version of the Cauchy formula in multi-variable complex analysis. Asa corollary, we give a lower bound for the dimension of certain Dolbeault cohomology interms of algebraic cycles modulo numerical equivalence.

In §4, we review the theory of rigid cohomology, and compute some logarithmic differentialforms on strictly semistable schemes in terms of rigid cohomology. These will be used later.

In §5, we prove Theorem 1.4. We work mainly in the étale topology, by using de Jong’salteration and the theory of rigid cohomology reviewed in the previous section. In particular,our terminology “weight decomposition” comes from the notion of weights in rigid cohomol-ogy. The theorem for analytic topology will be deduced from the one for étale topology notdifficultly.

In §6, we construct the so-called log-differential cycle class map, through Q-subsheaves ofde Rham cohomology sheaves spanned by logarithmic differential forms. The main result weshow is Theorem 6.6, an analogue of Theorem 1.8 in this context.

In the last section §7, we first reveal a connection between the tropical theory and thealgebraic de Rham theory in Theorem 7.1. In particular, this implies that the tropical cycleclass map and the log-differential cycle class map are essentially the same one. Based onthis, we deduce Theorems 1.8 & 1.9.

1.5. Conventions and Notation.

General conventions and notation:• For a product · · · ×W × · · · in a suitable category, we denote by

prW : · · · ×W × · · · → W

the canonical projection morphism to the factor W .• If X is a scheme over an affine scheme SpecA and B is an A-algebra, then weput XB = X ×SpecA SpecB. Such abbreviation will be applied only to schemes,neither formal schemes nor analytic spaces. If X is a scheme over SpecK for anon-Archimedean field K, then we write Xs for XK .

6 YIFENG LIU

• Let X be a site. Whenever we have a suitable notion of de Rham complex (Ω•X , d)on X, we denote by H•dR(X) := H•(X,Ω•X) the corresponding de Rham cohomologyof X, as the hypercohomology of the de Rham complex.• Let X be a site and K a complex of sheaves on X. We denote by H•(X,K ) thesheaf on X associated to the presheaf U 7→ H•(U,KU) := h•RΓ(U,KU).

Conventions and notation for non-Archimedean geometry:• Throughout the article, by a non-Archimedean field, we mean a complete topologicalfield whose topology is induced by a nontrivial non-Archimedean valuation | | of rank1. If the valuation is discrete, then we say that it is a discrete non-Archimedeanfield by abuse of terminology. In the main text, discrete non-Archimedean fields areusually denoted by lower-case letters like k, k′, etc. In addition, $ will always be auniformizer of a discrete non-Archimedean field, though we will still remind readersof this.• Throughout the article, we fix a finite field F. Denote by ZF the ring of Witt vectorsin F and QF the field of fractions of ZF. We fix a complete algebraic closure CF ofQF, both being non-Archimedean fields.• For a non-Archimedean field K with valuation | |, put

K = x ∈ K | |x| ≤ 1, K = x ∈ K | |x| < 1, K = K/K.

Denote by Ka an algebraic closure of K and Ka its completion. A residually algebraicextension ofK is an extensionK ′/K of non-Archimedean fields such that the inducedextension K ′/K is algebraic.• Let K be a non-Archimedean field. All K-analytic (Berkovich) spaces are assumedto be good [Ber93, 1.2.16], paracompact, Hausdorff and strictly K-analytic [Ber93,1.2.15].• Let K be a non-Archimedean field, and A an affinoid K-algebra. We then havethe K-analytic spaceM(A). Denote by A the subring of power-bounded elements,which is a K-algebra. Put As = A ⊗K K. We say that A is integrally smooth if Ais strictly K-affinoid and Spf A is a smooth formal K-scheme.• Let K be a non-Archimedean field. For a real number r > 0, we denote by D(0; r)Kthe open disc over K with center at zero of radius r as a K-analytic space. For realnumbers R > r > 0, we denote by B(0; r, R)K the open annulus over K with centerat zero of inner radius r and outer radius R as a K-analytic space. An open poly-disc(of dimension n) over K is the product of finitely many open discs D(0; ri)K (ofnumber n) over K.• Suppose that K ′/K is an extension of non-Archimedean fields. For a K-analyticspace X and a K ′-analytic space Y , we put

X⊗KK ′ = X ×M(K)M(K ′), Y ×K X = Y ×M(K′) (X⊗KK ′).For a formal K-scheme X and a formal K ′-scheme Y, we put

X⊗KK ′ = X×Spf K Spf K ′, Y×K X = Y×Spf K′ (X⊗KK ′).• If k is a discrete non-Archimedean field and X is a special formal k-scheme in thesense of [Ber96], then we have the notion Xη, the generic fiber of X, which is a k-analytic space; and Xs, the closed fiber of X, which is a scheme locally of finite typeover k; and a reduction map π : Xη → Xs. For a general non-Archimedean field K, we


say that a formal K-scheme X is special if there exist a discrete non-Archimedeanfield k ⊂ K and a special formal k-scheme X′ such that X ' X′⊗kK. For a specialformal K-scheme, we have similar notion π : Xη → Xs which is canonically defined.In this article, all formal K-schemes will be special. Note that if Z is a subschemeof Xs, then π−1Z is usually denoted as ]Z[Xη in rigid analytic geometry.• Let K be a non-Archimedean field and X a K-analytic space. For a point x ∈ X,one may associate nonnegative integers sK(x), tK(x) as in [Ber90, §9.1]. For readers’convenience, we recall the definition. The number sK(x) is equal to the transcendencedegree of H(x) over K, and the number tK(x) is equal to to the dimension of theQ-vector space

√|H(x)∗|/

√|K∗|, where H(x) is the completed residue field of x.

In the text, the field K will always be clear so will be suppressed in the notationsK(x), tK(x).

Acknowledgements. The author is partially supported by NSF grant DMS–1602149. Hethanks Walter Gubler, Phillipp Jell, Klaus Künnemann, and Weizhe Zheng for helpful discus-sions. He also thanks the anonymous referee for very careful reading and numerous helpfulcomments on the article.

2. Chow cohomology revisited

In this section, we study Chow cohomology of a smooth scheme via sheaves of MilnorK-theory.

Definition 2.1 (Sheaf of rational Milnor K-theory). Let (X,OX) be a ringed site. Wedefine the p-th sheaf of rational Milnor K-theory K p

X for (X,OX) to be the sheaf associatedto the presheaf assigning an open U in X to KM

p (OX(U)) ⊗Q ([Sou85, §6.1]). Here for a(commutative) ring R and an integer p ≥ 1, KM

p (R) is the abelian group generated by thesymbols f1, . . . , fp with f1, . . . , fp ∈ R∗, modulo the two relations

(1) f1, . . . , fif′i , . . . , fp = f1, . . . , fi, . . . , fp+ f1, . . . , f

′i , . . . , fp,

(2) f1, . . . , f, . . . , 1− f, . . . , fp = 0.We also adopt the convention that KM

0 (R) = Z and KMp (R) = 0 if p < 0.

Let k be a field. Let X be a smooth scheme of finite type over k. For every integer p ≥ 0,denote by X(p) ⊂ X the subset of points of codimension p. For x ∈ X, denote by k(x) theresidue field at x and ix : Spec k(x) → X the induced morphism. By [Sou85, Théorème 5],we have a functorial exact sequence

0→ K pX → K p,0

Xd−→ K p,1

Xd−→ K p,2

Xd−→ · · ·(2.1)

of Q-sheaves on X (in the Zariski topology) for every p ≥ 0, whereK p,qX =

⊕x∈X(i)

ix∗KMp−q(k(x))⊗Q.

The differential map d: K p,qX → K p,q+1

X in (2.1) is given by residue homomorphisms (see forexample [Ros96, §1]).

The sheaf K p,qX is obviously flasque. Therefore, we have a canonical isomorphism

Hq(X,K pX ) ∼=

ker(d: K p,qX (X)→ K p,q+1

X (X))im(d: K p,q−1

X (X)→ K p,qX (X))

.(2.2)

8 YIFENG LIU

In particular, we obtain the following.

Lemma 2.2 ([Sou85]). Let X be a smooth scheme of finite type over a field k. Then(1) For every p ≥ 0, we have a canonical isomorphism

Hp(X,K pX ) ' CHp(X)Q := CHp(X)⊗Q.

(2) For q > p ≥ 0, we have Hq(X,K pX ) = 0.

(3) For an irreducible closed subset Z of X of codimension p, we haveHq(X, iZ!i

!ZK p

X ) = 0if q 6= p, and

Hp(X, iZ!i!ZK p

X ) ∼= Q,where iZ : Z → X is the closed embedding.

Proof. Both (1) and (2) are direct consequences of (2.2). For (3), let U := X \ Z be thecomplement. Then we have the Gysin exact sequence

· · · → Hq(X, iZ!i!ZK p

X )→ Hq(X,K pX )→ Hq(U,K p

U )→ Hq+1(X, iZ!i!ZK p

X )→ · · ·as K p

X |U ∼= K pU . By (2.2), we know that the restriction map Hq(X,K p

X ) → Hq(U,K pU ) is

an isomorphism if q 6∈ p− 1, p, and is surjective if q = p. Thus, Hq(X, iZ!i!ZK p

X ) if q 6= p.When q = p, we have two cases:

• if Z is nontrivial in CHp(X)⊗Q, then we have canonical isomorphismsHp(X, iZ!i

!ZK p

X ) ∼= ker(Hp(X,K pX )→ Hp(U,K p

U )) ∼= KM0 (k(Z))⊗Q ∼= Q;

• if Z is trivial in CHp(X)⊗Q, then we have canonical isomorphismsHp(X, iZ!i

!ZK p

X ) ∼= coker(Hp−1(X,K pX )→ Hp−1(U,K p

U )) ∼= KM0 (k(Z))⊗Q ∼= Q.

The lemma is proved.

In what follows, we will denote bycluniv : CHp(X)Q → Hp(X,K p

X )the map obtained in Lemma 2.2 (1).

Let Z be a smooth closed subscheme of X of codimension p. For later use, we providean explicit description of cluniv(Z) as follows: Choose a finite affine open covering Uα ofX and a regular sequence fα1, . . . , fαp ∈ OX(Uα) such that Z ∩ Uα is defined by the ideal(fα1, . . . , fαp). Let Uαi be the nonvanishing locus of fαi. Then Uαi | 1 ≤ i ≤ p forms anopen covering of Uα\Z. Thus the element fα1, . . . , fαp ∈ KM

p (OX(⋂pi=1 Uαi)) gives rise toan element in Hp−1(Uα\Z,K p

X ) and hence an element c(Z)α in Hp(Uα, iZ!i!ZK p

X ). We havethe following lemma.

Lemma 2.3. Let the notation be as above. Assume that Z∩Uα is nonempty. Then under theisomorphism in Lemma 2.2 (3), the class c(Z)α ∈ Hp(Uα, iZ!i

!ZK p

X ) equals 1. In particular,it does not depend on the choice of fα1, . . . , fαp.

Proof. We may assume that Z∩Uα is irreducible for simplicity. To ease notation, we suppressthe subscript α in the proof.

We first construct a Dolbeault representative θ ∈ K p,p−1U (U\Z) of the (alternative) closed

Čech cocycle f1, . . . , fp ∈ K pU (⋂pi=1 Ui), where we have used the ordered covering U :=


U1, . . . , Up of U\Z. For I ⊂ 1, . . . , p, put UI = ⋂i∈I Ui. We inductively construct

elements θi ∈ Hp−i−1(U\Z,K p,iU ) represented by an (alternative) closed Čech cocycle

θi = θi,I ∈ K p,iU (UI) | |I| = p− i

for i = 0, . . . , p− 1. The class θ0 is simply

θ0,1,...,p = f1, . . . , fp ∈ K p,0U (U1,...,p).

Suppose that we have θi−1 for some 1 ≤ i ≤ p−1. As K p,i−1U is acyclic, the Čech cohomology

Hp−i(U,K p,i−1U ) is trivial. Thus there exists ϑi = ϑi,J ∈ K p,i−1

U (UJ) | |J | = p − i withδUϑi = θi−1, where δU denotes the Čech differential for the covering U . Now we set θi =dϑi := dϑi,J ∈ K p,i

U (UJ) | |J | = p − i. The last closed Čech cocycle θp−1 = θp−1,i ∈K p,p−1U (Ui) | i = 1, . . . , p can be glued to a Dolbeault representative θ ∈ K p,p−1

U (U\Z) weare looking for.

In practice, we may take

ϑ1,J =

f1, . . . , fp ∈ KMp (k(η))⊗Q ⊂ K p,0

U (UJ), J = 2, . . . , p,0, J 6= 2, . . . , p,

where η is the generic point of U2,...,p. Since f1, . . . , fp is a regular sequence, the intersectionf1 = 0 ∩ U2,...,p if of pure codimension 1. Denote by Γ1 the set of generic points off1 = 0 ∩U2,...,p, which is a finite subset of U (1)

2,...,p. Then we have a function µ1 : Γ1 → Zgiven by the vanishing order of f1. For j > 1, let f (1)

j be the residue of fj to the set Γ1.Thus, we have

θ1,I =

µ1 · f (1)2 , . . . , f (1)

p ∈ K p,1U (UI), I = 2, . . . , p,

0, I 6= 2, . . . , p.

By induction, we have

θi,I =

µi · f(i)i+1, . . . , f

(i)p ∈ K p,i

U (UI), I = i+ 1, . . . , p,0, I 6= i+ 1, . . . , p.

for i = 2, . . . , p − 1. We explain the notation inductively: Γi is the set of generic points off1 = · · · = fi = 0 ∩ Ui+1,...,p, which is a finite subset of U (i)

i+1,...,p; µi : Γi → Z is givenby the vanishing order of f (i−1)

i ; and f (i)j is the residue of f (i−1)

j to the set Γi for j > i. Inparticular, we may glue θp−1,1, . . . , θp−1,p together to give an element θ ∈ K p,p−1

U (U\Z).Then the lemma follows as f1 = · · · = fp = 0 coincides with Z ∩ U , and µp−1 · f (p)

p hasvanishing order 1 along it.

By the above lemma, we can glue the collection c(Z)α hence obtain an element c(Z) ∈H0(X,Hp(X, iZ!i

!ZK p

X )). By Lemma 2.2 (3) and the local to global spectral sequence, weobtain a canonical isomorphism H0(X,Hp(X, iZ!i

!ZK p

X )) ∼= Hp(X, iZ!i!ZK p

X ).

Lemma 2.4. Let the notation be as above. The image of the class c(Z) under the mapHp(X, iZ!i

!ZK p

X )→ Hp(X,K pX ) coincides with cluniv(Z).

Proof. The follows directly from Lemma 2.3 and Lemma 2.2.

10 YIFENG LIU

Remark 2.5. Suppose that k is of characteristic zero and X is a smooth scheme of finite typeover k. We have the algebraic de Rham complex

Ω•X : OX = Ω0X

d−→ Ω1X

d−→ Ω2X

d−→ · · · ,

and the algebraic de Rham cohomology HidR(X/k) := Hi(X,Ω•X). Moreover, we have the

algebraic de Rham cycle class mapcldR : CHp(X)Q → H2p

dR(X/k).One can construct cldR via cluniv as follows: we have a (Q-linear) map of complexes

K pX [−p]→ Ω•X

sending a section f1, . . . , fp ∈ K pX (U) to

df1

f1∧ · · · ∧ dfp

fp∈ Ωp,cl

X (U).

It induces the mapHp(X,K p) = H2p(X,K p

X [−p])→ H2p(X,Ω•X) = H2pdR(X/k).

Then cldR is simply the composition of cluniv with the above map.

3. Tropical cycle class map

In this section, we introduce the Dolbeault cohomology for non-Archimedean analyt-ic spaces, define the tropical cycle class map for separated smooth schemes over a non-Archimedean field, and study some basic properties.

First we recall some facts from the theory of real forms on non-Archimedean analyticspaces developed by Chambert-Loir and Ducros in [CLD12]. (See also [Gub13] for a slightlydifferent formulation.) Let K be a non-Archimedean field, and X a K-analytic space. Thereis a bicomplex (A •,•

X , d′, d′′) of sheaves of real vector spaces on (the underlying topologicalspace of) X, where A p,q

X is the sheaf of (p, q)-forms ([CLD12, §3.1]). It is known thatA p,qX = 0 unless 0 ≤ p, q ≤ dim(X). We have an integration functor∫

X: A n,n

X (X)c → R

for n = dim(X) and A n,nX (X)c denotes the set of (n, n)-forms with compact support.

Definition 3.1 (Dolbeault cohomology). Let X be a K-analytic space. We define theDolbeault cohomology of X to be

Hp,q(X) := ker(d′′ : A p,qX (X)→ A p,q+1

X (X))im(d′′ : A p,q−1

X (X)→ A p,qX (X))

.

By [Jel16, Corollary 4.6] and [CLD12, Corollaire 3.3.7], the complex (A p,•X , d′′) is a fine

resolution3 of ker(d′′ : A p,0X → A p,1

X ) for every p ≥ 0. In particular, we have a canonicalisomorphism

H•(X, ker(d′′ : A p,0X → A p,1

X )) ' Hp,•(X).On the other hand, we have the sheaf of rational Milnor K-theory K •

X for the ringedtopological space (X,OX) from Definition 2.1.

3Recall that we have assumed that all K-analytic spaces are good and paracompact in §1.5.


Definition 3.2. Let X be a K-analytic space. For every p ≥ 0, we define a (Q-linear) mapof sheaves

τ pX : K pX → ker(d′′ : A p,0

X → A p,1X )

as follows. For a symbol f1, . . . , fp ∈ K pX (U) with f1, . . . , fp ∈ O∗X(U), we have the

induced moment morphism (f1, . . . , fp) : U → (Ganm,K)p. Composing with the evaluation

map − log | | : (Ganm,K)p → Rp, we obtain a continuous map

tropf1,...,fp : U → Rp.

If we endow the target with coordinates x1, . . . , xp where xi = − log |fi|, then we define

τ pX(f1, . . . , fp) = d′x1 ∧ · · · ∧ d′xp ∈ ker(d′′ : A p,0X (U)→ A p,1

X (U)).It is easy to see that τ pX factors through the relations of Milnor K-theories, and thus inducesa map of corresponding sheaves. See also Remark 3.3. Finally, put

T pX = K p

X/ ker τ pX .It is canonically a subsheaf of ker(d′′ : A p,0

X → A p,1X ).

Remark 3.3. In Definition 3.2, the factorization of the first Milnor relation is due to theadditivity of differential forms; and the factorization of the second Milnor relation is due tothe fact that the image tropf1,...,fp(U) has dimension at most p− 1.

In fact, the factorization of Milnor relations is not relevant in Definition 3.2. We includethis observation in the definition simply for the preparation of defining cycle class maps inDefinition 3.6 later.

Proposition 3.4. Let K be a non-Archimedean field and X a K-analytic space. Then τ pXinduces an isomorphism

T pX ⊗Q R ' ker(d′′ : A p,0

X → A p,1X ).

Proof. It suffices to show that the induced mapτx : T p

X,x ⊗Q R → ker(d′′ : A p,0X,x → A p,1

X,x)(3.1)on stalks is an isomorphism for every point x ∈ X.

We fix a point x ∈ X. For every open neighborhood U of x in X and f1, . . . , fN ∈ O∗X(U),we have the induced tropicalization map

tropU : U f1,...,fN−−−−→ (Ganm,K)N − log | |−−−−→ TN ⊗Z R ' RN

where TN is the cocharacter lattice of GNm. It is proved by Berkovich that there is an integral

polyhedral complex4 CU such that tropU(U) is an open subset of CU . We may shrink U sothat one can choose CU such that it contains only one minimal polyhedron, denoted by σU ,and σU contains tropU(x). Such data (U ; f1, . . . , fN) will be called a basic chart at x. Let τbe an arbitrary polyhedron of CU under a basic chart. Then σU is a face of τ , and we denoteby L(τ) the underlying Q-linear subspace of TN,Q := TN ⊗Z Q of τ . We have an inclusion∑

τ∈CU

p∧L(τ) ⊂

p∧TN,Q

4In this article, a polyhedron in TN ⊗Z R is a subset defined by finitely many non-strict inequalities. Apolyhedral complex in TN ⊗Z R is integral if it consists of polyhedra with slopes in TN .

12 YIFENG LIU

of Q-vector spaces, and thus a surjective map

rU : HomQ

( p∧TN,Q,R

)→ HomQ

∑τ∈CU

p∧L(τ),R

.By [JSS15, Proposition 3.20], the canonical map

τU : HomQ

( p∧TN,Q,R

)→ ker

(d′′ : A p,0

CU (tropU(U))→ A p,1CU (tropU(U))

)factors through rU and induces an isomorphism

HomQ

∑τ∈CU

p∧L(τ),R

∼−→ ker(d′′ : A p,0


).

Here, A p,qCU is the sheaf of (p, q)-forms on CU (see [JSS15] for more details).

On the other hand, we have a Q-linear subspace T pX(U)f1,...,fN of T p

X(U) generated byτ pX(fi1 , . . . , fip) where fi1 , . . . , fip is a subset of f1, . . . , fN. Then the map

τ pX : T pX(U)f1,...,fN → ker

(d′′ : A p,0


)factors through HomQ

(∑τ∈CU

∧p L(τ),R)and induces isomorphisms

T pX(U)f1,...,fN

∼−→ HomQ

∑τ∈CU

p∧L(τ),Q

hence

T pX(U)f1,...,fN ⊗Q R ∼−→ ker

(d′′ : A p,0


).

Taking colimit over all basic charts at x, we conclude that τx (3.1) is an isomorphism. Theproposition follows.

Proposition 3.4 has the following corollary, which provide the Dolbeault cohomologyHp,q(X) a canonical rational structure.Corollary 3.5. Let K be a non-Archimedean field and X a K-analytic space. Then for allp, q ≥ 0, we have a canonical isomorphism

Hq(X,T pX)⊗Q R ' Hp,q(X).

In particular, the real vector space Hp,q(X) has a canonical rational structure.In what follows, we will always regard Hq(X,T p

X) as a subspace of Hp,q(X). Now we areready to define the tropical cycle class map.Definition 3.6 (Tropical Dolbeault cohomology and tropical cycle class map). Let K be anon-Archimedean field and X a separated scheme of finite type over K.

(1) For p, q ≥ 0, we define the tropical Dolbeault cohomology of X to beHp,q

trop(X ) := Hq(X an,T pX an).

It is a rational subspace of Hp,q(X an).(2) If X is moreover smooth, then we define the tropical cycle class map cltrop to be the

composition

CHp(X )Qcluniv−−−→ Hp(X ,K p

X )→ Hp(X an,K pX an)

Hp(X an,τpXan )−−−−−−−−→ Hp(X an,T p

X an) = Hp,ptrop(X ).


The following theorem can be regarded as a tropical version of the Cauchy formula inmulti-variable complex analysis.

Theorem 3.7. Let K be a non-Archimedean field and X a separated smooth scheme of finitetype over K of dimension n. Then for every algebraic cycle Z of X of codimension p, wehave the equality ∫

X ancltrop(Z) ∧ ω =

∫Zan

ω

for every d′′-closed form ω ∈ A n−p,n−pX an (X an)c with compact support. Here, if we write Z =∑

i aiZi where Zi’s are prime, then we define∫Zan

ω :=∑i

ai

∫Zani

ω.

Proof. By linearity, we may assume that Z is prime, that is, a reduced irreducible closedsubscheme of X of codimension p. Let Zsing ⊂ Z be the singular locus, which is a closedsubscheme of X of codimension > p. Put U = X\Zsing, Zsm = Z\Zsing, X = X an, U = Uan,and Z = Zan

sm. In particular, iZ : Z → U is a Zariski closed subset.In [CLD12], we have the bicomplex (D•,•U , d′, d′′) of currents and a map A •,•

U → D•,•U ofbicomplexes. In particular, we have a map T p

U → Dp,•U of complexes for every p ≥ 0 hence

the induced mapHp(U,T p

U )→ Hp(U,Dp,•U ).

Denote by cl′trop : CHp(U)Q → Hp(U,Dp,•U ) the composition of cltrop with the above map.

To ease notation, put

A p,q,cl? = ker(d′′ : A p,q

? → A p,q+1? ), Dp,q,cl

? = ker(d′′ : Dp,q? → Dp,q+1

? ).

We fix a form ω ∈ A n−p,n−p,clX (X)c. By [CLD12, Lemme 3.2.5], ω belongs to A n−p,n−p,cl

X (U)c.The proof consists of two parts. We first reduce the theorem to an explicit formula (3.2)

of local integration. Then we establish it.Step 1. We describe explicitly the class cl′trop(Zsm) ∈ Hp(U,Dp,•

U ) in terms of local co-homology. We choose a finite affine open covering Uα of U and fα1, . . . , fαp ∈ OU(Uα)such that Zsm ∩ Uα is defined by the ideal (fα1, . . . , fαp) such that the induced morphism(fα1, . . . , fαp) : Uα → Ap

K is smooth. Let Uαi be the nonvanishing locus of fαi. Put Uα = Uanα

and Uαi = Uanαi . Then Uαi | i = 1, . . . , p is an open covering of Uα\Z. Thus the element

τ pU(fα1, . . . , fαp) gives rise to an element in Hp−1(Uα\Z,A p,0,clU ) ' Hp−1(Uα\Z,A p,•

U ), andwe denote its image under the composite map

Hp−1(Uα\Z,A p,•U )→ Hp−1(Uα\Z,Dp,•

U ) δ′′−→ Hp(Uα, iZ!i!ZDp,•

U )

by c(Z)α, where δ′′ is the coboundary map in the corresponding Gysin exact sequence.By Lemma 2.3, c(Z)α does not depend on the choice of fα1, . . . , fαp. Therefore, c(Z)αgives rise to an element c(Z) ∈ H0(U,Hp(U, iZ!i

!ZDp,•

U )). By Lemma 3.8 below, we knowthat Hq(U, iZ!i

!ZDp,•

U ) = 0 for q < p, hence we have an isomorphism Hp(U, iZ!i!ZDp,•

U ) 'H0(U,Hp(U, iZ!i

!ZDp,•

U )) obtained from the local to global spectral sequence. By Lemma 2.4and construction, the image of c(Z) in Hp(U,Dp,•

U ) coincides with cl′trop(Zsm).Step 2. By Lemma 3.8 below, we have a canonical isomorphism

Hp(Uα, iZ!i!ZDp,•

U ) ' ker(Dp,p,clU (Uα)→ Dp,p

U (Uα\Z)) ⊂ Dp,p,clU (Uα).

14 YIFENG LIU

Let θα ∈ A p,p−1,clU (Uα\Z) be a Dolbeault representative of τ pU(fα1, . . . , fαp) as a cohomology

class in Hp−1(Uα\Z,A p,0,clU ), with induced class [θα] ∈ Hp−1(Uα\Z,Dp,•

U ). By partition ofunity, we may write ω = ∑

α ωα with ωα ∈ A n−p,n−pU (Uα)c. Note that∫

Xcltrop(Z) ∧ ω = 〈cl′trop(Zsm), ω〉U =

∑α

〈cl′trop(Zsm), ωα〉U =∑α

〈δ′′([θα]), ωα〉Uα

and ∫Zan

ω =∑α

∫Uα∩Z

ωα.

To prove the theorem, it suffices to show that

〈δ′′([θα]), ωα〉Uα =∫Uα∩Z

ωα

for every α. Here, 〈 , 〉 denotes the pairing between currents and forms.Step 3. In what follows, we suppress the subscript α. We summarize our data as follows:• an affine smooth scheme U over K of dimension n; put U = Uan,• a smooth irreducible closed subscheme Z of codimension p defined by theideal (f1, . . . , fp) where f1, . . . , fp ∈ OU(U) such that the induced morphism(f1, . . . , fp) : U → Ap

K is smooth; put Z = Zan,• θ ∈ A p,p−1,cl

U (U\Z) a Dolbeault representative of τ pU(f1, . . . , fp) as a cohomologyclass in Hp−1(U\Z,A p,•

U ), and• ω ∈ A n−p,n−p

U (U)c.Our goal is to show that

〈δ′′([θ]), ω〉U =∫Zω.(3.2)

Here we recall that [θ] ∈ Hp−1(U\Z,Dp,•U ) is the class induced by θ, and

δ′′ : Hp−1(U\Z,Dp,•U )→ Hp(U, iZ!i

!ZDp,•

U )is the coboundary map, in which the target Hp(Uα, iZ!i

!ZDp,•

U ) is canonically a subspace ofDp,p,clU (U). As Z is a closed Zariski subset of U of codimension p, the image of A n−p,n−p

U (U)cunder d′′ is in A n−p,n−p+1,cl

U (U\Z)c. By definition, the following diagram

Hp−1(U\Z,Dp,•U )

δ′′

× A n−p,n−p+1,clU (U\Z)c

〈 , 〉U // R

Dp,p,clU (U) × A n−p,n−p

U (U)c

d′′OO

〈 , 〉U // R

is commutative. Therefore, we have

〈δ′′([θ]), ω〉U =∫U\Z

θ ∧ d′′ω.

Thus it suffices to show that ∫U\Z

θ ∧ d′′ω =∫Zω.

Obviously, the equality does not depend on the choice of the Dolbeault representative.Step 4. Let Ui ⊂ U be the nonvanishing locus of fi. Then we have an open covering

U = Ui of U\Z, where Ui = Uani . For I ⊂ 1, . . . , p, put UI = ⋂

i∈I Ui.


Let us recall the construction of a Dolbeault representative θ. We inductively constructelements θi ∈ Hp−i−1(U\Z,A p,i,cl

U ) represented by an (alternative) closed Čech cocycle

θi = θi,I ∈ A p,i,clU (UI) | |I| = p− i

for i = 0, . . . , p− 1. The class θ0 is simply

θ0,1,...,p = τ pU(f1, . . . , fp) ∈ A p,0,clU (U1,...,p).

Suppose that we have θi−1 for some 1 ≤ i ≤ p−1. As A p,i−1U is acyclic, the Čech cohomology

Hp−i(U,A p,i−1U ) is trivial. Thus there exists ϑi = ϑi,J ∈ A p,i−1

U (UJ) | |J | = p − i withδUϑi = θi−1, where δU denotes the Čech differential for the covering U . Now we set

θi = d′′ϑi := d′′ϑi,J ∈ A p,i,clU (UJ) | |J | = p− i.

The last closed Čech cocycle θp−1 = θp−1,i ∈ A p,p−1,clU (Ui) | i = 1, . . . , p is simply a

Dolbeault representative of τ pU(f1, . . . , fp) ∈ Hp−1(U\Z,A p,•U ).

For ε > 0 and I ⊂ 1, . . . , p, put

V Iε = x ∈ U | fi(x) ∈ ∂D(0, ε), i ∈ I; fj(x) ∈ D(0, ε), j 6∈ I,

and Uε = U\V ∅ε . Here, D(0, ε) is the closed disc of radius ε with center at zero, and U\V ∅ε isthe closure of U\V ∅ε in U . As d′′ω ∈ A n−p,n−p+1

U (U\Z)c, there is a real number ε0 > 0 suchthat d′′ω = 0 on V ∅ε0 . Thus for every 0 < ε < ε0, we have∫

U\Zθ ∧ d′′ω =

∫U\V ∅ε

θ ∧ d′′ω = −∫U\V ∅ε

d′′(θ ∧ ω) = −∫Uε

d′′(θp−1 ∧ ω).(3.3)

Since Uε is a closed subset of U , the forms ω and hence θ ∧ ω have compact support on Uε.Step 5. Now we have to use integration on boundaries V I

ε and the corresponding Stokes’formula, for some fixed 0 < ε < ε0. We use the formulation of boundary integration throughcontraction as in [Gub13, §2]. We consider first a tropical chart

tropW : W → (Ganm,K)N − log | |−−−−→ RN ,

where W is an open subset of Uε. Since V Iε is a KI

ε -analytic space of dimension n − |I|for some extension KI

ε/K of non-Archimedean fields, the codimension of tropW (W ∩ V Iε ) in

tropW (W ) is at least |I|. We denote by σI the union of closed faces in tropW (W ∩ V Iε ) of

codimension exactly |I|. For every i ∈ I, we choose a tangent vector ωi for the closed faceσi of σ∅ of codimension 1, as defined in [Gub13, 2.8].

Suppose that I = m1, . . . ,mj where 1 ≤ m1 ≤ · · · ≤ mj ≤ p. If α is an (n, n − i)-superform on RN with compact support, then we define∫

σIα :=

∫σI〈α;−ωm1 , . . . ,−ωmj〉1,...,j.

It is easy too see that the above integral does not depend on the choice of ωi; however, itdoes depend on the order. We may patch the above integral to define the integral

∫V Iεα for

an (n, n − |I|)-form α on V Iε with compact support. The negative signs for ωi ensure that

we have the following Stokes’ formula∫V Iε

d′′α =∑j 6∈I

(−1)(j,I∪j)∫VI∪jε

α

16 YIFENG LIU

for an (n, n − |I| − 1)-form α on V Iε with compact support, for |I| ≥ 1. Here, (j, J) is the

position from the rear of the index j when J is ordered in the usual manner. However, forthe initial Stokes’ formula, we have∫

Uεd′′α =

∫∂Uε

α = −∑|I|=1

∫V Iε

α

for an (n, n− 1)-form α on Uε with compact support. Here, the inclusion ∂Uε ⊂⋃|I|=1 V

Iε is

obvious. The other inclusion ⋃|I|=1 VIε ⊂ ∂Uε is due to the compatibility of relative interior

under composition [Ber93, 1.5.5 (ii)]. In particular, we have

−∫Uε

d′′(θp−1 ∧ ω) = −∫∂Uε

θp−1 ∧ ω =∑|I|=1

∫V Iε

θp−1,I ∧ ω.(3.4)

In general, for 1 ≤ i ≤ p− 1, we have∑|I|=i

∫V Iε

θp−i,I ∧ ω =∑|I|=i

∫V Iε

d′′ϑp−i,I ∧ ω

=∑|I|=i

∫V Iε

d′′(ϑp−i,I ∧ ω)

=∑|I|=i

∑j 6∈I

(−1)(j,I∪j)∫VI∪jε

ϑp−i,I ∧ ω

=∑|J |=i+1

∫V Jε

∑j∈J

(−1)(j,J)ϑp−i,J\j ∧ ω

=∑|J |=i+1

∫V Jε

(δUϑp−i)J ∧ ω

=∑|J |=i+1

∫V Jε

θp−(i+1),J ∧ ω.

Combining with (3.3), (3.4), we have∫U\Z

θ ∧ d′′ω =∫V1,...,pε

θ0,1,...,p ∧ ω =∫V1,...,pε

τ pU(f1, . . . , fp) ∧ ω(3.5)

for every 0 < ε < ε0.Step 6. By (3.5), the theorem is reduced to the formula∫

V1,...,pε

τ pU(f1, . . . , fp) ∧ ω =∫Zω(3.6)

for sufficiently small ε > 0. Note that ∂D(0, ε) is actually a point, which we denote by η(ε),we have a morphism f1 : U → A1,an

K . Denote by U1 := Uη(ε) the fiber of U over η(ε). Itis a smooth K1-analytic space purely of dimension n − 1 for some non-Archimedean fieldextensionK1/K determined by ε. Then we have the induced map f2 : U1 → A1,an

K1 hence thefiber U1,2 := (U1)η(ε). Inductively, we have U1,...,p, which is a smooth Kp-analytic spacepurely of dimension n − p for some non-Archimedean field extension Kp/K determined byε. The underlying topological space U1,...,p, which is canonically a subspace of U , coincideswith V 1,...,pε (see [Ber93, §1.4]). Moreover, from definition, we have∫

V1,...,pε

τ pU(f1, . . . , fp) ∧ ω =∫U1,...,p

ω.


Then (3.6) follows from successively application of Lemma 3.9.

Lemma 3.8. Let U be a K-analytic space without boundary, iZ : Z → U a Zariski closedsubset of codimension at least p for some p ≥ 0. Then i!ZDp,q

U = 0 for q < p, and we have acanonical isomorphism

Hp(U, iZ!i!ZDp,•

U ) ∼= ker(Dp,pU (U)→ Dp,p+1

U (U)⊕Dp,pU (U\Z)),

where the map in the kernel is induced by the differential d′′ to the first factor and therestriction to the second factor.

Proof. Let j : U\Z → U be the open compliment. Then j∗Dp,qU = Dp,q

U\Z . By the partitionof unity, we know that both Dp,q

U and Dp,iU\Z are acyclic. Thus Rj∗j∗Dp,q

U = j∗j∗Dp,q

U , whichis also acyclic. In particular, we have i!ZDp,q

U = ker(Dp,qU → j∗j

∗Dp,qU ) which is acyclic. By

[CLD12, Lemme 3.2.5], we have i!ZDp,qU = 0 for q < p.

Moreover, we have

Hp(U, iZ!i!ZDp,•

U ) = ker(i!ZDp,pU (U) d′′−→ i!ZDp,p+1

U (U)),which is canonically isomorphic to ker(Dp,p

U (U) → Dp,p+1U (U) ⊕ Dp,p

U (U\Z)). The lemma isproved.

Lemma 3.9. Let X be a smooth K-analytic space purely of dimension n, and f : X → A1,anK

a smooth morphism. For every ε > 0, let η(ε) ∈ A1,anK be the boundary point of the closed

disc D(0, ε). Then for every compactly supported form ω ∈ A n−1,n−1X (X)c, we have for

sufficiently small ε, ∫X0ω =

∫Xη(ε)

ω.

Here, X0 (resp. Xη(ε)) denotes the fiber of X over 0 (resp. η(ε)).

Proof. Since f is smooth, we have ∫X0ω = 〈δdiv(f), ω〉X .(3.7)

By the Poincaré–Lelong formula [CLD12, Théorème 4.6.5], we have

(3.7) = 〈d′d′′ log |f |, ω〉X =∫X

log |f | ∧ d′d′′ω(3.8)

by [CLD12, (4.4.3.3)]. Since d′′ω has compact support on X \X0, there exists ε0 > 0 suchthat d′′ω = 0 on X \Xε0 where Xε := x ∈ X | |f(x)| ≥ ε. Now for every 0 < ε < ε0, wehave

(3.8) =∫Xε

log |f | ∧ d′d′′ω = −∫Xε

d′ log |f | ∧ d′′ω =∫Xε

d′′(d′ log |f | ∧ ω)(3.9)

by the Stokes’ formula. By the Stokes’ formula again, we have

(3.9) =∫∂Xε

d′ log |f | ∧ ω,

which is nothing but ∫Xη(ε)

ω.

The lemma follows.

Theorem 3.7 has following corollaries.

18 YIFENG LIU

Corollary 3.10. Let K be a non-Archimedean field and X a proper smooth scheme over Kof dimension n. Then for every algebraic cycle Z of X of dimension 0, we have∫

X ancltrop(Z) = degZ.

Proof. Since X is proper, we may take ω to be the constant function 1 on X an in Theorem3.10. Then the conclusion follows as ∫

Zan1 = degZ.

Corollary 3.11. Let K be a non-Archimedean field and X a proper smooth scheme over K.Let NSp(X ) be the quotient of CHp(X ) modulo numerical equivalence. Then we have

dim Hp,ptrop(X ) ≥ dim NSp(X )⊗Q

for every p ≥ 0.

Proof. Without lost of generality, we may assume that X has dimension n ≥ p. It sufficesto show that for algebraic cycles Z1, . . . ,Zr on X of codimension p, if they are linear-ly independent in NSp(X ), then they are linearly independent in Hp,p

trop(X ). Suppose thatcltrop(Z1), . . . , cltrop(Zr) are linearly dependent in Hp,p

trop(X ). Then there are c1, . . . , cr ∈ Q,not all zero, such that Z := c1Z1 + · · ·+ crZr is zero in Hp,p

trop(X ), that is, cltrop(Z) = 0. AsZ is nonzero in NSp(X ), there is an algebraic cycle Z ′ on X of codimension n− p such thatdeg(Z · Z ′) 6= 0, where Z · Z ′ ∈ CHn(X ) is the intersection product of Z and Z ′. Since thetropical cycle class map preserves ring structure, we have

cltrop(Z) ∧ cltrop(Z ′) = cltrop(Z · Z ′).Thus, cltrop(Z · Z ′) = 0. However, by Corollary 3.10, we have∫

X ancltrop(Z · Z ′) =

∫(Z·Z′)an

1 = deg(Z · Z ′),

which is a contradiction. The corollary follows.

We finish this section by exhibiting a counterexample of the Künneth decomposition forH•,•trop.

Example 3.12. Let X1,X2 be two separated schemes of finite type over a non-Archimedeanfield K. Put X = X1 ×SpecK X2. We have a canonical map⊕

p1+p2=p,q1+q2=qHp1,q1

trop (X1)⊗Q Hp2,q2trop (X2)→ Hp,q

trop(X ).(3.10)

We say that the Künneth decomposition holds for (X1,X2) if the above map is an isomorphismfor every (p, q).

We now show an example that the Künneth decomposition fails for certain projectivesmooth schemes over K = CF. Take X1 = X2 to be an irreducible projective smoothcurve over CF of genus g ≥ 1 with good reduction. Then we have dim H0,0

trop(Xi) = 1;dim H1,1

trop(Xi) = 1 by Theorem 7.3 (2); dim H0,1trop(Xi) = dim H1,0

trop(Xi) = b1(X ani ) = 0 by

Remark 7.4 and the assumption that Xi has good reduction. Therefore, for (p, q) = (1, 1),the left-hand side of (3.10) has dimension 2. However by Corollary 3.11, the right-hand sideof (3.10) has dimension as least dim NS1(X )⊗Q, which is 3 as g ≥ 1.


4. Rigid cohomology and logarithmic differential forms

In this section, we review the theory of rigid cohomology developed in, for example,[Bert97] and [LS07]. Then we study the behavior of logarithmic differential forms in therigid cohomology.

Let R be the category of triples (K,X,Z) where K is a non-Archimedean field of char-acteristic zero; X is a scheme of finite type over a subfield of K; and Z is a Zariski closedsubset of X. A morphism from (K ′, X ′, Z ′) to (K,X,Z) consists of a field extension K ′/Kand a morphism X ′ → X ⊗

KK ′ whose restriction to Z ′ factors through Z ⊗

KK ′. Let V

be the category of pairs (K,V •) where K is a field and V • is a graded K-vector space. Amorphism from (K,V •) to (K ′, V ′•) consists of a field extension K ′/K and a graded linearmap V • ⊗K K ′ → V ′•.

We have a functor of rigid cohomology with support: Ropp → V sending (K,X,Z) toH•Z,rig(X/K). Put H•rig(X/K) = H•X,rig(X/K) for simplicity. The following lemma summa-rizes properties which will be used extensively in this article

Lemma 4.1. Let notation be as above.(1) Suppose that we have a morphism (K ′, X ′, Z ′)→ (K,X,Z) with X ′ ' X ⊗

KK ′ and

Z ′ ' Z ⊗KK ′. Then the induced map H•Z,rig(X/K) ⊗K K ′ → H•Z′,rig(X ′/K ′) is an

isomorphism of finite dimensional graded K ′-vector spaces.(2) For Y = X\Z, we have a long exact sequence:

· · · → HiZ,rig(X/K)→ Hi

rig(X/K)→ Hirig(Y/K)→ Hi+1

Z,rig(X/K)→ · · · .(4.1)(3) If both X, Z are smooth, and Z is of codimension r in X, then we have a Gysin

isomorphism HiZ,rig(X/K) ' Hi−2r

rig (Z/K).(4) Suppose that K is residually algebraic over QF. Then both X,Z can be defined over a

finite extension F′ of F, and the sequence (4.1) is equipped with the absolute Frobeniusaction of degree |F′| by functoriality. In particular, each item V in (4.1) admits adirect sum decomposition V = ⊕

w∈Z Vw where Vw is the maximal subspace of V suchthat all of its generalized eigenvalues under the previous Frobenius action are Weil|F′|w/2-numbers; it is clear that Vw is independent of the choice of F′.

(5) Suppose that X is smooth and Z is of codimension r, then HiZ,rig(X/K)w = 0 unless

i ≤ w ≤ 2(i− r). If X is moreover proper, then Hi(X/K) is of pure weight i.

Proof. Part (1) is due to [GK02, Corollary 3.8] and [Ber07, Corollary 5.5.2]. Parts (2) is astandard fact in rigid cohomology; see for example, [Bert97] and [LS07]. Part (3) is due to[Tsu99]. Part (4) is due to [Chi98, §1 & §2]. Part (5) is due to [Chi98, Theorem 2.3].

We will extensively use the notion of K-analytic germs ([Ber07, §5.1]), rather than K-dagger spaces, for a non-Archimedean field K. Roughly speaking, a K-analytic germ is apair (X,S) where X is a K-analytic space and S ⊂ X is a subset. We say that (X,S) is aK-affinoid germ if S is an affinoid domain. We say that (X,S) is smooth if X is smooth inan open neighborhood of S. We have the structure sheaf O(X,S), and the de Rham complexΩ•(X,S) when (X,S) is smooth. (See [Ber07, §5.2] for details.) In particular, we have thede Rham cohomology H•dR(X,S) when (X,S) is smooth. For a smooth K-analytic germ(X,S) where S =M(A) for an integrally smooth K-affinoid algebra A, we have a canonicalfunctorial isomorphism H•dR(X,S) ' H•rig(SpecAs/K) (see [Bert97, Proposition 1.10], whoseproof actually works for general K).

20 YIFENG LIU

The following lemma generalizes the construction in [GK02, Lemma 2].

Lemma 4.2. Let K be a non-Archimedean field of characteristic zero. Let (X1, Y1) and(X2, Y2) be two smooth K-affinoid germs. Then for every morphism φ : Y2 → Y1 of K-affinoid domains, one can associate in a functorial way a pullback map

φ∗ : H•dR(X1, Y1)→ H•dR(X2, Y2).It satisfies the following conditions:(i) if φ extends to a morphism (X2, Y2) → (X1, Y1) of germs, then φ∗ coincides with the

usual pullback;(ii) for a finite extension K ′ of K, if we write X ′i (resp. Y ′i ) for Xi⊗KK ′ (resp. Yi⊗KK ′)

for i = 1, 2 and φ′ for φ⊗KK ′, then φ′∗ coincides with the scalar extension of φ∗, inwhich we identify H•dR(X ′i, Y ′i ) with H•dR(Xi, Yi)⊗K K ′ for i = 1, 2;

(iii) if Y1 = M(A1) and Y2 = M(A2) for some integrally smooth K-affinoid algebrasA1 and A2, then φ∗ coincides with φ∗s : H•rig(SpecA1,s/K) → H•rig(SpecA2,s/K) un-der the canonical isomorphism H•dR(Xi, Yi) ' H•rig(SpecAi,s/K) for i = 1, 2, whereφs : SpecA2,s → SpecA1,s is the induced morphism.

Proof. Put X = X1×KX2, Y = Y1×K Y2, and ∆ ⊆ Y the graph of φ, which is isomorphic toY2 via the projection to the second factor. Denote by ai : X → Xi the projection morphism.We have maps

H•dR(X1, Y1) a∗1−→ lim−→V

H•dR(V ) a∗2←− H•dR(X2, Y2),

where V runs through open neighborhoods of ∆ in X. We show that a∗2 is an isomorphism.Then we define φ∗ as (a∗2)−1 a∗1. The functoriality of φ∗ is straightforward but tedious tocheck; we will leave it to readers.

The proof of properties is similar to that of [GK02, Lemma 2]. The claim that a∗2 is anisomorphism is local on (X2, Y2) by the Mayer–Vietoris sequence. Thus we may assume thatthere are elements t1, . . . , tm ∈ OX1(X1) such that dt1, . . . , dtm form a basis of Ω1(X1, Y1)over O(X1, Y1), and there exist a K-affinoid neighborhood Uε ⊂ X of ∆ with an elementε ∈ |K×|, and an isomorphism

Uε ∩ Y∼−→M(K〈ε−1Z1, . . . ε

−1Zm〉)×K ∆,in which Zi is sent to ti ⊗ 1 − 1 ⊗ φ∗ti. Note that K〈ε−1Z1, . . . ε

−1Zm〉 is an integrallysmooth K-affinoid algebra, and SpecK〈ε−1Z1, . . . ε

−1Zm〉s is isomorphic to AmK. Thus by

[GK02, Lemma 2], the restriction map H•dR(X2, Y2)→ H•dR(X,Uε∩Y ) is an isomorphism. Wemay choose a sequence of such Uε with

⋂ε Uε = ∆. Then lim−→ε

H•dR(X,Uε∩Y ) ' lim−→VH•dR(V )

and thus a∗2 is an isomorphism.Properties (i) and (ii) follow easily from the construction. We now check Property (iii),

as it is important for our later argument. The induced projection morphismM(K〈ε−1Z1, . . . ε

−1Zm〉)×K ∆ ' Uε ∩ Y → Yi

extends canonically to a morphism of formal K-schemesSpf(K〈ε−1Z1, . . . ε

−1Zm〉⊗KA∆) → Spf Ai ,where A∆ is the coordinate K-affinoid algebra of ∆ which is isomorphic to A2. Therefore,the restriction map H•dR(Xi, Yi)→ H•dR(X,Uε ∩ Y ) coincides with the map

a∗i,s : H•rig(SpecAi,s/K)→ H•rig(Spec(K〈ε−1Z1, . . . ε−1Zm〉⊗KA∆)s/K)


induced from the homomorphism Ai,s → (K〈ε−1Z1, . . . ε−1Zm〉⊗KA∆)s of K-algebras. Note

that a∗2,s is an isomorphism, and that (a∗2,s)−1 coincides with the restriction map

H•rig(Spec(K〈ε−1Z1, . . . ε−1Zm〉⊗KA∆)s/K)→ H•rig(SpecA2,s/K)

induced from the homomorphism (K〈ε−1Z1, . . . ε−1Zm〉⊗KA∆)s → A2,s sending ε−1Zi to 0

for all i. Property (iii) follows.

The following example will be used in the computation later.

Example 4.3. Let K be a non-Archimedean field of characteristic zero. For an integer t ≥ 0and an element $ ∈ K, define the formal K-scheme

EtK,$ := Spf K[[T0, . . . , Tt]]/(T0 · · ·Tt −$)

and let EtK,$ be its generic fiber. Let Et

K,$ be the K-affinoid algebra

K〈|$|−1t+1T0, · · · , |$|−

1t+1Tt, |$|

1t+1T−1

0 , · · · , |$|1t+1T−1

t 〉/(T0 · · ·Tt −$),

which is integrally smooth, as Spf(EtK,$) is the formal completion along a smooth open in

some semistable scheme over K. Moreover, M(EtK,$) is canonically a K-affinoid domain

in EtK,$, and the restriction map

H•dR(EtK,$)→ H•dR(Et

K,$,M(EtK,$)) ' H•rig(Spec(Et

K,$)s/K)

is an isomorphism by [GK02, Lemma 3]. If K is residually algebraic over QF, then we have

Hprig(Spec(Et

K,$)s/K) = Hprig(Spec(Et

K,$)s/K)2p

for every p ≥ 0.

Now we study the behavior of logarithmic differential forms in the rigid cohomology. Wefirst review the notion of strictly semistable schemes.

Definition 4.4 (Strictly semistable scheme). Let k be a discrete non-Archimedean field.We say that a scheme X over k is strictly semistable of dimension n if X is locally offinite presentation, Zariski locally étale over SpecK[T0, . . . , Tn]/(T0 · · ·Tr − $) for someuniformizer $ of k.

For every integer 0 ≤ r ≤ n, denote by X [r]s the union of intersection of r + 1 distinct

irreducible components of Xs. It is a closed subscheme of Xs whose irreducible componentswith their reduced structure are smooth.

Let k be a finite extension of QF. Let S be a proper strictly semistable scheme overk of dimension s such that every irreducible component of S [r]

s is geometrically irreduciblefor every r ≥ 0. We fix an irreducible component E of Ss and let E1, . . . , EM be all otherirreducible components that intersect E . For a subset I ⊂ 1, . . . ,M, put EI = (⋂i∈I Ei)∩E(in particular, E∅ = E) and E♥I = EI\S [|I|+1]

s . For two subsets I, J of 1, . . . ,M, we writeI ≺ J if I ⊂ J and numbers in J\I are all greater than those in I.

For I ⊂ 1, . . . ,M, we have the open immersion E♥I ⊂ EI\S [|I|+2]s , whose compliment is∐

I⊂J,|J |=|I|+1 E♥J . Thus we have maps

H•rig(E♥I /k)→⊕

I⊂J,|J |=|I|+1H•+1E♥J ,rig

(EI\S [|I|+2]s /k) ∼−→

⊕I⊂J,|J |=|I|+1

H•−1rig (E♥J /k),

22 YIFENG LIU

where the second map is the Gysin isomorphism. In the above composite map, denote by ξIJthe induced map from H•rig(E♥I /k) to the component H•−1

rig (E♥J /k) if I ≺ J , and the zero mapif not.

In general, for I ≺ J , there is a unique strictly increasing sequence I = I0 ≺ I1 ≺ · · · ≺I|J\I| = J and we define

ξIJ := ξI|J\I|−1J · · · ξII1 : H•rig(E♥I /k)→ H•−|J\I|rig (E♥J /k),

and ξIJ = 0 if I ≺ J does not hold. Together, for i ≤ j, they induce a map

ξij :⊕|I|=i

H•rig(E♥I /k)→⊕|J |=j

H•+i−jrig (E♥J /k),

such that ξij|H•rig(E♥I /k) is the direct sum of ξIJ for all J with |J | = j. First, we have thefollowing lemma.

Lemma 4.5. Let notation be as above. For every 0 ≤ p ≤ s, the restriction of

ξ0p : Hp

rig(E♥/k)→⊕|J |=p

H0rig(E♥J /k)

to Hprig(E♥/k)2p is injective.

Proof. By the long exact sequence of cohomology with support (4.1), the kernel of the mapξ0p is a weight preserving extension of k-vector spaces Hp+|I|

EI ,rig(E/k) for |I| < p. Therefore, thelemma follows since Hp+|I|

EI ,rig(E/k) is pure of weight p+ |I| < 2p by [Tsu99, Theorems 5.2.1 &6.2.5] (with constant coefficients).

Denote by Zr(E)♥ the abelian group generated by irreducible components of EI with|I| = r. Denote by [Z] the generator corresponding to a component Z of EI . Note that theset I is determined by Z, which we denote by I(Z). Put Z(E)♥ = ⊕M

r=0 Zr(E)♥. We define

a wedge product∧ : Z(E)♥ ⊗ Z(E)♥ → Z(E)♥,

which is group homomorphism uniquely determined by the following conditions:• Z1 ∧ Z2 = (−1)r1r2Z2 ∧ Z1, if Z1 ∈ Zr1(E)♥ and Z2 ∈ Zr2(E)♥;• [Z1] ∧ [Z2] = 0 if Z1 ∩ Z2 = ∅;• [Z1] ∧ [Z2] = [Z1 ∩ Z2] if Z1 ∩ Z2 6= ∅ and I(Z1) ≺ I(Z1) ∪ I(Z2).

It is easy to see that ∧ is associative and maps Zr1(E)♥⊗Zr2(E)♥ into Zr1+r2(E)♥. We havean (injective) class map

cl♥ : Z(E)♥ →⊕I

H0rig(E♥I /k) '

⊕I

kπ0(EI)

sending [Z] to 1 corresponding to the irreducible component of EI(Z).For an element f ∈ O∗(San

k , π−1E♥), that is, an invertible function on some open neigh-

borhood of π−1E♥ in Sank , we can associate canonically an element div(f) ∈ Z1(E)♥. In fact,

there exists an element c ∈ k× such that |cf | = 1 on π−1E♥. Thus the reduction cf is an ele-ment in O∗E(E♥), and we define div(f) to be the associated divisor of cf , which is an elementin Z1(E)♥. Obviously, it does not depend on the choice of c. Finally, note that by the defini-tion of rigid cohomology, we have a canonical isomorphism H•dR(San

k , π−1E♥) ' H•rig(E♥/k).


Proposition 4.6. Let notation be as above. Given f1, . . . , fp ∈ O∗(Sank , π

−1E♥), if we regarddf1f1∧ · · · ∧ dfp

fpas an element in Hp

dR(Sank , π

−1E♥) ' Hprig(E♥/k), then we have

ξ0p

(df1

f1∧ · · · ∧ dfp

fp

)= cl♥ (div(f1) ∧ · · · ∧ div(fp)) .(4.2)

Proof. The question is local around the generic point of every irreducible component of EIwith |I| = p. So we fix such an irreducible component Z (with |I(Z)| = p) and take anaffine open neighborhood S0 of the generic point of Z in S such that it admits a smoothmorphism

f : S0 → Spec k[T0, . . . , Tp]/(T0 · · ·Tp −$)where $ is a uniformizer of k, satisfying

• E = E0 and Ei (i = 1, . . . , p) are all the irreducible components of Ts that intersect E ,where Ei is defined by the ideal (f ∗Ti, $);• EI := ⋂

i∈I Ei is irreducible and nonempty for I ⊂ 1, . . . , p.Note that

• d(cf)cf

= dff;

• both sides of (4.2) are multi-linear in f1, . . . , fp ∈ O∗(San0,k, π

−1E♥); and• df

f= df ′

f ′in H1

rig(E♥/k) if |f | = |f ′| = 1 on π−1E♥ and f = f ′.Thus we may assume that fi = f ∗Ti. Then as both sides of (4.2) are functorial in f underpullback, we may assume that S0 = Spec k[T0, . . . , Tp]/(T0 · · ·Tp −$) and fi = Ti.

Put S ′ = Spec k[T1, . . . , Tp] and let g : S0 → S ′ be the morphism sending Ti to Ti (1 ≤ i ≤p). For I ⊂ 1, . . . , p, let E ′I be the closed subscheme of S ′s defined by the ideal ($,Ti |i ∈ I).Then g induces an isomorphism EI ' E ′I . Similarly, we have maps

ξ′IJ : H•rig(E ′♥I /k)→ H•+i−jrig (E ′♥J /k)

for I ⊂ J and ξ′ij for i ≤ j, where E ′♥I = g(E♥I ). It is easy to see that ξ′IJ = ξIJ if we identifyH•rig(E ′♥I /k) with H•rig(E♥I /k) through g∗. Therefore, it suffices to show the equality (4.2) forS ′, that is,

ξ′0p

(dT1

T1∧ · · · ∧ dTp

Tp

)= 1 ∈ H0

rig(E ′1,...,p/k) ' k.

However, S ′, which is isomorphic to Apk can be canonically embedded into the proper

smooth scheme Ppk over k. Thus, the rigid cohomology H•rig(E ′♥I /k) and the map ξ′ij can

be computed on Pp,ank . On the generic fiber S ′k, we similarly define TI to be the closed

subscheme Spec k[T1, . . . , Tp]/(Ti | i ∈ I) of S ′k for I ⊂ 1, . . . , p, and T ♥I = TI\⋃I J TJ .

We may similarly define maps αIJ : H•dR(T ♥I )→ H•−|J\I|dR (T ♥J ) and αij via algebraic de Rhamcohomology theory. Then we have canonical vertical isomorphisms rendering the diagram

H•dR(T ♥I )αIJ //

'

H•−|J\I|dR (T ♥J )'

H•rig(E ′♥I /k)ξ′IJ // H•−|J\I|rig (E ′♥J /k)

24 YIFENG LIU

commutative. From the standard computation in algebraic de Rham cohomology, we have

α0p

(dT1

T1∧ · · · ∧ dTp

Tp

)= 1 ∈ H0

dR(T1,...,p),

where T1,...,p is just the point of origin. Thus, the proposition is proved.

5. Weight decomposition of de Rham cohomology sheaves

In this section, we prove Theorem 1.4 for étale topology, and then deduce the one foranalytic topology. In particular, sheaves like OX , cX , and the de Rham complex (Ω•X , d) areunderstood in the étale topology, until we say otherwise.

Definition 5.1 (Marked pair). Let k be a discrete non-Archimedean field. A marked k-pair(X ,D) of dimension n and depth t consists of an affine strictly semistable scheme X over kof dimension n (Definition 4.4), and an irreducible component D of X [t]

s that is geometricallyirreducible.

We start from the following lemma, which generalizes [Ber07, Lemma 2.1.2].

Lemma 5.2. Suppose that K is embeddable into ka for some discrete non-Archimedean fieldk. Let X be a smooth K-analytic space, and x a point of X with s(x) + t(x) = dimx(X).Given a morphism of K-analytic spaces X → Yη, where Y is a special formal K-scheme,there exist

• a finite extension K ′ of K, a finite extension k′ of k contained in K ′,• a marked k′-pair (X ,D) of dimension dimx(X) and depth t(x),• an open neighborhood U of (X/D)η⊗k′K ′ in X an

K′ ,• a point x′ ∈ (X/D)η⊗k′K ′,• a morphism of K-analytic spaces ϕ : U → X, and• a morphism of formal K-schemes X/D⊗k′K ′ → Y,

such that the following are true:(i) ϕ is étale and ϕ(x′) = x;(ii) the induced morphism (X/D)η⊗k′K ′ → Yη coincides with the composition

(X/D)η⊗k′K ′ → Uϕ−→ X → Yη.

Proof. Put t = t(x), s = s(x), and n = t + s. By [Ber07, Proposition 2.3.1], by possiblyreplacing k and K by their finite extensions, we may replace X by (B ×k Y )⊗kK, whereB = ∏t

j=1B(0; rj, Rj)k for some 0 < rj < Rj and Y is a smooth k-analytic space of dimensions, and x projects to b ∈ B with t(b) = t and y ∈ Y with s(y) = s. Denote by P the k-schemeP1k with the point 0 on the special fiber blown up, and by P the formal completion of P

along the open subscheme Ps\π(0), π(∞), which is isomorphic to Spf k〈X, Y 〉/(XY −$)for some uniformizer $ of k. By further replacing k and K by their finite extensions suchthat |$| < Rj− rj for every j, we may assume that there is an embedding ∏tPη ⊂ B whoseimage contains b such that π(b) = 0, where 0 is the closed point in ∏tPs that is nodal inevery component. In particular ∏tPη is a neighborhood of b.

For Y , we proceed exactly as in the Step 1 of the proof of [Ber07, Lemma 2.1.2]. Weobtain, by shrinking Y if necessary, two k-affinoid domains Z ′ ⊂ W ′ ⊂ Y such that W ′ isa neighborhood of Z ′. We follow Berkovich’s construction in the beginning of Step 3 of theproof of [Ber07, Lemma 2.1.2] after Raynaud to obtain an integral scheme Y ′ proper and flat


over k with an embedding Y ⊂ Y ′anη , open subschemes Z ′ ⊂ W ′ ⊂ Y ′s, such that Z ′ = π−1Z ′

and W ′ = π−1W ′.Now we put two parts together. Define Y = ∏

tP × Y ′ where the fiber product is takenover k, andW = ∏

tPs×W ′ where the fiber product is taken over k. ThenW := ∏tPη×W ′

coincides with π−1W in Yank . Moreover, WK is a neighborhood of x where WK denotes the

inverse image of W in X = (B ×k Y )⊗kK. Note that we have the induced map α : WK →Yη of K-analytic spaces by restriction. By the same argument in Step 2 of the proof of[Ber07, Lemma 2.1.2], we have finitely many open affine subschemes Yi for 1 ≤ i ≤ lof Y such that WK,i := α−1Yi

η is an affinoid subdomain of WK and WK = ⋃li=1WK,i.

By [Ber07, Lemma 2.1.3 (ii)], we may assume that WK,i can be descent to an k-affinoidsubdomain Wi of W for every i by replacing k and K by their finite extensions if necessary.Making a finite number of additional blow-ups, we may also assume that there are opensubschemes Wi ⊂ W with Wi = π−1Wi and W = ⋃l

i=1Wi.Now we proceed as in Step 4 of the proof of [Ber07, Lemma 2.1.2]. Take an alteration

φ : X ′ → Y from a strictly semistable scheme X ′ over k after further replacing k and K bytheir finite extensions, and a point x′ ∈ X ′an

K such that φ(x′) = x. By a similar argument,one can show that π(x′) ∈ X ′s ⊗k K has dimension at least s. On the other hand, we haves(x′) ≥ s and t(x′) ≥ t. Thus, s(x′) = s and t(x′) = t. Denote by C the Zariski closureof π(x′) in X ′s, equipped with the reduced induced scheme structure. Suppose that C iscontained in t′ + 1 distinct irreducible components of X ′s. Then t′ ≤ t as the codimension inX ′s of the intersection of t′ + 1 distinct irreducible components is t′. We take an affine opensubscheme U ′ of X ′ satisfying: D′ := U ′∩C is open dense in C; φ(D′) is contained inW ; U ′ isétale over Spec k[T0, . . . , Tn]/(T0 · · ·Tt′ − π) for some uniformizer $ of k such that D′ is thezero locus of the ideal generated by (T0, . . . , Tt, π). Now we blow up the ideal generated by(Tt′+1, $), and then the strict transform of the ideal generated by (Tt′+2, $), and continue toobtain an affine strictly semistable scheme X over k such that the strict transform D of D′is an irreducible component of X [t]

s . Possibly after further replacing k and K by their finiteextensions, and replacing X by an affine open subscheme such that Xs ∩ D is dense in D,we obtain a marked k-pair (X ,D) of dimension n and depth t such that φ : X → Y is étaleon the generic fiber. Here, the further finite extension of ground fields is to ensure that D isgeometrically irreducible. Note that (φan

K )−1WK is a neighborhood of x′ containing π−1D asφ(D) ⊆ W . Here, x′ ∈ X an

K is an arbitrary preimage of the original x′ ∈ X ′anK , which exists

by construction.We take U to be an arbitrary open neighborhood of π−1D, and ϕ to be φan

K |U . By the sameargument in Step 5 of the proof of [Ber07, Lemma 2.1.2], φ induces a morphism of K-formalschemes X/φ−1W⊗kK → Y and thus a morphism X/D⊗kK → Y. The conclusions of thelemma are all satisfied by the construction.

From now on to the end of this section, we assume thatK is a residually algebraic extensionof QF.

Definition 5.3 (Fundamental chart). Let X be a K-analytic space and x ∈ X a point. Afundamental chart of (X;x) consists of data (DL, (Y ,D), (D, δ),W, α; y) where

• (Y ,D) is a marked k-pair of dimension t(x) + s(x) and depth t(x), where k is a finiteextension of QF;• DL is an open poly-disc over L of dimension dimx(X) − t(x) − s(x), where L issimultaneously a finite extension of K and a (residually algebraic) extension of k;

26 YIFENG LIU

• D is an integrally smooth affinoid k-algebra, and

δ : Spf D[[T0, . . . , Tt]]/(T0 · · ·Tt −$) ∼−→ Y/D(5.1)

is an isomorphism of formal k-schemes, where $ is a uniformizer of k;• W is an open neighborhood of (Y/D)η⊗kL = (π−1D)⊗kL in Yan

L ;• y is a point in DL×LW with t(y) = t(prW (y)) = t(x), s(y) = s(prW (y)) = s(x), andsuch that prW (y) belongs to (π−1D)⊗kL whose reduction is the generic point of D

L;

• α : DL ×LW → X is an étale morphism of K-analytic spaces such that α(y) = x.Note that the field k will be implicit from the notation (as it is not important).

The isomorphism (5.1) induces an isomorphism SpecDs ' D of k-schemes, and an iso-morphism

δ∗ : HpdR(DL ×L (W, (π−1D)⊗kL)) ∼−→

p⊕j=0

Hjrig(D/k)⊗k Hp−j

dR (Etk,$)⊗k L(5.2)

of L-vector spaces [GK02, Lemmas 2 & 3] and [Ber07, Corollary 5.5.2]. Here, Etk,$ is the

k-analytic space defined in Example 4.3. Denote by Hpw(DL, (Y ,D), (D, δ),W ) the subspace

of the left-hand side of (5.2) corresponding to the subspacep⊕j=0

Hjrig(D/k)w−2(p−j) ⊗k Hp−j

dR (Etk,$)⊗k L

on the right-hand side. In particular, all elements in Hp−jdR (Et

k,$) are regarded to be of weight2(p− j). Then we have a direct sum decomposition

HpdR(DL ×L (W, (π−1D)⊗kL)) =

⊕w∈Z

Hpw(DL, (Y ,D), (D, δ),W ).(5.3)

Finally, we denote by Hp(w)(DL, (Y ,D), (D, δ),W ) the subspace of Hp

dR(DL ×L W ) as theinverse image of Hp

w(DL, (Y ,D), (D, δ),W ) under the restriction map

HpdR(DL ×LW )→ Hp

dR(DL ×L (W, (π−1D)⊗kL)).

Remark 5.4. Note that the composite morphism of formal k-schemes

Spf((Etk,$)⊗kD)→ Spf D[[T1, . . . , Tt]]/(T1 · · ·Tt −$) δ−→ Y/D

induces an isomorphism

HpdR(W, (π−1D)⊗kL) ' Hp

rig((Etk,$)s ⊗k D/L).

Therefore, an element ω ∈ HpdR(DL ×L W ) belongs to Hp

(w)(DL, (Y ,D), (D, δ),W ) if andonly if the restriction of ω to 0 ×LW belongs to Hp

rig((Etk,$)s ⊗k D/L)w under the above

isomorphism, by Example 4.3.

Remark 5.5. Note that Hpw(DL, (Y ,D), (D, δ),W ) = 0 unless p ≤ w ≤ 2p, and the de-

composition (5.3) is stable under base change along a residually algebraic extension of K(and L accordingly). We warn that the decomposition (5.3) depends on all of the data(DL, (Y ,D), (D, δ),W ), not just the L-analytic germ DL ×L (W, (π−1D)⊗kL).


Lemma 5.6. For part of the data (DL, (Y ,D), (D, δ),W ) from Definition 5.3 and f ∈O∗(DL ×LW ), we have

dff∈ H1

(2)(DL, (Y ,D), (D, δ),W ).

Here, we regard dff, a priori a closed 1-form on DL×LW , as an element in H1

dR(DL×LW ).

Proof. We may assume L = K without lost of generality. The function f restricts to amorphism of formal K-schemes, denoted again by

f : Spf((Etk,$)⊗kD⊗kK)→ Spf K[[T ]].

On the generic fiber, since the Shilov boundary ofM(Etk,$⊗kD⊗kK) consists of one point

and the image of the induced morphism M(Etk,$⊗kD⊗kK) → D(0; 1)K does not con-

tain 0, the pullback of T on M(Etk,$⊗kD⊗kK) has constant norm. In other words,

the morphism M(Etk,$⊗kD⊗kK) → D(0; 1)K induced by f factors through a morphism

M(Etk,$⊗kD⊗kK)→M(K〈r−1T, rT−1〉) for a unique number r < 1 in

√|K×|. By replac-

ing K by a finite extension, we may assume r ∈ |K|. Then K〈r−1T, rT−1〉 is integrallysmooth, and we have SpecK〈r−1T, rT−1〉s ' (Gm)

K. Moreover,

H1rig(Gm/K)2 = H1

rig(Gm/K) ' H1dR(D(0, 1)K ,M(K〈r−1T, rT−1〉)) = KdT

T.

Thus Lemma 4.2 and Remark 5.4 imply the lemma.

Definition 5.7. Let X be a K-analytic space and x ∈ X a point.(1) Let fEt(X;x) be the category whose objects are fundamental charts of (X;x), and a

morphismφ : (DL2 , (Y2,D2), (D2, δ2),W2, α2; y2)→ (DL1 , (Y1,D1), (D1, δ1),W1, α1; y1)consists of extensions K ⊂ L1 ⊂ L2

5 such that k1 ⊂ k2, and a morphismΦ(φ) : DL2 ×L2 W2 → DL1 ×L1 W1

of L1-analytic spaces sending y2 to y1, and such thatΦ(φ)∗Hp

(w)(DL1 , (Y1,D1), (D1, δ1),W1) ⊂ Hp(w)(DL2 , (Y2,D2), (D2, δ2),W2)

for all p, w ∈ Z. Note that Φ(φ) needs not to respect each factor.(2) Let Et(X;x) be the category of étale neighborhoods of (X;x). Recall that its objects

are triples (Y, α; y) where α : Y → X is an étale morphism sending y ∈ Y to x, andmorphisms are defined in the obvious way. In the notation (Y, α; y), the morphismα will be suppressed if it is not relevant. For a presheaf F on Xet, the stalk of F atx is defined to be Fx := lim−→(Y,α;y) F (Y ) where the colimit is taken over the categoryEt(X;x).

(3) We have a functor Φ: fEt(X;x)→ Et(X;x) sending an object(DL, (Y ,D), (D, δ),W, α; y)

of fEt(X;x) to (DL ×LW,α; y), and a morphism φ to Φ(φ).

The following lemma generalizes [Ber07, Proposition 2.1.1].5Even when L1 = L2, we regard DL1 and DL2 as two different poly-discs.

28 YIFENG LIU

Lemma 5.8. Suppose that K is embeddable into CF and X is a smooth K-analytic space.Fix an arbitrary point x ∈ X and let (Y, α0; y0) be an object of Et(X;x). Then

(1) there exists an object (DL, (Y ,D), (D, δ),W, α; y) ∈ fEt(X;x) such that its imageunder Φ admits a morphism to (Y, α0; y0);

(2) given two morphisms βi : Φ(DLi , (Yi,Di), (Di, δi),Wi, αi; yi)→ (Y, α0; y0) in Et(X;x)for i = 1, 2, there exists an object (DL, (Y ,D), (D, δ),W, α; y) ∈ fEt(X;x) togetherwith morphisms φi to (Di, (YLi ,Di), (Di, δi),Wi, αi; yi) in fEt(X;x) for i = 1, 2 suchthat the following diagram

Φ(DL1 , (Y1,D1), (D1, δ1),W1, α1; y1)β1

**Φ(DL, (Y ,D), (D, δ),W, α; y)

Φ(φ1)33

Φ(φ2) ,,

(Y, α0; y0)

Φ(DL2 , (Y2,D2), (D2, δ2),W2, α2; y2)β2

44

commutes.

Proof. Let n be the dimension of X at x; we may assume by arguing locally that X is ofdimension n. Put t = t(x) and s = s(x).

For (1), by [Ber07, Proposition 2.3.1], after replacing K by a finite extension, we mayassume Y = DK ×K X ′ and y0 ∈ Y such that t(prX′(y0)) = t and s(prX′(y0)) = s, whereX ′ is a smooth K-analytic space of dimension s + t. Now we only need to apply Lemma5.2 to Y = Spf K, the pair (X ′;x′), and the structure morphism X ′ → Yη =M(K). Theexistence of (D, δ) is due to the argument in Part (iv) of the proof of [GK02, Theorem 2.3].

For (2), we put Di = DLi , and may assume K = L1 = L2. For i = 1, 2, we choosea relative compactification Yi → Yi over ki , where Yi is proper. Such compactification isneeded to construct a formal scheme Y (see below) in order to apply Lemma 5.2. Then Wi

is open in Yiη, where Yi = Yi⊗kiK. Consider the étale morphism

α′0 : Y ′ := (D1 ×K W1)×Y (D2 ×K W2)→ Y,

and a point y′0 ∈ Y ′ projecting to y1 (resp. y2) in the first (resp. second) factor. Again by[Ber07, Proposition 2.3.1], we may find an object of the form (DK ×K X ′, α′; y′) in Et(X;x)such that t(prX′(y′)) = t and s(prX′(y′)) = s as in (1) with a morphism to (Y ′, α′0; y′0). Nowwe apply Lemma 5.2 to X ′, the point prX′(y′), Y = Y1 ×K Y2, the morphism

X ′(β1,β2)−−−−→ W1 ×K W2 ⊂ Yη,

where βi equals the compositionX ′ ' 0 ×K X ′ ⊂ DK ×K X ′ → Di ×K Wi → Wi (i = 1, 2)

with the last arrow being the projection. We obtain a marked k-pair (Y ,D) of dimensions+ t and depth t, for some discrete non-Archimedean field k containing k1, k2 and containedin (possibly a finite extension of) K; an open neighborhoodW of (Y/D)η⊗kK in Yan

K , a pointu ∈ W , an étale morphism of K-analytic spaces ϕ : W → X ′ such that ϕ(u) = prX′(y′),and a morphism of formal K-schemes ψ = (ψ1, ψ2) : Y/D⊗kK → Y1 ×K Y2 compatiblewith ϕ. As ψi maps the generic point of D

Kto the generic point of (Di)K , we may replace


(Y ,D) by an affine open such that ψi(DK) ⊂ (Di)K for i = 1, 2. In particular, we havemorphisms ψi : Y/D⊗kK → Yi/Di⊗kiK

. Note that ψi does not necessarily descent to anyfinite extension of k. By the proof of [GK02, Theorem 2.3], there is an integrally smoothk-affinoid algebra D and an isomorphism δ as in (5.1). Finally, we take a point y ∈ DK×KWabove y′ such that prW (y) = u.

Now the object (DK , (Y ,D), (D, δ),W, α; y) has been constructed with the obvious α (withL = K possibly a finite extension of the starting one). Let Φ(φi) be the composite morphismDK ×K W → DK ×K X ′ → Di ×K Wi for i = 1, 2. It remains to show that(i) For i = 1, 2, every p, every w, and an element ω ∈ Hp

rig(Di/ki)w, we have(βi ϕ)∗(δ∗i )−1ω ∈ Hp

(w)((Y ,D), (D, δ),W ).(ii) For i = 1, 2 and an arbitrary coordinate T of Et

ki,$i(where $i is a uniformizer of ki),

we have(βi ϕ)∗(δ∗i )−1 dT

T∈ H1

(2)((Y ,D), (D, δ),W ).For (i), as we have morphisms of formal K-schemes

Spf((Etk,$)⊗kD⊗kK)→ Y/D⊗kK

ψi−→ Yi/Di⊗kiK → Spf Di ⊗kiK

,

Lemma 4.2 implies that (βi ϕ)∗(δ∗i )−1ω coincides with ϕ∗iω inHp

dR(W, (π−1D)⊗kK) ' Hprig((Et

k,$)s ⊗k D/K)in view of Remark 5.4, where

ϕi : (Etk,$)s ⊗k DK → (Di)K

is the induced morphism of (affine smooth) K-schemes. Thus (i) follows from weight preser-vation of rigid cohomology.

For (ii), we only need to apply Lemma 5.6 to the function T .

Now we are ready to define the desired direct summand (Ωp,clX /dΩp−1

X )w in the weightdecomposition of de Rham cohomology sheaves.Definition 5.9 (De Rham cohomology sheaves with weights). Suppose that K is residuallyalgebraic over QF and X is a smooth K-analytic space. Let p ≥ 1 be an integer.

For every object U of Xet, define (Ωp,clX /dΩp−1

X )(U)prew ⊂ (Ωp,cl

X /dΩp−1X )(U) to be the

image of elements ω ∈ Ωp,clX (U) such that for every point u ∈ U , there exists a fun-

damental chart (DL, (Y ,D), (D, δ),W, α; y) of (U ;u) such that α∗ω, regarded as an ele-ment in Hp

dR(DL ×L W ), belongs to Hp(w)(DL, (Y ,D), (D, δ),W ). The assignment U 7→

(Ωp,clX /dΩp−1

X )(U)prew defines a sub-presheaf (Ωp,cl

X /dΩp−1X )pre

w of Ωp,clX /dΩp−1

X .We define (Ωp,cl

X /dΩp−1X )w to be the sheafification of (Ωp,cl

X /dΩp−1X )pre

w , which is canonicallya subsheaf of Ωp,cl

X /dΩp−1X .

The following lemma can be proved by the same way as for [Ber07, Corollary 5.5.3].Lemma 5.10. Let K ′/K be an extension such that K ′ is embeddable into CF. Let X bea smooth K-analytic space and ς : X ′ := X⊗KK ′ → X the canonical projection. Then thecanonical map of sheaves on X ′et

ς−1(Ωp,clX /dΩp−1

X )⊗L K ′ → Ωp,clX′ /dΩp−1

X′

is an isomorphism, where L is the algebraic closure of K in K ′.

30 YIFENG LIU

The following theorem establishes the functorial weight decomposition of de Rham coho-mology sheaves in Theorem 1.4 in the case of étale topology.

Theorem 5.11. If K is embeddable into CF and X is a smooth K-analytic space, then thefollowing hold:

(1) under the situation of Lemma 5.10, we have

ς−1(Ωp,clX /dΩp−1

X )w ⊗L K ′ = (Ωp,clX′ /dΩp−1

X′ )w,for every w ∈ Z;

(2) the image of the composite map

(Ωp1,clX /dΩp1−1

X )w1 ⊗ (Ωp2,clX /dΩp2−1

X )w2 → Ωp1,clX /dΩp1−1

X ⊗ Ωp2,clX /dΩp2−1

X∧−→ Ωp1+p2,cl

X /dΩp1+p2−1X

is contained in the subsheaf (Ωp1+p2,clX /dΩp1+p2−1

X )w1+w2;(3) the sheaf (Ωp,cl

X /dΩp−1X )w is zero unless p ≤ w ≤ 2p;

(4) the canonical map ⊕w∈Z

(Ωp,clX /dΩp−1

X )w → Ωp,clX /dΩp−1

X

is an isomorphism;(5) for every morphism f : Y → X of smooth K-analytic spaces, we have

f#(f−1(Ωp,clX /dΩp−1

X )w) ⊂ (Ωp,clY /dΩp−1

Y )wfor every w ∈ Z. Here, f# denotes the canonical map f−1Ω•X → Ω•Y and inducedmaps of cohomology sheaves.

Proof. Part (1) follows from the definition and Remark 5.5. Part (2) follows from definitionand Lemma 5.8 (2).

For the remaining parts, it suffices to work on stalks. Thus we fix a point x ∈ X witht = t(x) and s = s(x).

For (3), take an element [ω] in the stalk of (Ωp,clX /dΩp−1

X )w at x for some w < p or w > 2p.We may assume that it has a representative ω ∈ Ωp,cl

X (U) for some étale neighborhoods (U ;u)of (X;x). By definition, we have a fundamental chart (DL, (Y ,D), (D, δ),W, α; y) of (U ;u)such that α∗ω = 0 in Hp

dR(DL ×L (W, (π−1D)⊗kL)) by Remark 5.5. Then there exists anopen neighborhood W− of (π−1D)⊗kL in W , such that α∗ω = 0 in Hp

dR(DL ×L W−). Inother words, [ω] = 0 in the stalk of (Ωp,cl

X /dΩp−1X )w at x as α is étale and we are working with

differential sheaves in étale topology.For (4), we first show that the map is injective. Let [ω] be an element in the stalk

Ωp,clX,x/dΩp−1

X,x . Suppose that we have [ω] = ∑[ω]1w = ∑[ω]2w in which both [ω]1w and [ω]2ware in the stalk of (Ωp,cl

X /dΩp−1X )w at x. We may choose an object (U ;u) ∈ Et(X, x) such

that [ω]iw has a representative ωiw ∈ (Ωp,clX /dΩp−1

X )(U)prew for i = 1, 2 and every w ∈ Z, and∑

ω1w = ∑

ω2w. In particular, [ω] has a representative ω := ∑

ω1w = ∑

ω2w on (U ;u). Fix

a weight w ∈ Z. It suffices to show that [ω1w] = [ω2

w] in the stalk at x. By Definition 5.9,there exist two fundamental charts (DLi , (Yi,Di), (Di, δi),Wi, αi; yi) of (U ;u) such that α∗iωiwbelongs to Hp

(w)(DLi , (Yi,Di), (Di, δi),Wi) for i = 1, 2. By Lemma 5.8, we may find anotherfundamental chart (DL, (Y ,D), (D, δ),W, α; y) ∈ fEt(U ;u) as in that lemma. Then we haveΦ(φi)∗ωiw ∈ Hp

(w)(DL, (Y ,D), (D, δ),W ) for both i = 1, 2. However, Φ(φ1)∗ω1w and Φ(φ2)∗ω2

w,after restriction to Hp

dR(DL×L (W, (π−1D)⊗kL)), must be equal, as they are both the weight


w component of α∗ω in HpdR(DL ×L (W, (π−1D)⊗kL)) under the decomposition (5.3). As

the map HpdR(DL×LW )→ (Ωp,cl

X /dΩp−1X )x factors through Hp

dR(DL×L (W, (π−1D)⊗kL)), wehave [ω]1w = [ω]2w. Finally, Lemma 5.12 below implies that the map in (4) is surjective aswell.

For (5), we take a point y ∈ Y such that f(y) = x. We may take a fundamental chart(DL, (Y ,D), (D, δ),W, α; y) of (X;x) and replace X by DL ×LW and x by y. By the sameproof of Lemma 5.8 (2), we may find a fundamental chart (DL′ , (Y ′,D′), (D′, δ′),W ′, α′; y′) of(Y ; y) such that (f α′)∗Hp

(w)(DL, (Y ,D), (D, δ),W ) ⊂ Hp(w)(DL′ , (Y ′,D′), (D′, δ′),W ′). This

confirms Part (5) since Hp(w)(DL, (Y ,D), (D, δ),W ) (resp. Hp

(w)(DL′ , (Y ′,D′), (D′, δ′),W ′))restricts to the weight w part in the stalk of Ωp,cl

X /dΩp−1X (resp. Ωp,cl

Y /dΩp−1Y ) at x (resp. y),

by Lemma 5.12 below.

The following lemma is the most crucial and difficult part in the proof of the weightdecomposition.

Lemma 5.12. Let the assumptions be as in Theorem 5.11. We take a point x ∈ X.For any fixed weight w, an object (DL, (Y ,D), (D, δ),W, α; y) ∈ fEt(X, x), and an elementω ∈ Hp

(w)(DL, (Y ,D), (D, δ),W ), the induced class [ω] ∈ Ωp,clX,x/dΩp−1

X,x belongs to the stalk of(Ωp,cl

X /dΩp−1X )w at x.

We first explain why the lemma is not immediate. For simplicity, let us assume that DL

is trivial. To prove the lemma, we have to find an étale neighborhood U of x such that ωlies in (Ωp,cl

X /dΩp−1X )(U)pre

w . However, no matter what U we take, there are always points uof U such that its image in W is not in (π−1D)⊗kL. On W$π−1D)⊗kL, there is a priori noobvious control of the form ω, so it is a question to show that one can choose a fundamentalchart at u such that the induced form has weight w.

Before the proof, let us explain the main steps. Again, we assume that DL is trivial atthis moment. In Steps 1 & 2, we make the germ (W, (π−1D)⊗kL) a better shape. We willconstruct a strictly semistable scheme S over k together with an irreducible component Eon the special fiber, and an étale neighborhood (W \, y$ of x above (W, y) equipped with amorphism to (π−1E , π−1E \) where E \ = E\S [1]

s . In Step 3, we study some basic structure ofthe boundary part π−1E\π−1E \. In Step 4, we choose an appropriate covering of π−1E withnice behavior of ω on some open neighborhood of π−1E \ in every member of the cover. In Step5, we show that ω has weight w (in certain sense) even on the boundary part π−1E\π−1E \.Finally, we conclude the lemma via functoriality in rigid cohomology in Step 6.

Proof. We may assume L = K and write D = DL. To simplify notation, we denote by Vthe K-affinoid domain (π−1D)⊗kK in W . By Example 4.3 and Lemma 5.6, we may assumethat the image of ω in Hp

dR(D ×K (W,V )) is in Hprig(D/K)w in view of the decomposition

(5.2).Step 1. We choose a smooth k-algebra D\ (of relative dimension s) such that its $-adic

completion is D, where we recall that $ is a uniformizer of the discrete non-Archimedeanfield k ⊂ K. In particular, we may identify (SpecD\)s with D, andM(D) with a k-affinoiddomain in (SpecD\)an

k . As in Lemma 4.2, we have germs (W,V ) and ((SpecD\)ank ,M(D))

and a morphism V → M(D)⊗kK induced from δ. We choose a neighborhood Uε of thegraph of the previous morphism in W ×k (SpecD\)an

k as in the proof of Lemma 4.2, such

32 YIFENG LIU

that the induced map

H•dR(W,V )→ H•dR(W ×k (SpecD\)ank , Uε ∩ (V ×kM(D)))

is an isomorphism. By a similar argument in the proof of [GK02, Lemma 2], we may replaceW by a smaller open neighborhood of V such that there is a morphism W → Uε sending Vinto Uε ∩ (V ×kM(D)) whose induced map

H•dR(W ×k (SpecD\)ank , Uε ∩ (V ×kM(D)))→ H•dR(W,V )

is the inverse of the previous isomorphism. In other words, we have a morphism δ′ : W →(SpecD\)an

K sending V intoM(D)⊗kK such that, although δ′|V might not coincide with theoriginal morphism V →M(D)⊗kK induced from δ, we still have that the induced map

H•rig(D/K) ' H•dR((SpecD\)ank ,M(D))⊗k K δ′∗−→ H•dR(W,V ) ' H•rig((Et

k,$)s ⊗k D/K)

coincides with the pullback map.Step 2. We choose a compactification (SpecD\)k → Tk over k, and define T to be the

k-scheme Tk∐

(SpecD\)k SpecD\. Apply [dJ96, Theorem 8.2] to the k-variety T and Z = ∅.We obtain a finite extension k′/k, an alteration S\ → Tk′ and a k′-compactification S\ → Swhere S is a projective strictly semistable scheme over k′ such that S\S\ is a strict normalcrossing divisor of S (concentrated on the special fiber). We may further assume that everyirreducible component of S [r]

s is geometrically irreducible for every r ≥ 0. To ease notation,we replace k by k′ and possiblyK by a finite extension. We may fix an irreducible componentE of Ss such that its generic point belongs to S\s and maps to the generic point of Ts ' D.Note that the complement of E \ := E ∩ S\s in E is exactly S [1]

s ∩ E . Denote by σE the uniquepoint in San

K who reduction is the generic point of EK. Then π−1E

Kis an open neighborhood

of σE .We have open subsets (S\)an

K and π−1EK

of SanK . Define W \ via the following Cartesian

diagram:

W \ //

(S\)anK ∩ π−1E

K

W

δ′ // T anK ,

and let δ\ : W \ → Sη⊗kK = (π−1E)⊗kK the induced morphism, where S := S/E . We choosea point y\ ∈ D ×K W \ which lifts y and such that δ$prW \(y$) = σE . The image of theform ω in Hp

rig(D/K)w induces a class [ω\] ∈ Hprig(E \/K)w via restriction along the alteration.

Therefore, there exists a monic polynomial P ∈ k[X] whose roots are all Weil |k|w/2-numberssuch that P (Fr∗)[ω\] = 0 where Fr denotes the relative Frobenius of E \/k. We fix an openneighborhood U of π−1E \ in Sη such that [ω\] has a representative ω\ ∈ Hp

dR(U⊗kK).By construction, we may remove a Zariski closed subset ofW \ of dimension at most s+t−1

such that the morphism W \ → W hence the composition D ×K W \ → D ×K Wα−→ X are

both étale. In particular, (D×K W \; y\) is an object of Et(X;x).Step 3. Now we define a continuous function

dE : Sη → R


valued in [0, 1] such that dE(x) = 0 if and only if π(x) ∈ E \, as follows: For a point x ∈ Sη,choose an open neighborhood Sx of π(x) in S such that Sx is étale over

Spf k[[t0]]〈t1, . . . , tr, tr+1, t−1r+1, . . . , ts, t

−1s 〉/(t0 · · · tr −$)

for some 0 ≤ r ≤ s; and if we write fj for the image of tj in Sx, then E \ ∩Sx is defined bythe equations f0 = 0 and f1 · · · fr 6= 0. Then we define

dE(x) = 1− log |f0(x)|log |$| .

It is independent of the choice of the étale coordinates hence is continuous. Moreover, wehave |$| ≤ |f0(x)| < 1 and the equality holds if and only if π(x) ∈ E \.

Let E1 be an open subscheme of E and put E \1 = E \ ∩ E1. For ε > 0, put(π−1E1)<ε := x ∈ π−1E1 | dE(x) < ε.

Then (π−1E1)<ε form a fundamental system of open neighborhoods of π−1E \1 in π−1E1. Infact, let U1 ⊂ π−1E1 be an open neighborhood of π−1E \1. Consider (π−1E1)≤1/2 := x ∈π−1E1 | dE(x) ≤ 1/2, which is a compact subset of π−1E1. Thus the image of the functiondE on (π−1E1)≤1/2\U1 is a compact subset of (0, 1/2]. Thus, there exists ε > 0 such thatdE(x) > ε if x 6∈ U1. In other words, U1 contains (π−1E1)<ε.Step 4. For every point e ∈ E , we fix an affine open neighborhood Se of e in S together

with an étale morphism toSpf k[[t0]]〈t1, . . . , tr, tr+1, t

−1r+1, . . . , ts, t

−1s 〉/(t0 · · · tr −$)

where r = r(e) ≥ 0 is the unique integer such that e ∈ S [r]s \S [r+1]

s , such that• if we denote fe,i the image of ti in Se, then E \e := E \ ∩Se is defined by the equationsfe,0 = 0 and fe,1 · · · fe,r 6= 0;• if we denote by Fe the subscheme of Se defined by fe,0 = · · · = fe,r = 0, then thereexist an integrally smooth k-affinoid algebra Fe together with an isomorphism, whichwe fix,

Spf F e [[te,0, . . . , te,r]]/(te,0 · · · te,r −$) ' Se/Fe(5.4)of formal k-schemes, sending te,i to fe,i for 0 ≤ i ≤ r.

Note that Fe necessarily contains e by our choice of r. For every point e ∈ E , we furtherfix an open neighborhood Ue of π−1E \e in Se,η contained in U , together with an absoluteFrobenius lifting φe : Ue → U satisfying properties(a) φ∗efe,i = f

|k|e,i for 1 ≤ i ≤ r (as in [Chi98, Lemma 3.1.1]);

(b) |(φ∗eg − g |k|)(x)| < 1 for all regular functions g on Se/E\e and all x ∈ Ue at which both gand φ∗eg are defined (as [Ber07, Lemma 6.1.1]);

(c) P (φ∗e)ω\ = 0 in HpdR(Ue⊗kK).

Since Ue ∩ π−1Ee, where Ee := E ∩ Se, is an open neighborhood of π−1E \e in π−1Ee, thereexists εe > 0 such that Ue contains (π−1Ee)<εe by Step 3.

By [GK02, Lemma 3], we know that the restriction mapH•dR(Sη, π

−1Fe)→ H•dR(Ue, Ue ∩ π−1Fe)is an isomorphism, which is isomorphic to H•rig(SpecFe,s/k)⊗k H•dR(Er

k,$) induced by (5.4).In particular, we have the notion of weights on H•dR(Sη, π

−1Fe)⊗k K.

34 YIFENG LIU

Step 5. For a point e ∈ E , the element ω\ ∈ HpdR(U⊗kK) induces an element ωe ∈

HpdR(Sη, π

−1Fe)⊗k K. We show that ωe has weight w for every e ∈ E .Without lost of generality, we assume ωe ∈ Hp

dR(Sη, π−1Fe). If r(e) = 0, that is, e ∈ E \,

then it is trivial as [ω\] ∈ Hprig(E \/K)w. Now we assume that r > 0.

To compute the weight, we use the Frobenius lifting φe : U ′e → Ue where U ′e ⊂ Ue is asmaller open neighborhood of π−1E \e. Assume that U ′e contains (π−1Ee)<ε

′e for some 0 < ε′e <

εe. We introduce more notations as follows: Fix an integer N > |k|(ε′e)−1. Take a totallyramified extension k+/k with an element $+ ∈ k+ such that $rN

+ = $. We consider thefollowing k+-affinoid algebras

F0 = Fe⊗kk+〈τ1, τ−11 , . . . , τr, τ

−1r 〉,

F1 = Fe⊗kk+

⟨te,0

$rN−r+

,$rN−r

+

te,0,te,1$+

,$+

te,1, . . . ,

te,r$+

,$+

te,r

⟩/(te,0 · · · te,r −$),

F2 = Fe⊗kk+

⟨te,0

$rN−r|k|+

,$rN−r|k|+

te,0,te,1

$|k|+

,$|k|+

te,1, . . . ,

te,r

$|k|+

,$|k|+

te,r

⟩/(te,0 · · · te,r −$).

Note that F0 is integrally smooth. We have natural isomorphisms

ρ1 : F1∼−→ F0, te,i 7→ $+τi, 1 ≤ i ≤ r, te,0 7→ $rN−r

+

r∏i=1

τ−1i ;

ρ2 : F2∼−→ F0, te,i 7→ $

|k|+ τi, 1 ≤ i ≤ r, te,0 7→ $

rN−r|k|+

r∏i=1

τ−1i .

For α = 1, 2, we define a formal k+-scheme Sα via the following pullback diagram

Sα//

Se/Fe⊗kk+(5.4)

Spf F α // Spf F e [[te,0, . . . , te,r]]/(te,0 · · · te,r −$)⊗k k+

so that Sα,η is canonically a k+-affinoid domain in U ′α⊗kk+ by our choice of N . Moreover,ρα induces an isomorphism, denoted again by ρα,

ρα : Spf F 0∼−→ Sα

of formal k+-schemes. Properties (a) and (b) of the Frobenius lifting φe implies that itinduces by restriction a morphism φe : S1,η → S2,η, and the composition

ρ−12,η φe ρ1,η : M(F0)→M(F0)

is a Frobenius lifting. By Lemma 4.2, we have induced isomorphismsρ∗1 : H•dR(U ′e⊗kk+,S1,η) ∼−→ H•rig(SpecF0,s/k+),

(ρ−12 )∗ : H•rig(SpecF0,s/k+)→ H•dR(U ′e⊗kk+,S2,η)

On the other hand, by [GK02, Lemma 3], we have isomorphismsrα : H•dR(Sη, π

−1F)⊗k k+∼−→ H•dR(U ′e⊗kk+,Sα,η)

for α = 1, 2 via restriction satisfying r2 = (ρ−12 )∗ ρ∗1 r1. In particular, we may equip

H•dR(U ′e⊗kk+,Sα,η) with a weight decomposition inherited from that of H•dR(Sη, π−1F).


Recall that we have ωe ∈ HpdR(Sη, π

−1Fe) obtained from ω\ by restriction. Let ω0 bethe unique element in Hp

rig(SpecF0,s/k+) such that (ρ−12 )∗ω0 = ωe. By Property (c) of the

Frobenius lifting φe, we have that P (ρ∗1φ∗e (ρ−12 )∗)ω0 = 0. However, ρ−1

2 φeρ1 : Spf F 0 →Spf F 0 is a Frobenius lifting of the Frobenius endomorphism of SpecF0,s over k+ = k.Therefore, ω0 and hence ωe have weight w.Step 6. To conclude the lemma, it remains to show that for every point u ∈ W \, there

exists a fundamental chart (DL′ , (Y ′,D′), (D′, δ′),W ′, α′; y′) of (W \;u) such that α′∗(δ\)∗ω\belongs to Hp

(w)(DL′ , (Y ′,D′), (D′, δ′),W ′).Note that δ$u) belongs to Sη⊗kK. Let e be the image of π(δ\(u)) in E . Then we can find

a fundamental chart (DL′ , (Y ′,D′), (D′, δ′),W ′, α′; y′) of (W \;u) with the induced morphism(W ′, (π−1D′)⊗k′L′)→ (Sη, π

−1Fe)⊗kL′.Then the claim that α′∗(δ$∗ω\ belongs to Hp

(w)(DL′ , (Y ′,D′), (D′, δ′),W ′) follows from Re-mark 5.4, Lemma 5.6, and the claim in Step 5.

Remark 5.13. From the proof of Theorem 5.11, we know that the support of (Ωp,clX /dΩp−1

X )wis contained in the subset x ∈ X | s(x) ≥ 2p− w, s(x) + t(x) ≥ p.

Now we are ready to prove Theorem 1.4. We begin with the case of étale cohomology andthen the case of analytic topology.

Proof of Theorem 1.4 in étale topology. Recall that sheaves like OX , cX , and the de Rhamcomplex (Ω•X , d) are understood in the étale topology.

The direct sum decomposition has been proved in Theorem 5.11 (4). Property (i) followsfrom Theorem 5.11 (3).

For Property (ii), the inclusion ΥpX ⊂ (Ωp,cl

X /dΩp−1X )2p follows from Theorem 5.11 (2) and

Lemma 5.6. Now we show that (Ω1,clX /dOX)2 ⊂ Υ1

X . We check the inclusion on stalks. Takea point x ∈ X with s = s(x) and t = t(x). For every class [ω] in the stalk of (Ω1,cl

X /dOX)2at x, we may find a fundamental chart (DL, (Y ,D), (D, δ),W,Z, α; y) of (X;x) such that[ω] has a representative ω ∈ H1

(2)(DL, (Y ,D), (D, δ),W ). Note that the decomposition (5.2)specializes to the decomposition

H1dR(DL ×L (W, (π−1D)⊗k L)) = H1

rig(D/L)⊕ H1dR(Et

k,$⊗kL).

If the restriction of ω to H1dR(D ×L (W, (π−1D) ⊗k L)) belongs to H1

dR(Etk,$⊗kL), then we

are done. Otherwise, ω restricts to H1rig(D/L)2. It suffices to show that classes in H1

rig(D/L)2can be represented by logarithmic differential of invertible functions étale locally, up to aconstant multiple.

We repeat certain process in Step 2 of the proof of Lemma 5.12 as follows. Choose a smoothk-algebra D\ (of dimension s) such that its $-adic completion is D, a compactification(SpecD\)k → Sk over k, and define S to be the k-scheme Sk

∐(SpecD\)k SpecD\. Then we

obtain a finite extension k′/k, an alteration S\ → Sk′ and a k′-compactification S\ → Swhere S is a projective strictly semistable scheme over k′ such that S\S\ is a strict normalcrossing divisor of S. We may further assume that all irreducible components of S [r]

s aregeometrically irreducible for every r ≥ 0. To ease notation, we replace k by k′ and possiblyL by a finite extension. We may fix an irreducible component E of Ss such that its genericpoint belongs to S\s and maps to the generic point of Ss ' D. Thus there is a unique pointσE ∈ (S\)η such that π(σE) is the generic point of E .

36 YIFENG LIU

Now we apply the setup in the beginning of this section to S and E . Note that E ∩ S\scoincides with E♥. It suffices to show that every class in H1

rig(E♥/k)2 can be represented bythe logarithmic differential of an invertible function on some étale neighborhood of σE . PutE [r] = E ∩ S [r]

s for r ≥ 1. We have E [1]\E [2] = ∐Mi=1 E♥i. Consider the following Gysin exact

sequenceH1

rig(E/k)→ H1rig(E♥/k)→ H2

E [1],rig(E/k)→ H2rig(E/k).(5.5)

We claim that the restriction map H2E [1],rig(E/k) → H2

E [1]\E [2],rig(E\E [2]/k) is an isomorphism.In fact, it fits into another Gysin exact sequence

H2E [2],rig(E/k)→ H2

E [1],rig(E/k)→ H2E [1]\E [2],rig(E\E [2]/k)→ H3

E [2],rig(E/k)

in which H2E [2],rig(E/k) = H3

E [2],rig(E/k) = 0 by the semi-purity theorem (see [Fuj02]6) as thecodimension of E [2] in E is at least 2. By the purity theorem (Lemma 4.1 (3)), we have acanonical isomorphism

H2E [1]\E [2],rig(E\E [2]/k) ' H0

rig(E [1]\E [2]/k) 'M⊕i=1

H0rig(E♥i/k),

where the right-hand side is canonically isomorphic to Z1(E)♥⊗k. Therefore, we may rewrite(5.5) as

H1rig(E/k)→ H1

rig(E♥/k)→ Z1(E)♥ ⊗ k → H2rig(E/k).

in which the last map Z1(E)♥ ⊗ k → H2rig(E/k) is nothing but the Chern class map (see

[Pet03, §6]). As Hirig(E/k) is of pure weight i by Lemma 4.1 (5), we have the isomorphism

H1rig(E♥/k)2

∼−→ ker(Z1(E)♥ ⊗ k → H2rig(E/k)).(5.6)

Now take a divisor D = ∑Mi=1 ci[Ei] with ci ∈ Z such that its Chern class in H2

rig(E/k) 'H2

cris(E/k) is trivial. Then by [Pet03, §5] and [Del81, Remarque 3.5] there exists some integerµ > 0 such that µD is algebraically equivalent to zero, and in particularOE(µD) is an elementin Pic0

E/k(k). Since Pic0E/k is a projective scheme over the finite field k, one may replace µ by

some multiple such that OE(µD) is a trivial line bundle. Therefore, there exists a functionf ∈ O∗E(E♥) with div(f) = µD. We may choose a finite set of affine open subschemes SpecRi

of S such that (SpecRi)s form an open covering of E♥ and f |(SpecRi)s lifts to an invertiblefunction fi ∈ R∗i . Note that for every i, Ui := (SpecRi)an ∩ π−1E is an open neighborhoodof σE . Since on Ui ∩ Uj we have that |fi/fj| is constant, the collection dfi

fi gives rise to

a section of Υ1San in an open neighborhood of π−1E♥, whose image is µD under the map

H1rig(E♥/k)→ H2

E [1],rig(E/k) ' Z1(E)♥ ⊗ k by Proposition 4.6. Thus, (ii) is proved.For Property (iii), when X has dimension 1, it follows from (the proof of) [Ber07, Theorem

4.3.1]. In general, it suffices to show that (Ω1,clX /dOX)1 ⊂ ΨX by [Ber07, Theorem 4.5.1 (i)]

and Theorem 5.11 (4). However, this follows from the definition of ΨX , Theorem 5.11 (5),and the case of curves.

Proof of Theorem 1.4 in analytic topology. Now sheaves like OX , cX , ΥpX , and the de Rham

complex (Ω•X , d) are understood in the analytic topology. The corresponding objects in theétale topology will be denoted by OXet , cXet , Υp

Xet, and (Ω•Xet

, d).6Although in [Fuj02], the author did not prove the semi-purity theorem for the rigid cohomology. But it

is a formal argument from the purity theorem, which is known for the rigid cohomology.


Note that we have a canonical morphism ν : Xet → X of sites, and OX = ν∗OXet , cX =ν∗cXet , Ωp

X = ν∗ΩpXet

for every p ≥ 0. We claim that the canonical map

Ωp,clX /dΩp−1

X → ν∗(Ωp,clXet/dΩp−1

Xet)

is an isomorphism. It will follow from:(a) Ωp,cl

X = ν∗Ωp,clXet

as subsheaves of ΩpX ;

(b) dΩp−1X = ν∗dΩp−1

Xetas subsheaves of Ωp

X ; and(c) R1ν∗dΩp−1

Xet= 0.

Assertion (a) is obvious. Both (b) and (c) will follow from the general fact that Riν∗F = 0for i > 0 and any sheaf of Q-vector spaces F on Xet. In fact for every x ∈ X, wehave (Riν∗F )x = Hi(H(x), i−1

x F ), where H(x) is the completed residue field of x andix : M(H(x))→ X is the canonical morphism, and we know that the profinite Galois coho-mology Hi(H(x), i−1

x F ) is torsion hence trivial for i > 0.Now for w ∈ Z, we define (Ωp,cl

X /dΩp−1X )w = ν∗(Ωp,cl

Xet/dΩp−1

Xet)w. Then we have a decompo-

sitionΩp,clX /dΩp−1

X =⊕w∈Z

(Ωp,clX /dΩp−1

X )w,

stable under base change and functorial in X and satisfying Property (i).For Property (ii), we have the inclusion Υp

X ⊂ ν∗ΥpXet

as subsheaves of Ωp,clX /dΩp−1

X , whichis canonically isomorphic to ν∗(Ωp,cl

Xet/dΩp−1

Xet). Thus, we have the inclusion of sheaves Υp

X ⊂(Ωp,cl

X /dΩp−1X )2p. When p = 1, we have to show that ν∗Υ1

Xet⊂ Υ1

X . We check this on the stalkat an arbitrary point x ∈ X. Take an element [ω] in (ν∗Υ1

Xet)x. We may assume that it has a

representative ω ∈ Ω1X(U) for some open neighborhood U of x that satisfies α∗ω = df ′

f ′+ dg′

for some finite étale surjective morphism α : U ′ → U and f ′ ∈ O∗(U ′), g′ ∈ O(U ′). Then wehave ω = deg(α)−1 df

f+ dg where f (resp. g) is the multiplicative (resp. additive) trace of f ′

(resp. g′) along α.

6. Cohomological triviality before tropicalization

In this section, we study the sheaf ΥpX (in analytic topology) in more details. We show that

it also has a canonical rational structure. Then we construct the so-called log-differentialcycle class map and prove Theorem 6.6.

Definition 6.1. LetK be a non-Archimedean field of characteristic zero. LetX be a smoothK-analytic space. For every p ≥ 0, we define a (Q-linear) map

λpX : K pX → Ωp,cl

X /dΩp−1X

of sheaves on X as follows. For a symbol f1, . . . , fp ∈ K pX (U) with f1, . . . , fp ∈ O∗X(U),

we putλpX(f1, . . . , fp) = df1

f1∧ · · · ∧ dfp

fp,

where the right-hand side is regarded as an element in Ωp,clX (U) and hence in (Ωp,cl

X /dΩp−1X )(U).

It is easy to see that λpX factors through the relations of Milnor K-theories, and thus inducesa map of corresponding sheaves. Finally, put

L pX = K p

X/ kerλpX .

38 YIFENG LIU

It is canonically a subsheaf of Ωp,clX /dΩp−1

X .

Proposition 6.2. Let K be a non-Archimedean field embeddable into CF, and X a smoothK-analytic space. Then the canonical map L p

X ⊗Q cX → ΥpX is an isomorphism of sheaves

on X for every p ≥ 0.

Proof. By definition, it suffices to show that the map L pX ⊗Q cX → Υp

X is injective on stalks.Thus we fix a point x ∈ X with s = s(x) and t = t(x). Take an element

F =M∑l=1

blλpX(F l) ∈ cX(U)⊗Q L p

X(U)

such that F = 0 in ΥpX(U), where U is a connected open neighborhood of x, and bl ∈ cX(U),

F l ∈ K pX (U). It suffices to show that possibly after shrinking U , the elements λpX(F l) are

linearly dependent in ΥpX(U) over Q.

Write F l = ∑Nlα=1 c

lαf lα1, . . . , f

lαp where clα ∈ Q and f lαβ ∈ O∗X(U). We apply Lemma 6.3

to the finite collection f lαβ and adopt the notation there. Then by Künneth decomposition,Example 4.3, and Proposition 4.6, we have for every subset Γ ⊂ 1, . . . , t with |Γ| ≤ p that

M∑l=1

bl

Nl∑α=1

clα∑

ι : Γ→1,...,pε(ι)

∏γ∈Γ

dlαι(γ)γcl♥ ∧β 6∈ι(Γ)

div(glαβ) ∈ ⊕

I,|I|=p−|Γ|H0

rig(E♥I /L)(6.1)

vanishes, for some finite extension of non-Archimedean fields L/cX(U). Here, ε(ι) ∈ ±1is certain sign determined by ι, and the multiple wedge product is taken in the increasingorder of the relevant indices.

Note that H0rig(E♥I /L) is canonically isomorphic to Q⊕π0(E♥I ) ⊗Q L, and for every l,

Nl∑i=1

clα∑ι

ε(ι)∏γ∈Γ

dlαι(γ)γcl♥ ∧β 6∈ι(Γ)

div(glαβ) ∈ ⊕

I,|I|=p−|Γ|Q⊕π0(E♥I ).

Thus, there exist b′l ∈ Q, not all being zero, such that (6.1) vanishes for every Γ if we replacebl by b′l.

This implies that there is an open neighborhood V ′ of y contained in V such that µ∗λpX(F ′)is exact where F ′ = ∑M

l=1 b′lλpX(F l), by Lemma 4.5. By shrinking V ′, we may assume that the

restricted morphism µ : V ′ → U ′ is finite étale, where U ′ := µ(V ′) is an open neighborhoodof x in U . Write µ∗λpX(F ′)|V ′ = dω′ for some (p − 1)-form ω′ on V ′. Then λpX(F ′)|U ′ =(deg µ)−1dω, where ω is the trace of ω′ along µ : V ′ → U ′. The proposition follows.

Lemma 6.3. Let K be a non-Archimedean field embeddable into CF, and X a smooth K-analytic space. Let x ∈ X be a point with s = s(x) and t = t(x). For finitely many elementsfα ∈ O∗X(U) where U is an open neighborhood of x in X, there exist

• a proper strictly semistable scheme S over k of dimension s such that every irre-ducible component of S [r]

s is geometrically irreducible for every r ≥ 0, where k is afinite extension of QF,• an irreducible component E of Ss,• an open neighborhood W of (π−1E♥)⊗kL in San

L where L is a finite extension of Kcontaining k, and E♥ = E\S [1]

s ,• a closed subset Z of Sk of dimension at most s− 1,


• a point y ∈ V := DL ×L∏tγ=1B(0; rγ, Rγ)L ×LW which projects to σE in W , where

DL is a poly-disc of dimension s over L, and σE ∈ W is the unique point whosereduction is the generic point of E

L,

• a morphism µ : V → U that is étale away from DL×L∏tγ=1 B(0; rγ, Rγ)L×L(W∩Zan

L ),such that µ(y) = x,• integers dα1, . . . , dαt and functions gα ∈ O∗(W ), such that

µ∗dfαfα−

d(ν∗gα

∏tγ=1 T

dαγγ

)ν∗gα

∏tγ=1 T

dαγγ

is an exact 1-form on V for every α. Here, Tγ is the coordinate function onB(0; rγ, Rγ)L for 1 ≤ γ ≤ t, which will be regarded as a function in O∗(V ) viathe obvious pullback; and ν : V → W is the projection morphism.

In particular, |µ∗fα · (ν∗gα∏tγ=1 T

dαγγ )−1| is a constant on V .

Proof. The existence of the data except for the last part follows from [Ber07, Propositions2.1.1, 2.3.1]. The existence of integers dα1, . . . , dαt and functions gα ∈ O∗(W ) (after possiblyshrinking W ) is fulfilled by (5.2), Example 4.3 and Theorem 1.4 (ii).

The last assertion is due to [Ber07, Theorem 4.3.1 (i)].

Definition 6.4 (Log-differential cycle class map). Let K be a non-Archimedean field ofcharacteristic zero and X a smooth scheme of finite type overK. We define the log-differentialcycle class map cllog to be the composition

CHp(X )Qcluniv−−−→ Hp(X ,K p

X )→ Hp(X an,K pX an)

Hp(X an,λpXan )−−−−−−−−→ Hp(X an,L p

X an).Suppose that X is a geometrically connected proper smooth scheme over K of dimension

n. We have a pairing〈 , 〉X : Hp(X an,L p

X an)× Hn−p(X an,L n−pX an )→ K(6.2)

coming from the composite mapHp(X an,L p

X an)× Hn−p(X an,L n−pX an )→ Hn(X an,L n

X an)→ Hn(X an,ΩnX an/dΩn−1

X an ) ' K

in which the last isomorphism comes from the following lemma.Lemma 6.5. Let K be a non-Archimedean field of characteristic zero. Let X be a geomet-rically connected proper smooth scheme over K of dimension n. Then we have a canonicalisomorphism Hn(X an,Ωn

X an/dΩn−1X an ) ' K.

Proof. Put X = X an. By the spectral sequence Ep,q2 = Hp(X,Ωq,cl

X /dΩq−1X ) ⇒ Hp+q(X,Ω•X)

and the GAGA comparison isomorphism H•dR(X) ' H•dR(X/K), it suffices to show thatHi(X,F ) = 0 for i > n and every abelian sheaf F on X, which holds by [Ber93, Proposition1.2.18]. Then we have canonical isomorphisms Hn(X,Ωn

X/dΩn−1X ) ' H2n

dR(X/K) ' K.

The following theorem is an analogous version of Theorem 1.8 for cllog. We will eventuallyreduce Theorem 1.8 to this one.Theorem 6.6. Let k be a finite extension of QF and X a geometrically connected propersmooth scheme over k of dimension n. Let Z be an algebraic cycle of X of codimension psuch that cldR(Z) = 0. Then

〈cllog(Z), ω〉X = 0for every ω ∈ Hn−p(X an,L n−p

X an ).

40 YIFENG LIU

Before the proof, we review some facts about cup products from [SP, 01FP]. Let X bea topological space, k a field, n ≥ 0 an integer. Let Ω be a sheaf of k-vector spaces on X.Suppose that we have two bounded complexes F •,G • of sheaves of k-vector spaces on X,with a map of complexes of sheaves of k-vector spaces

χ : Tot(F • ⊗k G •)→ Ω[n].Then we have a bilinear pairing

∪χ : Hi(X,F •)× H2n−i(X,G •)→ H2n(X,Ω[n]) = Hn(X,Ω)for every i ∈ Z. Now suppose that we have four bounded complexes F •

1 ,F•2 ,G

•1 ,G

•2 of

sheaves of k-vector spaces on X, maps α1 : F •1 → F •

2 , α2 : G •2 → G •1 , andχ1 : Tot(F •

1 ⊗k G •1 )→ Ω[n], χ2 : Tot(F •2 ⊗k G •2 )→ Ω[n],

such that χ1 (idF•1⊗ α2) = χ2 (α1 ⊗ idG •2

). Then we have the following commutativediagram

Hi(X,F •1 )

Hi(X,α1)

× H2n−i(X,G •1 )∪χ1 // Hn(X,Ω)

Hi(X,F •2 ) × H2n−i(X,G •2 )

H2n−i(X,α2)

OO

∪χ2 // Hn(X,Ω)

(6.3)

for every i ∈ Z.Proof of Theorem 6.6. Put X = X an. Note that we have the following commutative diagram

Hp(X,T pX)× Hn−p(X,T n−p

X ) ∪ //

Hn(X,T nX )

Hp(X,Ωp,clX /dΩp−1

X )× Hn−p(X,Ωn−p,clX /dΩn−p−1

X ) ∪ // Hn(X,ΩnX/dΩn−1

X ).

The goal is to show that cllog(Z) ∪ ω equals zero in Hn(X,ΩnX/dΩn−1

X ). Denote by ζ theimage of cllog(Z) in Hp(X,Ωp,cl

X /dΩp−1X ), and regard ω as in Hn−p(X,Ωn−p,cl

X /dΩn−p−1X ). We

show ζ ∪ ω = 0.To explain the idea, we first give a “fake” proof. We have the conjugate spectral sequence

Ep,qr associated to the de Rham complex (Ω•X , d), abutting to H•dR(X) = H•(X,Ω•X) with

Ep,q2 = Hp(X,Ωq,cl

X /dΩq−1X ) =

2q⊕w=q

Hp(X, (Ωq,clX /dΩq−1

X )w).

We make the following assumption: the differential map dp,qr of the spectral sequence pre-serves the above direct sum decomposition with respect to the weight w for r ≥ 2. Then wehave an induced map Hp(X, (Ωq,cl

X /dΩq−1X )2q)→ Hp+q

dR (X). It is not hard to see that the pair-ing (6.2) factors through the pairing H2p

dR(X) × H2n−2pdR (X) → K for de Rham cohomology.

Therefore, the conclusion follows.Unfortunately, we do not know whether the above assumption is true or not. Therefore,

we need to consider a different de Rham complex in order to have certain Frobenius action onthe entire complex so that the weight decomposition will be carried to the entire conjugatespectral sequence. The ad hoc de Rham complex uses log crystalline sites, which will beconstructed through Steps 1–3 below. Steps 4–5 complete the remaining argument for theproof.


Step 1. To construct the ad hoc de Rham complex, we need the adic topology of X. By[Sch12, Theorem 2.24], we may associate to a K-analytic space X an adic space Xad, andwe have a canonical continuous map γX : Xad → X of topological spaces which makes X amaximal Hausdorff quotient of Xad. Let (Ω•Xad , d) be the de Rham complex on Xad. Thenwe have a canonical map

γ−1X (Ω•X , d)→ (Ω•Xad , d)

of complexes of sheaves of k-vector spaces on Xad. Denote by ζad (resp. ωad) the image of ζ(resp. ω) under the canonical map

Hi(X,Ωi,clX /dΩi−1

X )→ Hi(Xad,Ωi,clXad/dΩi−1

Xad)for i = p (resp. i = n− p). Note that when i = n, the above map is an isomorphism, by thesame argument for Lemma 6.5.

We claim that there exists an alteration f : X ′ → X possibly after replacing k by a finiteextension, such that f ∗ωad is in the image of the canonical map

H2n−2p(X ′ad, τ≤n−pΩ•X′ad)→ Hn−p(X ′ad,Ωn−p,clX′ad /dΩn−p−1

X′ad ),(6.4)where X ′ = X ′an.

Assuming the above claim, we deduce the theorem as follows. Applying (6.3) to X ′ad andthe sheaf Ω := Ωn

X′ad/dΩn−1X′ad , we obtain the following commutative diagram

Hp(X ′ad,Ωp,clX′ad/dΩp−1

X′ad)α1

× Hn−p(X ′ad,Ωn−p,clX′ad /dΩn−p−1

X′ad ) // Hn(X ′ad,Ω)

H2p(X ′ad, τ≥pΩ•X′ad) × H2n−2p(X ′ad, τ≤n−pΩ•X′ad)

α2

OO

β2

// Hn(X ′ad,Ω)

H2p(X ′ad,Ω•X′ad)

β1

OO

× H2n−2p(X ′ad,Ω•X′ad) // Hn(X ′ad,Ω)

in which the maps among various complexes of sheaves are defined in the obvious way. Bythe above claim, there exists ω′ ∈ H2n−2p(X ′ad, τ≤n−pΩ•X′ad) such that α2(ω′) = f ∗ωad. Thus,

f ∗ζad ∪ f ∗ωad = f ∗ζad ∪ α2(ω′) = α1(f ∗ζad) ∪ ω′.(6.5)Note that we have the following commutative diagram

CHp(X ′)Q //

cldR

Hp(X ′,Ωp,clX′ /dΩp−1

X′ ) //

Hp(X ′ad,Ωp,clX′ad/dΩp−1

X′ad)α1

H2p(X ′,Ω•X ′) // H2p(X ′ad,Ω•X′ad) β1 // H2p(X ′ad, τ≥pΩ•X′ad)

by Remark 2.5. As cldR(Z) = 0, we have cldR(f ∗Z) = 0. Thus α1(f ∗ζad) = 0 and f ∗ζ∪f ∗ω =0 by (6.5). However, as f ∗ : Hn(Xad,Ωn

Xad/dΩn−1Xad)→ Hn(X ′ad,Ωn

X′ad/dΩn−1X′ad) is injective, we

have ζ ∪ ω = 0. The theorem is proved.Step 2. Now we focus on the claim in Step 1. We first introduce some new sheaves on

X. For a (proper flat) integral model Y of X , define K pX,Y to be the sheaf on Ys associated

to the presheafU 7→ lim−→

π−1U⊂UKMp (OX(U))⊗Q, U ⊂ Ys

42 YIFENG LIU

where the colimit is taken over all open neighborhoods U of π−1U in X. Denote γY : Xad →Ys the induced continuous map. We have a canonical map K p

X,Y → γY∗γ−1X K p

X induced fromthe canonical map lim−→π−1U⊂U K

Mp (OX(U)) ⊗ Q → lim−→π−1U⊂U K p

X (U). We claim that theinduced map

lim−→Yγ−1Y K p

X,Y → γ−1X K p

X ,(6.6)

where the filtered colimit is taken over all integral models Y of X, is an isomorphism ofsheaves on Xad. In fact, denote by K pre,p

X the presheaf U 7→ KMp (OX(U)) ⊗ Q on X.

Then K pX,Y is simply (γY∗γ−1

X K pre,pX )+, where + denotes sheafification. Since sheafification

commutes with pullback and taking colimit, we have

lim−→Yγ−1Y K p

X,Y∼−→ lim−→

Yγ−1Y (γY∗γ−1

X K pre,pX )+ ∼−→

lim−→Yγ−1Y γY∗γ

−1X K pre,p

X

+

.

By [Sch12, Theorem 2.22], we have a cofiltered limit Xad ' lim←−Y Ys induced by γY . ApplyingLemma 6.7 below to the presheaf γ−1

X K pre,pX on Xad, we havelim−→

Yγ−1Y γY∗γ

−1X K pre,p

X

+∼−→ (γ−1

X K pre,pX )+ ∼−→ γ−1

X (K pre,pX )+ = γ−1

X K pX .

Thus (6.6) is an isomorphism.Put Ω†,•X,Y = γY∗γ

−1X Ω•X . Then we have a complex of sheaves of k-vector spaces (Ω†,•X,Y , d)

on Ys. Again by Lemma 6.7, the canonical maplim−→Yγ−1Y (Ω†,•X,Y , d)→ γ−1

X (Ω•X , d)(6.7)

is an isomorphism.For every p ≥ 0, we have a canonical map

λpX,Y : K pX,Y → Ω†,p,cl

X,Y /dΩ†,p−1X,Y

similar to Definition 3.2. Denote by L pX,Y the image sheaf of λpX,Y in the above map. Passing

to the quotient of isomorphisms (6.6) and (6.7), we obtain a canonical isomorphismlim−→Yγ−1Y L p

X,Y ' γ−1X L p

X .(6.8)

Step 3. By [Ber93, Proposition 1.3.6 (i)] and the fact that the topos on Xad is equivalentto the G-topos on X under γX , we know that the canonical map

Hn−p(X,L n−pX )→ Hn−p(Xad, γ−1L n−p

X )is an isomorphism. On the other hand, by the isomorphism (6.8) and [SP, 0A37], thecanonical map

lim−→Y

Hn−p(Ys,L n−pX,Y )→ Hn−p(Xad, γ−1L n−p

X )

is an isomorphism. Together, we may assume that ω is in the image of the mapHn−p(Ys,L n−p

X,Y )→ Hn−p(X,L n−pX ).

for some integral model Y . By [dJ96, Theorem 8.2], possibly after replacing k by a finiteextension, we have a projective strictly semistable scheme Y ′ over k with an alteration


f : Y ′ → Y . Put X ′ = Y ′an. If we put Ω•X′,Y ′ = γY ′∗Ω•X′ad , then (Ω•X′,Y ′ , d) is a complex ofsheaves of k-vector spaces on Y ′s and we have a canonical map

(Ω†,•X′,Y ′ , d)→ (Ω•X′,Y ′ , d).(6.9)Note that we have the exact sequence

0→ τ≤n−p−1Ω•X′ad → τ≤n−pΩ•X′ad → Ωn−p,clX′ad /dΩn−p−1

X′ad [p− n]→ 0of complexes of sheaves on X ′ad, which induces an short exact sequenceH2n−2p(X ′ad, τ≤n−pΩ•X′ad)→ Hn−p(X ′ad,Ωn−p,cl

X′ad /dΩn−p−1X′ad )→ H2n−2p+1(X ′ad, τ≤n−p−1Ω•X′ad).

Thus ω is in the image of (6.4) if and only if ω maps to zero in H2n−2p+1(X ′ad, τ≤n−p−1Ω•X′ad).Since we have assumed that ω comes from Hn−p(Ys,L n−p

X,Y ), it suffices to show that thecomposite map

Hn−p(Y ′s,Ln−pX′,Y ′)→ Hn−p(X ′,L n−p

X′ )→ Hn−p(X ′ad,Ωn−p,cl

X′ad /dΩn−p−1X′ad )→ H2n−2p+1(X ′ad, τ≤n−p−1Ω•X′ad)

vanishes. However, the above composite map coincides with the following one

Hn−p(Y ′s,Ln−pX′,Y ′)→ H2n−2p+1(Y ′s, τ≤n−p−1Ω†,•X′,Y ′)

→ H2n−2p+1(Y ′s, τ≤n−p−1Ω•X′,Y ′)→ H2n−2p+1(X ′ad, τ≤n−p−1Ω•X′ad).Therefore, the claim in Step 1 is reduced to the vanishing of the map

Hn−p(Y ′s,Ln−pX′,Y ′)→ H2n−2p+1(Y ′s, τ≤n−p−1Ω•X′,Y ′).(6.10)

Step 4. The advantage of (Ω•X′,Y ′ , d) is that the entire complex admits a canonicalFrobenius action as we explain as follows. Fix a uniformizer of k; let Y ′× and Spec(k)× bethe log schemes equipped with the canonical log structure as in [HK94, (2.13.2)]. Then theinduced morphism Y ′× → Spec(k)× is log smooth as Y ′ is strictly semistable over k. LetY ′×s → Spec(k)× be the induced morphism of log schemes equipped with the pullback logstructure as in [HK94, (2.13.2)]. Finally, let Spf W (k)× be the formal log scheme whose logstructure is the canonical lifting of Spec(k)× as in [HK94, Definition (3.1)]. Here, we use theZariski topology in the construction of log schemes and log crystal sites instead of the étaleone in [HK94]. There is a canonical morphism u : (Y ′×s / Spf W (k)×)log-cris → Y ′s of sites. By[HK94, (5.4) & Proposition (2.20)], we have a canonical isomorphism

Ru∗Olog-crisY ′×s / Spf W (k)×

⊗W (k) k ' (Ω•X′,Y ′ , d)

in the derived category of abelian sheaves on Y ′s, where Olog-crisY ′×s / Spf W (k)×

denotes the structuresheaf in the log crystal site. Note that when applying [HK94, Proposition (2.20)], we use thetrivial covering as Y ′× → Spec(k)× is log smooth. Since OY ′×s /Spf W (k)× admits a Frobeniusaction over Spec k, we obtain a Frobenius action on the entire complex (Ω•X′,Y ′ , d) in thederived category.

For w ∈ Z, denote by (Ωp,clX′,Y ′/dΩp−1

X′,Y ′)w the maximal subsheaf of Ωp,clX′,Y ′/dΩp−1

X′,Y ′ generatedby sections of generalized weight w. We claim that(a) the image of the canonical map K p

X′,Y ′ → L pX′,Y ′ → Ωp,cl

X′,Y ′/dΩp−1X′,Y ′ is contained in the

subsheaf (Ωp,clX′,Y ′/dΩp−1

X′,Y ′)2p for every p;

44 YIFENG LIU

(b) the image of the canonical map Ω†,p,clX′,Y ′/dΩ†,p−1

X′,Y ′ → Ωp,clX′,Y ′/dΩp−1

X′,Y ′ is contained in thesubsheaf ⊕2p

w=0(Ωp,clX′,Y ′/dΩp−1

X′,Y ′)w for every p.Then the triviality of the map (6.10) follows easily from an argument of spectral sequences.In fact, we have a map of spectral sequences †Ep,q

r → Ep,qr abutting to the canonical map

H•(Y ′s,Ω†,•X′,Y ′)→ H•(Y ′s,Ω•X′,Y ′) induced from (6.9), whose second page consists of canonical

maps Hp(Y ′s,Ω†,q,clX′,Y ′/dΩ†,q−1

X′,Y ′)→ Hp(Y ′s,Ωq,clX′,Y ′/dΩq−1

X′,Y ′). Consider the following commutativediagram

Hn−p(Y ′s,Ln−pX′,Y ′) // †En−p,n−p

2//

†dn−p,n−p2

En−p,n−p2

dn−p,n−p2

†En−p+2,n−p−12

// En−p+2,n−p−12 .

By (a), (b) and the fact that (Ω•X′,Y ′ , d) has a Frobenius action, the composite map

Hn−p(Y ′s,Ln−pX′,Y ′)→ En−p+2,n−p−1

2

obtained from the above diagram factors through

(En−p,n−p2 )2n−2p →

2n−2p−2⊕w=0

(En−p+2,n−p−12 )w,

hence must be zero. By a similar argument, we know that for every r ≥ 2, the (inductivelydefined) map

Hn−p(Y ′s,Ln−pX′,Y ′)→ En−p,n−p

rdn−p,n−pr−−−−−→ En−p+r,n−p−r+1

r

vanishes. Therefore, (6.10) vanishes as a consequence of spectral sequence.Step 5. The last step is devoted to the proof of the two claims (a) and (b) in Step 4. We

remark that they are not formal consequences of Theorem 1.4.By definition, Ωp,cl

X′,Y ′/dΩp−1X′,Y ′ is the sheaf on Y ′s associated to the presheaf U 7→ Hp

dR(π−1U),and Ω†,p,cl

X′,Y ′/dΩ†,p−1X′,Y ′ is the sheaf on Y ′s associated to the presheaf U 7→ Hp

dR(X ′, π−1U) by[Ber07, Lemma 5.2.1]. We check (a) and (b) on stalks and thus fix a point x ∈ Y ′s.

To prove (a), it suffices to consider the case where p = 1. Let U ⊂ Y ′ be an openaffine neighborhood of x. Take f ∈ O∗(X ′, π−1Us); we have to show that the image ofdff

in H1dR(π−1Us) is of generalized weight 2 for a possibly smaller open neighborhood U of

x. First, by [Gub13, Proposition 7.2] and possibly shrinking U , we may replace f by therestriction of an (algebraic) function f ∈ O(Uk) without changing |f |, hence the image of df

f

in H1dR(X ′, π−1Us) by [Ber07, Theorem 4.2.1], and the image of df

fin H1

dR(π−1Us). Since Y ′is projective, we may choose a closed embedding Y ′ → PN

k into a projective space. Choosean open affine neighborhood V of x in PN

k such that V ∩ Y ′ ⊂ U and f |V∩Y ′ = g|V∩Y ′ forsome g ∈ O∗(Vk). Since PN

k is smooth, by Lemma 5.6, dggbelongs to H1

rig(Vs/k)2 and thus itsimage in H1

dR(π−1Vs) is of generalized weight 2. By functoriality of log crystal sites for themorphism Y ′ → PN

k , we conclude that the image of dff

in H1dR(π−1(V ∩Y ′)s) is of generalized

weight 2. Here, π−1(V ∩ Y ′)s is the inverse image in X ′.Claim (b) is a consequence of [GK05, Theorem 0.1] and [Chi98, Theorem 2.3]. In fact,

we have a functorial map of spectral sequences ′Ep,qr → ′′Ep,q

r abutting to H•dR(X ′, π−1U) →


H•dR(π−1U) with the first page being′Ep,q

1 = Hqrig(U (p)

s / Spf W (k)×)⊗W (k) k →

′′Ep,q1 = Hq

log-cris(U (p)s / Spf W (k)×)⊗

W (k) k,

where U (p)s is the disjoint union of irreducible components of U [p]

s , equipped with the inducedlog structure from Y ′×s . By [GK05, Theorem 3.1, Lemma 4.6] and [Chi98, Theorem 2.3], weknow that the weights of (the finite dimensional k-vector space) ′Ep,q

1 are in the range [p, 2p],and thus the weights of Hp

dR(X ′, π−1U) are in the range [0, 2p].

Lemma 6.7. Let I be a cofiltered category. Let i 7→ Xi be a diagram of topological spacesover I. Let X = lim←−iXi be the limit with projection maps γi : X → Xi. Let P be a presheafon X. Then the canonical map (

lim−→i

γ−1i γi∗P

)+

→P+

is an isomorphism. Here, we regard γ−1i and γi∗ as pullback and pushforward of presheaves

respectively, and + denotes sheafification.

Proof. Consider the collection B of open subsets of X of the form γ−1i Ui for i ∈ I and an

open subset Ui of Xi. Then B is a basis of the topology on X. It suffices to show that forevery U ∈ B, the canonical map(

lim−→i

γ−1i γi∗P

)(U)→P(U)

is an isomorphism. This is obvious as I is cofiltered.

7. Cohomological triviality for tropical cycle classes

In this section, we study tropical cycle class maps and prove Theorem 1.8, by first estab-lishing the relation of maps τ pX and λpX in the following result.

Theorem 7.1. Let K be a non-Archimedean field embeddable into CF, and X a smoothK-analytic space. Then ker τ pX = kerλpX . In other words, we have a canonical isomorphism

T pX ' L p

X

of sheaves on X for every p ≥ 0.

Proof. It suffices to check the equality on stalks. Thus we fix a point x ∈ X with s = s(x)and t = t(x).

Let U be an open neighborhood of x. Take an element

F =N∑α=1

cαfα1, . . . , fαp ∈ K pX (U)

where cα ∈ Q and fαβ ∈ O∗X(U). We apply Lemma 6.3 to the finite collection fαβ andadopt the notation there. In particular, |µ∗fαβ · (ν∗gαβ

∏tγ=1 T

dαβγγ )−1| is a positive constant

on V , which we denote by cαβ.Define three tropical charts as follows.

46 YIFENG LIU

• The first one uses fαβ (1 ≤ α ≤ N, 1 ≤ β ≤ p), which induce a moment morphismU → (Gan

m,K)Np, and thus a tropicalization map

tropU : U → (Ganm,K)Np − log | |−−−−→ RNp.

• The second one uses functions gαβ (1 ≤ α ≤ N, 1 ≤ β ≤ p), which induce a momentmorphism W → (Gan

m,L)Np, and thus a tropicalization map

tropW : W → (Ganm,L)Np − log | |−−−−→ RNp.

• The third one uses functions ν∗gαβ (1 ≤ α ≤ N, 1 ≤ β ≤ p) and Tγ (1 ≤ γ ≤ t),which induce a moment morphism V → (Gan

m,L)Np+t, and thus a tropicalization map

tropV : V → (Ganm,L)Np+t − log | |−−−−→ RNp+t.

We have a commutative diagram

WtropW // RNp

V

µ

ν

OO

tropV // RNp+t

µ

ν

OO

UtropU // RNp

in which µ sends a point (xαβ, xγ) ∈ RNp+t = RNp ×Rt to (yαβ) where

yαβ = − log cαβ + xαβ +t∑

γ=1dαβγxγ,

and ν is the projection onto the first Np factors. Note that

τ pX(F ) =N∑α=1

cα

p∧β=1

d′yαβ,

and thus

µ∗τ pX(F ) =N∑α=1

cα

p∧β=1

d′xαβ +t∑

γ=1dαβγd′xγ

as a (q, 0)-superform on RNp+t.

It is elementary to see that

µ∗λpX(F ) =∑

Γ⊂1,...,t,|Γ|≤pν∗ωΓ ∧

∧γ∈Γ

dTγTγ

,where

ωΓ =N∑α=1

cα∑

ι : Γ→1,...,qε(ι)

∏γ∈Γ

dαι(γ)γ

∧β 6∈ι(Γ)

dgαβgαβ

.Here, ε(ι) ∈ ±1 is certain sign determined by ι, and all multiple wedge products are takenin the increasing order of the relevant indices.


On the tropical side, we have

µ∗τ pX(F ) =∑

Γ⊂1,...,t,|Γ|≤pν∗ωΓ ∧

∧γ∈Γ

d′xγ

,where

ωΓ =N∑α=1

cα∑

ι : Γ→1,...,qε(ι)

∏γ∈Γ

dαι(γ)γ

∧β 6∈ι(Γ)

d′xαβ

.We show (kerλpX)x ⊂ (ker τ pX)x. We assume that λpX(F ) is an exact form on U , and we

need to show that τ pX(F ) = 0 on a possibly smaller open neighborhood of x. Since µ∗λpX(F )is exact, by Example 4.3, we know that ωΓ induces the zero class in Hq−|Γ|

dR (W, (π−1E♥)⊗kL)for every Γ. Therefore by Lemma 7.2 below, there exists a compact neighborhood W ′ of(π−1E♥)⊗kL contained in W such that ωΓ = 0 on tropW (W ′) for every Γ. Therefore,µ∗τ pX(F ) = 0 on µ−1W ′, and we are done since µ is an affine surjection.

We show (ker τ pX)x ⊂ (kerλpX)x. We assume that τ pX(F ) = 0 on U . Then by the similarargument as above, we may conclude that there is an open neighborhood V ′ of y contained inV such that µ∗λpX(F ) is exact. By shrinking V ′, we may assume that the restricted morphismµ : V ′ → U ′ is finite étale, where U ′ := µ(V ′) is an open neighborhood of x in U . Writeµ∗λpX(F )|V ′ = dω′ for some (p − 1)-form ω′ on V ′. Then λpX(F )|U ′ = (deg µ)−1dω, where ωis the trace of ω′ along µ : V ′ → U ′. So λpX(F ) is zero in the stalk at x.

The theorem is proved.

Lemma 7.2. Let k be a finite extension of QF and K a non-Archimedean field containingk. Suppose that we have

• a proper strictly semistable scheme S over k of dimension s such that every irre-ducible component of S [r]

s is geometrically irreducible for every r ≥ 0,• an irreducible component E of Ss,• an open neighborhood W of (π−1E♥)⊗kK in San

K where E♥ = E\S [1]s ,

• invertible functions g1, . . . , gh ∈ O∗(W, (π−1E♥)⊗kK), which induce the tropicaliza-tion map tropW : W → Rh,• c1, . . . , cN ∈ Q and β1, . . . , βN : 1, . . . , p → 1, . . . , h.

Put

ω =N∑α=1

cα

(dgβα(1)

gβα(1)∧ · · · ∧

dgβα(p)

gβα(p)

)and

ω =N∑α=1

cα(d′xβα(1) ∧ · · · ∧ d′xβα(p)

),

where x1, . . . , xh are the natural coordinates on Rh.Then ω induces the zero class in Hp

dR(W, (π−1E♥)⊗kK) if and only if ω = 0 on tropW (W ′)for some compact neighborhood W ′ of (π−1E♥)⊗kK contained in W .

Proof. We adopt the notation in §4. Let e ∈ E be a point. There is a unique subsetI(e) of 1, . . . ,M such that e ∈ E♥I(e). We choose an affine open neighborhood Se of etogether with functions fe,0 and fe,i in O(Se) for i ∈ I(e) such that Se ∩ EI is defined bythe ideal (fe,0, fe,i | i ∈ I) for every subset I ⊂ I(e). We may further assume that Se ∩ EIis irreducible for every I. Put Ee = Se ∩ E and Ee,i = Se ∩ Ei. For 1 ≤ β ≤ h, suppose

48 YIFENG LIU

div(gβ)|Ee = ∑i∈I(e) aβ,e,iEe,i for aβ,e,i ∈ Z. Then |gβ ·

∏i∈I(e) f

−aβ,e,ie,i | is equal to a constant

cβ,e ∈ R>0 on some compact neighborhood We (for all β) of π−1(Se∩E♥)Kin W ∩π−1(Ee)K .

Put

Z =N∑α=1

cα(div(gβα(1)) ∧ · · · ∧ div(gβα(p))

).

Then the restriction of D to Ee is equal to

Ze :=∑

i1<···<ip⊂I(e)be,i1<···<ip

(div(fe,i1) ∧ · · · ∧ div(fe,ip)

),

where βe,i1<···<ip ∈ Q is uniquely determined by cα and aβ,e,i via an explicit formula whichwe omit.

On the other hand, on We, the tropicalization map tropW factors through another one

tropWe: We → RI(e)

induced by functions fi for i ∈ I(e). In other words, we have an affine map νe : RI(e) →Rh determined by aβ,e,i and cβ,e, whose explicit formula is left to the reader, such thattropW |We = νe tropWe

. Moreover, it is straightforward to check that we have

ν∗e ω =∑

i1<···<ip⊂I(e)be,i1<···<ip

(d′xe,i1 ∧ · · · ∧ d′xe,ip

)where xe,i for i ∈ I(e) are the natural coordinates on RI(e).

We first consider the “only if” direction: Assume that ω induces the zero class inHp

dR(W, (π−1E♥)⊗kK). By Proposition 4.6, we have Z = 0. Thus Ze = 0 for every e ∈ E .As

div(fe,i1) ∧ · · · ∧ div(fe,ip) = Se ∩ Ei1,...,ip 6= 0,we have be,i1<···<ip = 0 for all subsets i1 < · · · < ip ⊂ I(e). Therefore, ν∗e ω = 0. Now wetake a finite subset E ⊂ E such that Ee |e ∈ E form a covering of E . ThenW ′ := ⋃

e∈EWe isa compact neighborhood of (π−1E♥)⊗kK contained inW , and moreover ω = 0 on tropW (W ′).

We then consider the “if” direction: Assume ω = 0 on tropW (W ′) for some compactneighborhood W ′ of (π−1E♥)⊗kK contained in W . Then for every e ∈ E , the imagetropWe

(We ∩W ′) always contains a p-dimensional cell that is parallel to the plane spannedby xe,i1 , . . . , xe,ip for every subset i1 < · · · < ip ⊂ I(e). In particular, we must havebe,i1<···<ip = 0 for all subsets i1 < · · · < ip ⊂ I(e). This implies that Z = 0. By Propo-sition 4.6, Lemma 4.5, and Theorem 1.4 (ii), we know that ω induces the zero class inHp

dR(W, (π−1E♥)⊗kK).The lemma is proved.

The following theorem shows the finiteness of H1,1trop and studies the tropical cycle class of

line bundles.

Theorem 7.3. Let X be a proper smooth scheme over CF. Then(1) H1,1

trop(X ) is of finite dimension.(2) dim H1,1

trop(X ) = 1 if X is an irreducible curve.(3) For a line bundle L on X whose (algebraic) de Rham Chern class cldR(L) ∈

H2dR(X/CF) is trivial, we have cltrop(L) = 0.


Proof. We put X = X an. By Theorem 7.1, we have a canonical isomorphism H1,1trop(X ) '

H1(X,L 1X). By Proposition 6.2 and Theorem 1.4, we have

H1(X,Ω1,clX /dOX) = H1(X,ΨX)⊕ H1(X,L 1

X)⊗Q CF.

For (1), it suffices to show that dim H1(X,Ω1,clX /dOX) < ∞. In fact, we have a

spectral sequence Ep,qr abutting to H•dR(X) = H•(X,Ω•X) with the second page terms

Ep,q2 = Hp(X,Ωq,cl

X /dΩq−1X ). Thus, it suffices to show that both H3(X,CF) and H2(X,Ω•X)

are finite dimensional. Since the homotopy type of X is a finite CW complex by [HL16],dim Hi(X,CF) <∞ for every i ∈ Z. By GAGA, Hi(X,Ω•X) is canonically isomorphic to thealgebraic de Rham cohomology Hi

dR(X/CF) for every i, and thus finite dimensional.For (2), we know that dim H1(X,Ω1,cl

X /dOX) = 1 from the discussion for (1). Thusdim H1,1

trop(X ) ≤ 1. However, from Corollary 3.11, we have dim H1,1trop(X ) ≥ 1. Thus (2)

follows.For (3), by Theorem 7.1, it suffices to show that cllog(L) is zero in H1(X,Ω1,cl

X /dOX). Notethat the map CH1(X )Q → H1(X,Ω1,cl

X /dOX) factors as

CH1(X )Q → H1(X ,Ω1,clX /dOX )→ H1(X,Ω1,cl

X /dOX)in which the first map factors through the algebraic equivalence (up to torsion) by [BO74,(0.5)]. Thus, cllog(L) = 0 since L has the trivial de Rham Chern class.

Remark 7.4. Suppose that X is an irreducible proper smooth curve over CF. Then we alsohave dim H1,0

trop(X ) = dim H0,1trop(X ) = b1(X an). In fact, by the argument for Theorem 7.3, we

have dim H1,0trop(X ) = dim H0(X an,Υ1

X an) = b1(X an) by [Ber07, Corollary 4.3.5 (iii)].

From now on, we fix an embedding R → CF.

Definition 7.5 (Tropical trace map). Let X be a proper smooth CF-analytic space ofdimension n. We have a total integration map∫

X: Hn(X,T n

X )→ Hn,n(X)→ R.

By the isomorphism T nX ' L n

X in Theorem 7.1, Lemma 6.2, and by extending the abovemap linearly over CF, we obtain a CF-linear map

TrtropX : Hn(X,Υn

X) ' Hn(X,L nX)⊗Q CF → CF,

called the tropical trace map for X.

Remark 7.6. We expect that dim Hn(X,T nX ) = 1 in the setup of Definition 7.5. Then it is

an interesting question to ask the value of∫X on Hn(X,T n

X ). Suppose that X = X ana for

X in Proposition 7.7, then we know (assuming dim Hn(X,T nX ) = 1) that the restriction of∫

X to Hn(X,T nX ) takes values in Q by Corollary 3.10. Therefore, the tropical trace map

TrtropX does not depend on the choice of the embedding R → CF, and factors through the

canonical isomorphism Hn(X, (ΩnX/dΩn−1

X )2n) ∼= CF in Lemma 6.5.

The following proposition can be regarded as certain algebraicity property of the tropicaltrace map, which is defined via transcendental way.

Proposition 7.7. Let k ⊂ CF be a discrete non-Archimedean subfield, and X a geometricallyconnected proper smooth scheme over k of dimension n. If we put Xa = X ⊗k CF, then the

50 YIFENG LIU

tropical trace map TrtropX an

afactors through the canonical map

Hn(X ana ,Υn

X ana

)→ Hn(X ana , (Ωn

X ana/dΩn−1

X ana

)2n).(7.1)

In particular, Hn(X ana , (Ωn

X ana/dΩn−1

X ana

)w) ' CF if w = 2n, and is trivial otherwise.

Proof. Put X = X ana . The last assertion follows from the previous one and the following

facts:• Trtrop

X is nonzero as one can write down an (n, n)-form on X with nonzero totalintegral;• Hn(X,Ωn

X/dΩn−1X ) = ⊕

w∈Z Hn(X, (ΩnX/dΩn−1

X )w) by Theorem 1.4;• Hn(X,Ωn

X/dΩn−1X ) ' CF by Lemma 6.5; and

• the image of Hn(X,ΥnX) → Hn(X,Ωn

X/dΩn−1X ) is contained in Hn(X, (Ωn

X/dΩn−1X )2n)

by Theorem 1.4.To prove the factorization of Trtrop

X , we may replace X by Y through an alteration Y → Xwith Y smooth and geometrically connected, due to the functoriality of Trtrop

• and the factthat the pullback map Hn(X an

a ,ΩnX an

a/dΩn−1

X ana

) → Hn(Yana ,Ωn

Yana/dΩn−1

Yana

) is an isomorphism.Thus, we may assume that X is projective.

Define ΞnX to be the quotient sheaf in the following exact sequence

0→ ΥnX → (Ωn

X/dΩn−1X )2n → Ξn

X → 0.(7.2)

It is functorial in X. One can show that the sheaf ΞnX is supported on x ∈ X | s(x) ≥ 2.

This suggests that one should expect Hi(X,ΞnX) = 0 for i ≥ n − 1, which suffices for the

proposition. In fact, such vanishing result can be proved if we have semistable resolutioninstead of alteration. In the absence of semistable resolution, we need an ad hoc argumentas follows.

Take an arbitrary cohomology class α ∈ Hn−1(X,ΞnX). Since X is (Hausdorff and) com-

pact, by [SP, 09V2, 01FM], there is a finite open covering U = Ui | i = 1, . . . ,M ofX such that α is represented by an (alternative) Čech cocycle α = αI ∈ Ξn

X(UI) | I ⊂1, . . . ,M, |I| = n on U , where UI = ⋂

i∈I Ui as always. By refining U , we may assumethat αI is in the image of the map (Ωn

X/dΩn−1X )(UI)pre

2n → ΞnX(UI) for every I (See Definition

5.9 for the notation). By Lemma 7.8 below and possibly replacing k by a finite extensioninside CF, we have a proper flat integral model Y of X such that if Z1, . . . ,ZM are al-l reduced irreducible components of Ys, then the covering (π−1Zi)⊗kCF | i = 1, . . . ,Mrefines U (here, M might be different from the previous one). By [dJ96, Theorem 8.2], pos-sibly after further replacing k by a finite extension inside CF, we have a (proper) strictlysemistable scheme Y ′ over k with an alteration Y ′ → Y . For simplicity, we may also assumethat every irreducible component of Y ′[r]s is geometrically irreducible for every r ≥ 0. Inparticular, if we denote by Z ′1, . . . ,Z ′M ′ all irreducible components of Y ′s, then the covering(π−1Z ′i)⊗kCF | i = 1, . . . ,M ′ refines f−1U := f−1Ui | i = 1, . . . ,M.

Put X ′ = (Y ′ ⊗k CF)an. We claim that f ∗α = 0, where f ∗α is the canonical image off−1α in Hn−1(X ′,Ξn

X′). Assume the claim. We prove the factorization of TrtropX . Take an

element ω in the kernel of (7.1). We have to show that TrtropX (ω) = 0. By (7.2), we know that

ω = δ(α) for some α ∈ Hn−1(X,ΞnX) where δ : Hn−1(X,Ξn

X) → Hn(X,ΥnX) is the induced

coboundary map. By the claim, we know that f ∗ω = 0. In other words, TrtropX′ (f ∗ω) = 0.

However, since TrtropX′ (f ∗ω) = deg(f) Trtrop

X (ω), we have TrtropX (ω) = 0. Thus the proposition

is proved modulo the previous claim.


To prove the claim, we may replace f ∗α by α and assume that Y ′ = Y to ease notation.Moreover, we may assume that α is represented by a Čech cocycle with respect to theopen covering U = Ui | i = 1, . . . ,M in which Ui = (π−1Zi)⊗kCF. We study a typicaln-fold intersection of U . Without lost of generality, we consider U1,...,n := ⋂n

i=1 Ui. If⋂ni=1Zi = ∅, then U1,...,n = ∅. So we may assume that ⋂ni=1Zi = ∐L

j=1 Cj, where each Cjis a geometrically irreducible proper smooth curve over k. Then U1,...,n = ∐L

j=1 UCj whereUCj := (π−1Cj)⊗kCF.

Now we take a typical member C in C1, . . . , CL. Put C♥ = C\Y [n]s . Then we have

C\C♥ = C ∩ Y [n]s =

C∐c=1Pc

where Pc is isomorphic to Spec k. We may cover C♥ by affine opens Cb | 1 ≤ b ≤ B suchthat π−1Cb ' En−1

k,$ ×kM(Cb) for some integrally smooth affine k-affinoid algebra Cb suchthat Cb ∼= Spec(Cb)s, where Et

k,$ is defined in Example 4.3. Applying Theorem 1.4 (ii) toM(Cb) and by Example 4.3, we may find an open neighborhood Vb of π−1Cb in UC and anelement βb ∈ Υn

X(Vb) such that α1,...,n |Vb −βb induces the zero class in HndR(Vb). On the

other hand, π−1Pc is isomorphic to Enk,$ for every 1 ≤ c ≤ C. Thus by Example 4.3, we

may find an element γc ∈ ΥnX(Wc) where Wc := (π−1Pc)⊗kCF such that α1,...,n |Wc −γc

induces the zero class in HndR(Wc). Note that Vb | 1 ≤ b ≤ B ∪ Wc | 1 ≤ c ≤ C form

an open covering of UC. The existence of βb and γc (for an arbitrary subset of 1, . . . ,Mof cardinality n and an arbitrary C) ensures that we can refine the covering U such that αIbecomes trivial everywhere in the refined one. In particular, α = 0 in Hn−1(X,Ξn

X).

Lemma 7.8. Let K be a non-Archimedean field and X a projective scheme over K. Forevery finite open covering U = Ui |i = 1, . . . ,M of (X⊗K Ka)an, there are a finite extensionK ′/K in Ka and a proper flat integral model Y of X ⊗K K ′ over K ′ such that for everyirreducible component Z of Ys, there is some Ui ∈ U containing (π−1Z)⊗K′Ka.

Proof. The proof of [Pay09, Theorem 4.2] in fact implies the canonical isomorphism

(X ⊗K Ka)an ∼= lim←−ι

Trop(X ⊗K Ka, ι)

of topological spaces, where the limit is taken over all projective embeddings ι : X ⊗K K1 →PN(ι)K1 defined over finite extensions K1/K in Ka. Here, we adopt the notation in [Pay09].

In particular, Trop(X ⊗K Ka, ι) is a closed subset of Trop(PN(ι)Ka ). By an easy topological

argument, we may find a single projective embedding ι : X ⊗K K1 → PNK1 with a finite open

covering U ′ of Trop(PN

Ka) such that the induced open covering under the above isomorphismrefines U . By successive blow-up of linear subspaces in the special fiber of PN

K1, we can find

a further finite extension K ′/K1 in Ka and a proper flat integral model Y ′ of PNK′ over K ′

such that for every irreducible component Z ′ of Y ′s, there is some U ′i ∈ U ′ containing theimage of (π−1Z ′)⊗K′Ka in Trop(PN

Ka). Finally, we take Y to be the closure of ι(X ⊗K K ′)in Y ′.

Proof of Theorem 1.8. By Lemma 7.9 below and possibly replacing k by a finite extension inCF, we may assume ω ∈ A n−p,n−p

X an (X an). Put X = X an. We regard ω as the induced class in

52 YIFENG LIU

Hn−p,n−p(X). By Corollary 3.5, we may further assume that ω is an element in Hp(X,T pX).

By Theorem 3.7, we need to show that∫X

cltrop(Z) ∧ ω = 0.

However, by Proposition 7.7, it suffices to show that 〈cllog(Z), ω〉X = 0 where cllog is definedin Definition 6.4. However, this follows from Theorem 6.6.

Lemma 7.9. Let the assumption and notation be as in Theorem 6.6. Then for every p, q ≥ 0,the canonical map

lim−→k′

A p,qX ank′

(X ank′ )→ A p,q

X ana

(X ana )

is an isomorphism, where the colimit is taken over all finite extensions k′ of k in CF.

Proof. The lemma follows from the fact that for every f ∈ O∗(X ana , V ) where V is a rational

affinoid domain, there is a function g ∈ O∗(X ank′ , V

′) for some k′ such that ς−1k′ (V ′) = V

and f−1 · ς∗k′g has norm 1 on some open neighborhood of V , where ςk′ : X ana → X an

k′ is thecanonical projection. Here, we have used [Ber07, Lemma 2.1.3 (ii)].

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Department of Mathematics, Northwestern University, Evanston IL 60208, United StatesE-mail address: [email protected]














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