Structure Theorem of Kummer Etale K-Group II

Kei Hagihara

Received: May 15, 2013

Communicated by Lars Hesselholt

Abstract. In this article, we investigate the lambda-ring structure of Kummer etale K-groups for some class of logarithmic schemes, up to torsion. In particular, we give a logarithmic analogue of Chow groups for the logarithmic schemes, and describe its structure.

2010 Mathematics Subject Classification: Primary 19D55; Secondary 14F20. Keywords and Phrases: algebraic K-theory, logarithmic geometry, algebraic cycles.

1 Introduction

In [Hag03], for a wide class of logarithmic varieties over a separably closed field, we gave an explicit description of their Kummer etale K-groups in terms of the usual K-groups of the associated stratifications. However, this description is still unsatisfactory in that it disregards λ-ring structures, with which every Kummer etale K_{0}-group is naturally endowed through exterior product operations. Since, already in the classical case, these structures play essential roles, for instance, for a definition of (rational) motivic cohomology, this defect should be overcome. So in this paper, for a logarithmic variety satisfying some good conditions, we construct an isomorphism between its Kummer etale K-group and usual Kgroups associated with its stratification, preserving their λ-ring structures up to torsion. More precisely, we consider the following situation. Let k be a field of characteristic p, X a scheme smooth, separated and of finite type over k, D a strictly normal crossing divisor on X, and {D_{i}}_{i∈I} its irreducible components. We also denote by X the log scheme associated with (X, D). Then the main theorem of this article is as follows:

Theorem 1.1. (= Theorem 3.23) We have an isomorphism of rings

K_{0}(X_{Ket}) ⊗_{ℤ} ℚ ⟶^{≅} \underleftarrow{lim}_{C_{I}} K_{0}(D_{J_{2}}) ⊗_{ℤ} ℚ[(ℤ_{(p)}/ℤ)^{⊕J_{1}}],

which is compatible with the actions of Adams operations (For their actions on the right hand side, see below).

Here, for J = {i_{1}, ⋯, i_{r}} ⊂ I, we put D_{J} = D_{i_{1}} ∩⋯∩D_{i_{r}} (Note that D_{∅} = X), and we set

C_{I} = {(J_{1}, J_{2})|J_{1} ⊂ J_{2} ⊂ I}

Structure Theorem of Kummer etale K-group II 1347

and regard it as an ordered set by defining (J_{1}, J_{2}) ≥ (J′_{1}, J′_{2}) to be J′_{1} ⊂ J_{1} ⊂ J_{2} ⊂ J′_{2}. The transition morphism in the limit is induced by the natural closed inclusion D_{J_{2}} ⊃ D_{J′_{2}} and a projection (ℤ_{(p)}/ℤ)^{⊕J_{1}} → (ℤ_{(p)}/ℤ)^{⊕J′_{1}}. We consider ℚ[(ℤ_{(p)}/ℤ)^{⊕J}] to be a group ring endowed with ring endomorphisms {Ψ^{m}}_{m>0}, which we call Adams operations, defined by Ψ^{m}([α]) = [mα] for m ∈ ℕ, m > 0 and α ∈ (ℤ_{(p)}/ℤ)^{⊕J}. The actions of Adams operations on the right hand side are naturally induced from the usual ones on K_{0}(D_{J_{2}})_{ℚ} and on ℚ[(ℤ_{(p)}/ℤ)^{⊕J_{1}}] (For more details, see Subsection 3.3). By this theorem, we can determine completely the actions of Adams operations, and accordingly the λ-ring structure up to torsion, on the left hand side. We will give some remarks on the proof. As in [Hag03], the fundamental idea of the proof is a local-global argument, that is, the reduction to the local case where we can apply various results in equivariant K-theory. However, there are some differences to be mentioned, between the strategy there and that in this article. In [Hag03], we focused only on the (Abelian) group structure of the Kummer etale K-group, so we could use a “localisation sequence” for K′-groups and reduce the theorem to the case where the underlying scheme is of the form Spec L for a field L. However, this method does not work well in the analysis of the λ-ring structure. In fact, this difficulty is not so serious because we have enough tools in simplicial homotopy theory to reduce to the case of the (Henselian) local ring, where we can apply equivariant K-theory effectively. The more serious difficulty is the high complexity of the Kummer etale K-group of each irreducible component of the divisor. This is already seen in the calculation of Kummer etale K-group of a log point. Let (Spec ℂ)^{log} be a log scheme associated with a monoid map from e^{ℕ} (this denotes the free (multiplicative) monoid with generator e) to ℂ which send every element except the unit to 0. Then its Kummer etale K-group is isomorphic to a colimit of (usual) equivariant K-groups K_{i}(Spec ℂ[x]/(x^{n}), μ_{n}), which are very complicated because of the existence of nilpotent elements in ℂ[x]/(x^{n}). To avoid this difficulty, we introduce the notion of a “pellicular Kummer etale ringed topos” of an fs log scheme X, which we denote by (X_{Ket}, \overline{𝒪} _{X_{Ket}}). This has the same underlying topos as the usual Kummer etale topos, but its structure sheaf \overline{𝒪} _{X} associates with each “open set” a ring Γ(U, 𝒪_{U})^{red} (not a ring Γ(U, 𝒪_{U})). Upon this notion we can develop general theory of \overline{𝒪} _{X}-modules and define the corresponding K-(and K′-)groups, which we denote by \overline{K} (resp. \overline{K}′) and call pellicular Kummer etale K-(resp. K′-)groups. The advantage of this new K-group is that it is extremely easier to compute than the usual Kummer etale K-group. For instance, in the case of a log point above, its pellicular Kummer etale K-group is a colimit of equivariant Kgroups K_{i}(Spec ℂ, μ_{n}), where the action of μ_{n} is trivial (The name “pellicular”, which means “film-like”, is adopted on the ground that \overline{𝒪} _{P_{Ket}} is a “very thin” quotient of 𝒪_{P_{Ket}}, for instance, if P is a log point). It is easily checked that this is isomorphic to K_{i}(Spec ℂ) ⊗ ℤ[ℚ/ℤ]. Fortunately, the totality of the pellicular Kummer etale K-groups of D_{J}’s has sufficient information for the recovery of the Kummer etale K-group of the ambient log scheme. Another difficulty is the problem of how we carve the factor which “purely” corresponds to each D_{J} out of the Kummer etale K-group of the ambient log scheme. For instance, let X be a curve with a divisor D consisting of one point, and i : D → X the natural closed immersion. Then it is (or at least seems) highly difficult to separate the information corresponding only to X (i.e. the usual K-group of the underlying scheme of X) from the Kummer etale K-group of X without ignoring the λ-ring structure. In order to get over this difficulty, we prove an “inversion formula”, which enables us to interpret a Kummer etale K-group as a generalised cohomology of another Kummer etale topos with coefficients in Kummer etale K. Using it, we can construct some maps which seem to go in the “opposite” ways, but which still respect λ-ring structures, up to torsion. For example, in the above case, we can define a map from the rational Kummer etale K-groups of X to the usual rational K-group of X compatibly with their λ-ring structures. The existence of this map already seems very non-trivial even for K_{0} of curves. Note that, as careful readers can notice immediately, such maps appear implicitly in the statement of the structure theorem. With these tools at hand, the proof (as well as the formulation) of the main theorem is only an easy exercise of simplicial homotopy theory, equivariant K-theory and logarithmic geometry. Finally we give two remarks. First note that, as a result of the above theorem, for some logarithmic varieties we can define the notion of what we might call “Kummer etale Chow groups” as eigenspaces of Adams operations and can describe them in terms of the usual Chow groups. As an application, by using these objects we can formulate and prove a Riemann-Roch-type theorem for Kummer etale K-groups. These topics and their arithmetic applications will be given in the forthcoming paper. Secondly, as the readers see immediately, most of our results also hold for higher Kummer etale K-theory. However, in this paper we concentrate on the K_{0}- case since our motivation comes from its application to number theory via the aforementioned Riemann-Roch-type theorem for the Kummer etale K_{0}-group and the Kummer etale Chow groups.

Now let us mention the organisation of this paper. First, in Section 2, we review logarithmic geometry and Kummer etale topos, and in Section 3, we introduce some new notions on logarithmic schemes, especially, those of pellicular Kummer etale ringed topos, and formulate the main theorem in a form suitable for some local-to-global arguments. In addition, in this section, we reduce the theorem to some key lemmata. These will be proven in Section 5, after some consideration on the general theory of pellicular Kummer etale ringed topos in Section 4. The author is very grateful to Professors Kazuya Kato and Iku Nakamura for their ceaseless encouragement and interest in this work, and to the referee for the useful comments. This work was partially supported by Global COE Program at University of Tokyo and by JSPS KAKENHI 21674001, 23224001, 23740030. In addition, it was also supported in part by KAKENHI 26247004, as well as the JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry” and the KiPAS program 2013–2018 of the Faculty of Science and Technology at Keio University.

2 Preliminaries

For a ring A, we denote by A^{red} the quotient ring of A by its nilradical. For a scheme X, we denote by 𝒪_{X_{Zar}} its structure sheaf, and by Vect(X_{Zar}), Coh(X_{Zar}), Qcoh(X_{Zar}) and Mod(X_{Zar}) the category of coherent locally free 𝒪_{X_{Zar}}-modules, coherent sheaves of 𝒪_{X_{Zar}}-modules, quasi-coherent sheaves of 𝒪_{X_{Zar}}-modules and 𝒪_{X_{Zar}}-modules, respectively. In the rest of this section, we recall some definitions and propositions in logarithmic geometry. For the notions which are not given in this section and the proofs of propositions omitted here, see [Kat89], [Kat94], [Nak92], [Nak97] and [Hag03].

2.1 Logarithmic geometry

2.1.1 Monoid theory

In this paper, a monoid means a commutative one with a unit (unless otherwise mentioned), and a homomorphism of monoids is always assumed to preserve the units. The symbol ℕ means the set of integers i such that i ≥ 0 and is regarded as a monoid by addition, although we write the operations in monoids multiplicatively very often. When given a monoid P, we call a monoid Q equipped with a monoid homomorphism P → Q a P -monoid. A subset I of a monoid P is called an ideal if a ∈ P and x ∈ I implies ax ∈ I. An ideal is called a prime ideal if its complement is a submonoid of P. A submonoid of P is called a face if it is the complement of a prime ideal. We call a monoid isomorphic to ℕ^{r} for some r a finitely generated free monoid. As is easily seen, there is a natural bijection between the set of prime ideals in a finitely generated free monoid and the power set of the basis of the monoid. We often denote the finitely generated free monoid with basis e_{1}, …, e_{r} by e_{1}^{ℕ} ⋯ e_{r}^{ℕ}. When we are given a monoid M and elements m_{1}, …, m_{r} ∈ M, the notation m_{1}^{ℕ} ⋯ m_{r}^{ℕ} is also used to represent the submonoid of M generated by m_{1}, …, m_{r}. For a monoid P, we denote by P^{×} the submonoid of P consisting of invertible elements, that is, elements that divide the unit. P is called sharp if and only if P^{×} = {1}. By P^{gp} we mean its group completion, which is naturally defined. For an Abelian group A, we often use the notation P_{A} instead of P^{gp} ⊗_{ℤ} A. In particular, P_{ℤ} means P^{gp}. A monoid P is called integral if ac = bc always implies a = b for any element a, b and c in P, or equivalently, if P → P^{gp} is injective. An integral monoid P is called saturated if any element a ∈ P^{gp} satisfying a^{n} ∈ P for some n ∈ ℕ belongs to P. A finitely generated saturated monoid is also called an fs monoid. For a monoid P, P^{sat} denotes its saturation, which is defined by the universality. More precisely, the functor (−)^{sat} is defined to be a functor left adjoint to the inclusion functor from the category of saturated monoids to that of monoids. Recall that its existence is assured and we have a natural map P → P^{sat}. For a saturated monoid P and saturated P -monoids Q and R, we set Q ⊕_{P}^{sat} R = (Q ⊕_{P} R)^{sat}, the cofiber product in the category of saturated monoids. For a monoid P and a natural number n, P^{1/n} is defined to be a P -monoid such that P → P^{1/n} is isomorphic to the n-th power map (−)^{n} : P → P, and for a natural number m we set P^{m′−div} = colim_{(n,m)=1}P^{1/n}. If m is fixed (e.g. m is the characteristic exponent of a fixed scheme) and no risk of confusion is induced, we also use the notation P^{div′}. We also set ℚ′ = ℤ^{div′} and (ℚ/ℤ)′ = ℚ′/ℤ. A monoid homomorphism ϕ : P → Q from P to Q is called a projection if there exist a monoid R and an isomorphism f : Q × R → P such that ϕ ∘ f equals pr_{1}, the projection to Q.

2.1.2 Logarithmic schemes

A logarithmic scheme (or a log scheme) is a pair of a scheme X and a homomorphism of etale sheaves of monoids α_{X} : M_{X} → 𝒪_{X}, with 𝒪_{X} regarded as a sheaf of monoids by multiplication, such that α_{X}^{−1}(𝒪_{X}^{×}) → 𝒪_{X}^{×} is an isomorphism. We always regard the sheaf of invertible functions 𝒪_{X}^{×} as a subsheaf of M_{X} and set \overline{M}_{X} = M_{X}/𝒪_{X}^{×}. We often write X to represent a log scheme (X, M_{X}), and then, for a log scheme X, |X| denotes its underlying scheme or its underlying topological space (in the preceding paper [Hag03], the notation

X^{∘} is used for underlying schemes). The morphism α_{X} is called the log structure of X. When there is no risk of confusion, a log scheme X is called Noetherian, quasicompact, regular and so on, if its underlying scheme |X| is so. Similarly, we often write “x ∈ X” instead of “x ∈ |X|”, and the notation X_{Zar} is used to mean |X|_{Zar}. A morphism from a log scheme (X, M_{X}) to a log scheme (Y, M_{Y}) is a pair of a morphism of schemes ϕ : X → Y and a morphism of monoid sheaves ϕ^{∗}M_{Y} → M_{X} compatible with the maps to 𝒪_{X} (For the definition of the pullback of the log structure ϕ^{∗}M_{Y} → 𝒪_{X}, see [Kat89]). For a morphism f of log schemes, we denote by |f| the underlying morphism between schemes. We often say, for example, “f is of finite type” instead of saying “|f| is of finite type”. A morphism f : X → Y between log schemes is called strict if for all x ∈ X, \overline{M}_{Y,\overline{f(x)}} → \overline{M}_{X,\overline{x}} is an isomorphism. A strictly closed (or open) subscheme of a log scheme (X, M_{X}) is a closed (or open) subscheme i : Y ↪ X with the induced log structure i^{∗}M_{X}. For a monoid P we denote by (Spec ℤ[P], P̃), or simply by Spec ℤ[P], the log scheme whose underlying scheme is Spec ℤ[P] and which is endowed with the log structure induced by the natural homomorphism of monoids P → ℤ[P]. A log scheme is called fs if etale locally it has a strict morphism to Spec ℤ[P] with P an fs monoid. This strict morphism is called a chart. We denote by ×^{fs} the fiber product in the category FsLogSch of fs log schemes, to distinguish it from ×, the one in the category of log schemes or of schemes. Note that, in general, |X ×_{S}^{fs} Y | ≇ |X ×_{S} Y | ≅ |X| ×_{|S|} |Y | for log schemes X and Y over S.

2.1.3 Standard coverings

Definition 1. Let X be an fs log scheme, P a sharp fs monoid, X → Spec ℤ[P] a chart and n a natural number.

1. We set X_{n} = X ×_{Spec ℤ[P]}^{fs} Spec ℤ[P^{1/n}].

Note that, since the group I_{n} = Hom (P^{1/n}/P, ℤ[ζ_{n}]^{×}) acts linearly on a ring ℤ[ζ_{n}][P^{1/n}] by

φ(a[p]) = (φ(p)a)[p],

for φ ∈ I_{n}, a ∈ ℤ[ζ_{n}] and p ∈ P^{1/n}, if |X| is a scheme over ℤ[ζ_{n}], we have a natural action of the group on X_{n} over X.

2. We set X̃_{n} = X ×_{Spec ℤ[P]}^{fs} Spec ℤ[ζ_{n}][P^{1/n}].

Note that, letting Γ_{n} = Gal(ℚ(ζ_{n})/ℚ), we have a natural action of the group I_{n} ⋊Γ_{n} on X̃_{n} over X, where the semi-direct product is constructed by the action of Γ_{n} on I_{n} such that γ(φ) = γ ∘ φ for γ ∈ Γ_{n} and φ ∈ I_{n}.

Proposition 2.1. In the above situation, we have

Spec ℤ[P^{1/n}] ×_{Spec ℤ[P]}^{fs} Spec ℤ[P^{1/n}] ≅ Spec ℤ[P^{1/n} ⊕_{P}^{sat} P^{1/n}]
≅ Spec ℤ[P^{1/n} ⊕ (P^{1/n}/P)].

Here the second isomorphism is induced by a monoid isomorphism from P^{1/n} ⊕_{P}^{sat} P^{1/n} to P^{1/n} ⊕ (P^{1/n}/P) characterised by the property that (a, b) ∈ P^{1/n} ⊕ P^{1/n} is mapped to (ab, b mod P) via the composite with a natural map P^{1/n} ⊕ P^{1/n} → P^{1/n} ⊕_{P}^{sat} P^{1/n}. In particular,

1. if |X| is a scheme over Spec ℤ[ζ_{n}, 1/n], we have a canonical isomorphism

X_{n} ×_{X}^{fs} X_{n} ≅ ∐_{α∈I_{n}}(X_{n})_{α},

where (X_{n})_{α} is a copy of X_{n}, so that the composite of the natural inclusion i_{α} : (X_{n})_{α} → X_{n} ×_{X}^{fs} X_{n} and the projection pr_{1} : X_{n} ×_{X}^{fs} X_{n} → X_{n} (resp. pr_{2}) is the identity (resp. the action of α ∈ I_{n}), and

2. if |X| is a scheme over Spec ℤ[1/n], we have a canonical isomorphism

X̃_{n} ×_{X}^{fs} X̃_{n} ≅ ∐_{α∈I_{n}⋊Γ_{n}}(X̃_{n})_{α},

in the similar way.

Proof. Straightforward.

2.1.4 (weak) Logarithmic regularity

Finally, we recall the notion of log regularity and generalise it slightly (cf.

[Kat94] or [Hag03] Section 2.5).

Definition 2. Let X be a locally Noetherian fs log scheme.

1. We say X is log regular at x ∈ |X| if the following two conditions are satisfied:

(a) 𝒪_{X,\overline{x}}/I(\overline{x}, M) is a regular local ring.

(b) dim 𝒪_{X,\overline{x}} = dim 𝒪_{X,\overline{x}}/I(\overline{x}, M) + rank_{ℤ}\overline{M}_{X,\overline{x}}^{gp}.

Here I(\overline{x}, M) is an ideal of 𝒪_{X,\overline{x}} generated by the image of M_{X,\overline{x}} \ 𝒪_{X,\overline{x}}^{×} and dim denotes the Krull dimension.

X is said to be log regular if it is log regular at x for all x ∈ X.

2. For a natural number r, we denote by (Spec ℤ)^{log} the log scheme induced by a morphism of monoids ℕ^{r} → ℤ which maps any element except the identity to zero. For a log regular log scheme T, consider the log scheme T′ = T ×_{Spec ℤ} (Spec ℤ)^{log}, where Spec ℤ is endowed with the trivial log structure.

X is said to be weakly log regular if it is locally isomorphic to a log scheme defined as above.

2.2 Kummer etale morphism

For the definition of logarithmic etale and smooth morphisms, see [Kat89]. Here we restrict ourselves to the review of the definition of a Kummer etale morphism. For more details see [Nak92] or [Nak97]. A morphism ϕ : P → Q of integral monoids is said to be of Kummer type if it is injective and satisfies the following condition: for any element q ∈ Q there exists a positive integer n such that q^{n} ∈ Image ϕ. A morphism f : X → Y of fs log schemes is said to be of Kummer type if for all x ∈ X, \overline{M}_{Y,\overline{f(x)}} → \overline{M}_{X,\overline{x}} is of Kummer type. A morphism of fs log schemes is called Kummer etale, or shortly Ket, if it is logarithmic etale and of Kummer type. Recall that a strict morphism is Kummer etale if and only if its underlying morphism of schemes is etale. So we often call a strict Kummer etale morphism classically etale, or more simply, etale. The propositions below, due to Nakayama, play essential roles in the followings.

Proposition 2.2. ( [Nak92] 6.4.2) Let U and X fs log schemes and f : U ⟶ X a Kummer etale morphism. Assume U is quasi-compact, X is equicharacteristic and there exists a chart X ⟶ Spec ℤ[P] with P fs and sharp. Then there exists a positive integer n, invertible on U, such that the pull-back of f on X_{n},f_{n} : U ×_{X}^{fs} X_{n} ⟶ X_{n} is classically etale.

Proof. We can take a natural number n invertible in X such that M_{U}^{gp}/f^{∗}M_{X}^{gp} becomes zero by multiplication by n. Set U_{n} = U ×_{X}^{fs} X_{n}. Then by considering stalks of \overline{M}-sheaves at each point and by using Proposition 2.1.1 in [Nak97], we can reduce this proposition to the next lemma.

Lemma 2.3. Let n be a natural number and R ← P → Q a diagram of saturated sharp monoids. Assume that

1. P → Q is an inclusion and Q^{n} ⊂ P

2. Every element in P becomes n-divisible when mapped to R.

Then we have an isomorphism R →^{≅} (R ⊕_{P}^{sat} Q)/(R ⊕_{P}^{sat} Q)^{×}.

Proof. It suffices to note that, in the category of saturated sharp monoids, P → Q is an epimorphism and that P → R factors through P → Q.

Proposition 2.4. (cf. [Nak92] 4.2.4.1) Let U and X be fs log schemes and f : U → X a Kummer etale morphism. Take a point u ∈ U and put x = f(u). Assume that we are given a chart of X, ϕ : X → Spec ℤ[P], such that the canonical map P → \overline{M}_{X,\overline{x}} is an isomorphism. Then we have an fs monoid Q and a morphism h : P → Q of Kummer type and the following diagram

U′ ⟶ V ⟶ Spec ℤ[Q]
g↓  ↓  ^{Spec ℤ[h]}↓
U ⟶^{f} X ⟶^{ϕ} Spec ℤ[P],

where

1. the right square is Cartesian,

2. the number of Coker h^{gp} is invertible on U′,

3. u belongs to the image of g and

4. U′ is classically etale over V and over U.

Proof. By the direct application of Theorem 3.5 in [Kat89] we can immediately construct the diagram satisfying all the conditions but the saturatedness of Q. In addition, investigating its proof, we also see that Q becomes saturated automatically when U is assumed to be fs.

Corollary 2.5. Let X be an fs log scheme and {U_{i} → X}_{i∈I} a Kummer etale covering of X. Then we have

1. an etale covering {X_{j} → X}_{j∈J} by affine log schemes X_{j} over Spec ℤ[1/n_{j}, ζ_{n_{j}}] with n_{j} ∈ ℤ,

2. a map ϕ : J → I,

and for each j ∈ J,

3. a chart X_{j} → Spec ℤ[P_{j}] with P_{j} an fs and sharp monoid,

4. an etale covering {V_{j} → (X_{j})_{n_{j}}} (recall that (X_{j})_{n_{j}} = X_{j} ×_{Spec ℤ[P_{j}]}^{fs}

Spec ℤ[P_{j}^{1/n_{j}}]), and

5. an X-morphism V_{j} → U_{ϕ(j)}

such that, for each i ∈ I, a set of morphisms {V_{j} → U_{i}}_{j∈ϕ^{−1}(i)} is a Kummer etale covering (in particular {V_{j} → X}_{j∈J} refines {U_{i} → X}_{i∈I}).

Proof. We can deduce it easily from the above proposition (cf. the proof of Corollary 2.7 in [Hag03]).

2.3 Kummer etale K-theory

Let X be an fs log scheme.

Definition 3. 1. We define a category Ket/X to be the full subcategory of FsLogSch/X consisting of fs log schemes X′ Kummer etale over X. We endow it with the topology by regarding a family of Kummer etale morphisms {f_{i} : X′_{i} → X′} such that X′ = ⋃ f_{i}(X′_{i}) as a covering. Indeed, we can check that this becomes a site ([Nak92], [Nak97]), so we denote by X_{Ket} this site and by (X_{Ket}) the associated topos.

2. The ringed topos ((X_{Ket}), 𝒪_{X_{Ket}}) is defined as follows:

(a) A topos (X_{Ket}) is the one defined as above.

(b) A ring object 𝒪_{X_{Ket}} in (X_{Ket}) is the rule which associates to each X′, Kummer etale over X, the ring Γ(|X′|, 𝒪_{|X′|}) (This is indeed a sheaf, as is shown in Proposition 3.1 in [Hag03]).

We also denote it by (X_{Ket}, 𝒪_{X}) if no confusion occurs. We have the natural notion of 𝒪_{X_{Ket}}-modules and define Mod(X_{Ket}) to be the category of 𝒪_{X_{Ket}}- modules on a ringed topos (X_{Ket}, 𝒪_{X}). The natural morphism of ringed topoi from (X_{Ket}, 𝒪_{X}) to (X_{Zar}, 𝒪_{|X|}) is denoted by ε_{X}, the subscript X being often omitted. Now we recall the definitions of Kummer etale K-theory and its variant (cf. Subsection 3.3 of [Hag03]).

Definition 4. Let X be an fs log scheme. An object ℱ in Mod(X_{Ket}) is called a Kummer etale vector bundle (or a Ket vector bundle, for short) if, Kummer etale locally, it is isomorphic to a finite direct sum of 𝒪_{X_{Ket}}. The 𝒪_{X_{Ket}}-module ℱ is said to be a Kummer etale coherent sheaf if it is so in the sense of J. -P. Serre, that is, if ℱ is Ket locally finitely generated and any Ket locally given morphism 𝒪_{U_{Ket}}^{n} → ℱ |_{U} has a (Kummer etale locally) finitely generated kernel. We denote by Vect(X_{Ket}) and Coh(X_{Ket}) the categories consisting of Ket vector bundles and Ket coherent sheaves, respectively.

As is seen in Subsections 3.3 and 3.4 of [Hag03], the category Vect(X_{Ket}) becomes an exact category in a natural way, and Coh(X_{Ket}) is an Abelian category if the functor ε_{X}^{∗} : Mod(X_{Zar}) → Mod(X_{Ket}) is exact (e.g. if \overline{M}_{X,x̄} is finitely generated free for all x ∈ |X|), and |X| is Noetherian.

Remark 2.6. In fact, the definition of a Kummer etale coherent sheaf given in [Hag03] is different from the one given here. However, these two notions coincide for fs log schemes treated in this paper (cf. Corollary 3.10 and Proposition 3.12 of [Hag03]).

Definition 5. Let K(X_{Ket}) be the simplicial set associated with the exact category Vect(X_{Ket}) ([Qui73]) and K_{i}(X_{Ket}) its i-th homotopy group. These are called a Kummer etale K-theory spectrum and a Kummer etale K-group, respectively. We also define a Kummer etale K′-theory spectrum K′(X_{Ket}) and a Kummer etale K′-group K′_{i}(X_{Ket}) by using Coh(X_{Ket}) in the case where ε_{X}^{∗} is exact and X is Noetherian. Also, we denote by Pic (X_{Ket}) the group of the isomorphic classes of Ket line bundles (i.e. Kummer etale vector bundles of rank one).

The aim of this article is to determine the λ-ring structure of K_{0}(X_{Ket}) for suitable fs log schemes up to torsions.

3 The statement of the main theorem

In this section we state the main theorem in Subsection 3.3 after introducing some notions in logarithmic geometry in Subsections 3.1 and 3.2.

3.1 Logarithmic schemes with standard frames

Here we recall the notion of F -framed log schemes (introduced in Section 5 of [Hag03]) and some constructions of log schemes associated with them. Throughout this subsection, fix a finitely generated free monoid F.

Definition 6. 1. An F -framed log scheme is a pair of a log scheme X and a morphism of monoids θ : F ⟶ Γ(X, \overline{M}_{X}) such that for all x ∈ X the composite F ⟶ Γ(X, \overline{M}_{X}) ⟶ \overline{M}_{X,\overline{x}} (which we denote by θx̄) is isomorphic to a projection ℕ^{m} → ℕ^{n} with m ≥ n. Such a morphism θ is called a frame. We also call the pair (X, θ) a log scheme with a standard frame if we are not interested in F.

2. Let (X, θ), (Y, θ′) be F -framed log schemes. An F -framed morphism from (X, θ) to (Y, θ′) is a morphism of log schemes ϕ : X ⟶ Y such that

F  F
θ′↓  ^{θ}↓
Γ(Y, \overline{M}_{Y}) ⟶^{ϕ^{∗}} Γ(X, \overline{M}_{X})

is commutative.

For example, the log scheme Spec ℤ[F] can be naturally regarded as F -framed.

Remark 3.1. 1. It is easily checked that if X is F -framed, we have \overline{M}_{X,x} ≅ \overline{M}_{X,x̄} for each x ∈ X.

2. An F -framed morphism becomes strict automatically.

Mainly we work with the category of F -framed log schemes for a fixed monoid F. The proposition below is easily checked.

Proposition 3.2. Let (X, θ) be an F -framed log scheme and Y ⟶ X a strict morphism of log schemes. Then Y has a canonical F -framed log structure.

In particular, for a subscheme Y of |X| we can define an F -framed log structure canonically, and if a log scheme X has a chart X → Spec ℤ[F] with F finitely generated free, then X is naturally F -framed. The next proposition is Proposition 3.12 of [Hag03].

Proposition 3.3. Let (X, θ) be an fs log scheme with a standard frame. Then for all x ∈ X, 𝒪_{X,x(log)} is flat over 𝒪_{X,x}.

Next we introduce some definitions concerning the monoid F.

Definition 7. Let F be a finitely generated free monoid and 𝔭, 𝔮 and 𝔮′ its prime ideals.

1. We define F (𝔭) to be the unique face satisfying F (𝔭) ⊕ (F \ 𝔭) = F and put 𝔭^{∨} = F \ F (𝔭). For a face N of F, we also put N^{∨} = F \ ((F \ N)^{∨}).

2. We denote by 𝔮 ∨ 𝔮′ (and 𝔮 ∧ 𝔮′) the minimal prime ideal containing 𝔮 and 𝔮′ (resp. the maximal prime ideal contained in 𝔮 and 𝔮′). Note that 𝔮 ∨ 𝔮′ coincides with the set-theoretic join 𝔮 ∪ 𝔮′, while 𝔮 ∧ 𝔮′ does not with 𝔮 ∩ 𝔮′.

3. For a natural number m, we set F^{𝔮−m′−div} = (F \ 𝔮)^{m′−div} ⊕ F (𝔮). We also use the notation F^{𝔮−div′}, omitting m, if there is no risk of confusions.

4. We set

Λ[𝔮] = ℤ[F (𝔮)^{gp} ⊗_{ℤ} ℚ/ℤ].

As usual, for the characteristic exponent p of a (fixed) scheme, Λ′[𝔮] is defined similarly by replacing ℚ/ℤ by ℚ′/ℤ = ⊕_{l≠p}ℚ_{l}/ℤ_{l}.

The followings are immediately checked (under the assumption that the ambient monoid is finitely generated free).

Proposition 3.4. Let F be as above and 𝔮 and 𝔮′ prime ideals.

1. 𝔮^{∨∨} = 𝔮.

2. If 𝔮 ⊂ 𝔮′, then 𝔮^{∨} ⊃ 𝔮′^{∨} and F (𝔮) ⊂ F (𝔮′).

3. F \ 𝔮 = F (𝔮^{∨}).

4. F (𝔮 ∨ 𝔮′) = F (𝔮) + F (𝔮′), F (𝔮 ∧ 𝔮′) = F (𝔮) ∩ F (𝔮′).

5. (𝔮 ∨ 𝔮′)^{∨} = 𝔮^{∨} ∧ 𝔮′^{∨}, (𝔮 ∧ 𝔮′)^{∨} = 𝔮^{∨} ∨ 𝔮′^{∨}.

Remark 3.5. For prime ideals 𝔮 ⊂ 𝔮′, the inclusion F (𝔮) ⊂ F (𝔮′) has a unique left inverse which is also a projection. This will also play an important role later.

Now, for a given log scheme with a standard frame, we have two recipes for constructing other log schemes. First we “remove” the log structure of an F -framed log scheme “along” a prime ideal of F in the following way.

Definition 8. Let (X, θ) be an F -framed fs log scheme and 𝔮 a prime ideal of F. Then we denote by the log scheme X^{𝔮} the one having the same underlying scheme as X, endowed with the log structure M_{X^{𝔮}} defined by the Cartesian diagram

\xymatrix{M_{X^{𝔮}} \ar[d] \ar@{^{(}->}[r] & M_{X} \ar[d] \\ θ′((F \ 𝔮)_{X}) \ar@{^{(}->}[r] & \overline{M}_{X},}

where θ′((F \ 𝔮)_{X}) is the image of the morphism θ′ from the constant monoid sheaf (F \ 𝔮)_{X} to the monoid sheaf \overline{M}_{X} adjoint to the morphism of monoids θ : F \ 𝔮 → Γ(X, \overline{M}_{X}).

The proposition below is easily checked:

Proposition 3.6. 1. We have an isomorphism of log schemes X^{∅} ≅ X. On the other hand, if 𝔪 is the maximal ideal of F, then X^{𝔪} is the trivial log scheme with underlying scheme |X|.

2. For 𝔮 ⊂ 𝔮′, we have a natural morphism of log schemes X^{𝔮} → X^{𝔮′}.

3. The log scheme X^{𝔮} naturally has an (F \ 𝔮)-framed structure. Moreover, for prime ideals 𝔮 ⊂ 𝔮′ of F, we have a commutative diagram

\xymatrix{F \ 𝔮′ \ar@{^{(}->}[r] \ar[d] & F \ 𝔮 \ar[d] \\ Γ(X^{𝔮′}, \overline{M}_{X^{𝔮′}}) \ar[r] & Γ(X^{𝔮}, \overline{M}_{X^{𝔮}})}

of natural monoid homomorphisms.

4. Let 𝔮 be a prime ideal of F, and 𝔮′ one of F \ 𝔮, which implies that 𝔮 ∪ 𝔮′ is a prime ideal of F and we can define (X^{𝔮})^{𝔮′}. Then we have a natural isomorphism (X^{𝔮})^{𝔮′} ≅ X^{𝔮∪𝔮′}.

5. Let X and Y be F -framed fs log schemes, 𝔮 a prime ideal of F and f : X → Y is a strict F -framed morphism. Then the induced morphism X^{𝔮} → Y^{𝔮} is also strict.

The next proposition makes the investigations of X^{𝔮} easier.

Proposition 3.7. Let (X, θ) be an F -framed locally Noetherian fs log scheme, x ∈ |X| and 𝔮 a prime ideal of F. Then we have

1. a Zariski neighbourhood U ⊂ |X| of x,

2. an F -framed strict morphism U → Spec ℤ[F], and

3. an (F \ 𝔮)-framed strict morphism U^{𝔮} → Spec ℤ[F \ 𝔮]

such that

U ⟶ Spec ℤ[F]
↓  ↓
U^{𝔮} ⟶ Spec ℤ[F \ 𝔮]

is commutative, where U, Spec ℤ[F], U^{𝔮} and Spec ℤ[F \𝔮] are naturally endowed with an F - or (F \ 𝔮)-framed structure, and the vertical maps are natural ones. In particular, X^{𝔮} is also an fs log scheme, and X has, Zariski locally, a chart.

Proof. By the definition of the frame, we have a commutative diagram

\xymatrix{F′ \ar@{^{(}->}[r]^{incl_{1}} \ar[drr] & F′ ⊕ F′′ \ar[r]^{≅} & F \ar[d]^{θx̄} \\  &  & \overline{M}_{X,x̄},}

where the oblique morphism (named \overline{γ}_{x}) is an isomorphism, and the composite \overline{γ}_{x}^{−1} ∘θx̄ is a projection. Since the map M_{X,x} → \overline{M}_{X,x}(≅ \overline{M}_{X,x̄}) is surjective by the vanishing of R^{1}ε_{X∗}^{et}𝒪_{X_{et}}^{×}, where ε_{X}^{et} : X_{et} → X_{Zar} is the natural morphism of sites, the map \overline{γ}_{x} can be lifted to a monoid homomorphism γ_{x} : F′ → M_{X,x}. Moreover, γ_{x} can be extended to γ_{U} : F′ → Γ(U, M_{X}) for some Zariski neighbourhood U ⊂ |X| of x such that

1. (F′_{U_{et}})^{a} → M_{X}|_{U_{et}}, where the domain is the log structure associated with a pre-log structure F′_{U_{et}}, and

2. Γ(U, \overline{M}_{X}) → \overline{M}_{X,x}

are isomorphisms. Note here that, in particular, the composite

\overline{γ}_{U} : F′ → Γ(U, M_{X}) → Γ(U, \overline{M}_{X})

is also an isomorphism. Denoting by f the chart U → Spec ℤ[F′] obtained above, we define U′ to be the log scheme whose underlying scheme is |U| and whose log structure is induced from that on Spec ℤ[F′ \ (F′ ∩ 𝔮)] via the morphism of schemes

|U| ⟶^{|f|} |Spec ℤ[F′]| ⟶ |Spec ℤ[F′ \ (F′ ∩ 𝔮)]|.

In addition, we define an (F \ 𝔮)-frame F \ 𝔮 → Γ(U′, \overline{M}_{U′}) by composing the natural F′ \ (F′ ∩ 𝔮)-frame of U′ (induced from that of Spec ℤ[F′ \ (F′ ∩ 𝔮)]) with the projection F \ 𝔮 ≅ F′ \ (F′ ∩ 𝔮) ⊕ F′′ \ (F′′ ∩ 𝔮) → F′ \ (F′ ∩ 𝔮). Thus we have a commutative diagram of log schemes

\xymatrix{U \ar[r] \ar[d] & Spec ℤ[F′] \ar[r] \ar[d] & Spec ℤ[F] \ar[d] \\ U′ \ar[r] & Spec ℤ[F′ \ (F′ ∩ 𝔮)] \ar[r] & Spec ℤ[F \ 𝔮].}

Here the right horizontal maps are induced from projections, so become strict, and the composite of the lower horizontal morphisms is by definition (F \ 𝔮)- framed. Moreover, the composite of the upper horizontal morphisms is F -framed, that is, the given frame θ : F → Γ(U, \overline{M}_{X}) can be expressed as the composite of the projection proj : F → F′ and \overline{γ}_{U} : F′ → Γ(U, \overline{M}_{X}). Indeed, the two maps \overline{γ}_{U} ∘ proj and θ coincide with θx̄ when composed with the isomorphism Γ(U, \overline{M}_{X}) ≅ \overline{M}_{X,x̄}. Therefore it suffices to show that there exists an isomorphism U′ ≅ U^{𝔮} of (F \𝔮)- framed log schemes, which is compatible with a natural morphism U → U^{𝔮} and U → U′ defined above. From the commutative diagram

\xymatrix{F′ \ (F′ ∩ 𝔮) \ar[r] \ar[d] & Γ(U′, \overline{M}_{U′}) \ar[d] \\ F′ \ar[r] & Γ(U, \overline{M}_{U}),}

we obtain a commutative diagram of sheaves of monoids on |U|_{et}

\xymatrix{(F \ 𝔮)_{|U|} \ar[r] \ar[d] & F′ \ (F′ ∩ 𝔮)_{|U|} \ar[r] \ar[d] & \overline{M}_{U′} \ar[d] \\ F_{|U|} \ar[r] & F′_{|U|} \ar[r] & \overline{M}_{U}.}

The upper horizontal morphisms are surjective, so it is sufficient to prove the injectivity of \overline{M}_{U′} → \overline{M}_{U}. For that, it is enough to prove that, for any point y ∈ |U|, defining Q and Q′ so that both squares in the diagram

\xymatrix{Q′ \ar@{^{(}->}[r] \ar@{^{(}->}[d] & Q \ar[r] \ar@{^{(}->}[d] & 𝒪_{U,ȳ}^{×} \ar@{^{(}->}[d] \\ F′ \ (F′ ∩ 𝔮) \ar@{^{(}->}[r] & F′ \ar[r]^{γ_{U,ȳ}} & M_{U,ȳ}}

are Cartesian, the natural map (F′ \ (F′ ∩ 𝔮))/Q′ → F′/Q is injective. It is now clear since F′ is finitely generated free and both F′ \ (F′ ∩ 𝔮) and Q are faces of F′.

Secondly, given an F -framed log scheme, we can “stratify” the log scheme “along” a prime ideal of F as follows:

Definition 9. Let X be an F -framed log scheme and a ∈ Γ(X, \overline{M}_{X}). We say that the element a is pseudo-zero if, for every geometric point x̄ of X and a lift ã ∈ M_{X,x̄} of ax̄ ∈ \overline{M}_{X,x̄}, we have α_{X,x̄}(ã) = 0 as an element of 𝒪_{X,x̄}, where α_{X} : M_{X} → 𝒪_{X} is the logarithmic structure of X.

Clearly, if a is pseudo-zero, for any geometric point x̄ and any lift ã ∈ M_{X,x̄} we have α_{X,x̄}(ã) = 0. Note also that, for an element b ∈ Γ(X, M_{X}), if its image b̄ ∈ Γ(X, \overline{M}_{X}) is pseudo-zero, then α_{X}(b) is zero as an element of Γ(X, O_{X}).

Proposition-Definition 3.8. Let (X, θ) be an F -framed fs log scheme and 𝔭 a prime ideal of F. Then there exists the unique strictly closed log subscheme Y such that, for every log scheme over X, f : T → X, f factors (uniquely) through Y if and only if, for every element m ∈ 𝔭, f^{∗}(θ(m)) ∈ Γ(T, \overline{M}_{T}) is pseudo-zero. We denote this log scheme by X[𝔭].

Proof. Since it suffices to consider etale locally on X, we may assume that X is affine and has a chart X → Spec ℤ[F] that is F -framed (Proposition 3.7). Then the proposition follows from the following lemma, which is easily proven.

Lemma 3.9. Let (X, θ) be an F -framed fs log scheme with the underlying scheme Spec A and assume that θ is lifted to θ̃: F → Γ(X, M_{X}). Then X[𝔭] is Spec A/I, where I is an ideal generated by α_{X}(θ̃(m)) (m ∈ 𝔭), endowed with the log structure induced from X.

Note that X[𝔭] also becomes an F -framed fs log scheme in the natural way. The followings are easily checked.

Proposition 3.10. Let X be an F -framed fs log scheme, and 𝔮, 𝔭 and 𝔭′ be prime ideals of F. Then

1. X[∅] = X.

2. If 𝔭 ⊃ 𝔭′, then X[𝔭] ⊂ X[𝔭′].

3. (X[𝔭])[𝔭′] = X[𝔭 ∨ 𝔭′].

4. Let f : Y → X be a (strict) F -framed morphism between F -framed log schemes. Then for any prime 𝔭 of F, we have

X[𝔭] ×_{X}^{fs} Y = Y [𝔭].

In particular, if Y be a strictly closed subscheme of X, then Y [𝔭] = Y ∩ X[𝔭].

5. Assume that 𝔭 ∧ 𝔮 = ∅. Then the natural morphism X[𝔭]^{𝔮} → X^{𝔮} induces an isomorphism X[𝔭]^{𝔮} ≅ (X^{𝔮})[𝔭 \ 𝔮] with 𝔭 \ 𝔮 regarded as a prime ideal of F \ 𝔮.

Remark 3.11. As is easily checked by using Lemma 3.9, the logarithmic subscheme V_{X}(𝔭) introduced in Section 5 of [Hag03] is the maximal reduced strictly closed subscheme of X[𝔭].

The following proposition is now easily checked.

Proposition 3.12. Let X be a weakly log regular and regular fs log scheme, and assume that X is F -framed. Then, for prime ideals 𝔭 and 𝔮 of F, X[𝔭]^{𝔮} is also weakly log regular and regular. If X is log regular and regular, so is X^{𝔮}.

3.2 Pellicular Kummer etale K-theory

In this subsection we introduce the notions of “pellicular Kummer etale ringed topos” and its K-theory, which play essential roles throughout this paper.

Definition 10. Let X be an fs log scheme.

1. We define a ring object \overline{𝒪} _{X_{Ket}} in the topos (X_{Ket}) as follows:

For an object X′ in Ket/X, \overline{𝒪} _{X_{Ket}}(X′) = Γ(|X′|, 𝒪_{|X′|}^{red})

(This is indeed a sheaf, as is proven below). We call the pair ((X_{Ket}), \overline{𝒪} _{X_{Ket}}) a pellicular Kummer etale (or shortly, pellicular Ket) ringed topos, and often denote it simply by (X_{Ket}, \overline{𝒪} _{X}). Two natural morphisms of ringed topoi (X_{Ket}, \overline{𝒪} _{X}) → (X_{Ket}, 𝒪_{X}) and (X_{Ket}, \overline{𝒪} _{X}) → (X_{Zar}, 𝒪_{X}) are denoted by ε̄_{X}^{red} and by ε̄_{X}, respectively. We denote by \overline{Mod}(X_{Ket}) the category of \overline{𝒪} _{X}-modules on a site X_{Ket}.

2. An object ℱ in \overline{Mod}(X_{Ket}) is called a pellicular Ket vector bundle (resp. a pellicular Ket quasi-coherent sheaf) if, Kummer etale locally, it is isomorphic to a finite direct sum of \overline{𝒪} _{X} (resp. to the module of the form (\overline{𝒪} _{X})^{I} → (\overline{𝒪} _{X})^{J} → ℱ → 0). The \overline{𝒪} _{X}-module ℱ is said to be a pellicular Ket coherent sheaf if it is so in the sense of J. -P. Serre, that is, if ℱ is Kummer etale locally finitely generated and any Ket locally given morphism \overline{𝒪} _{U}^{n} → ℱ |_{U} has a (Kummer etale locally) finitely generated kernel.

3. We denote by \overline{Vect}(X_{Ket}), \overline{Coh}(X_{Ket}) and \overline{Qcoh}(X_{Ket}) the full subcategory of \overline{Mod}(X_{Ket}) consisting of pellicular Ket vector bundles, pellicular Ket coherent sheaves and pellicular Ket quasi-coherent sheaves, respectively.

4. A pellicular Ket line bundle is defined to be a pellicular Ket vector bundle of rank one and \overline{Pic}(X_{Ket}) denotes the group consisting of isomorphic classes of pellicular Ket line bundles.

A pellicular Ket (quasi-)coherent sheaf of \overline{𝒪} _{X}-modules is simply called a (quasi- )coherent \overline{𝒪} _{X}-module, too.

Definition 11. Let X be an fs log scheme. We denote by \overline{K}(X_{Ket}) a simplicial set made from the exact category \overline{Vect}(X_{Ket}) via Quillen’s recipe. Note that it can be regarded as the 0-th space of a fibrant spectrum (for instance, use [Wal85] and a “fibrant replacement” in 5.2 in [Tho85]). We also denote this spectrum by \overline{K}(X_{Ket}), if there is no risk of confusion, and we call it a pellicular Ket K-theory spectrum.

As is easily seen, if X is log regular, then the category \overline{Vect}(X_{Ket}) is equivalent to Vect(X_{Ket}), so we have a natural weak equivalence \overline{K}(X_{Ket}) ≅ K(X_{Ket}). Thus we will concentrate on the analysis of pellicular Ket K-theory in the following.

3.3 Structure theorem

We begin with introducing and fixing some notations. In the following, we follow Thomason’s terminology in [Tho85] (See also [Jar97]). First of all, throughout this section, we will make the following convention:

Convention 3.13. F is a finitely generated free monoid and X is an F -framed weakly log regular fs log scheme such that |X| is Noetherian, separated, regular and of finite Krull dimension, equi-characteristic of characteristic exponent p.

Note that, under these assumptions, X[𝔭] also satisfies the same conditions for any prime ideal 𝔭 (Proposition 3.12). Next, we fix a presheaf of fibrant spectra \overline{K}_{Ket} on Ket/X such that \overline{K}_{Ket}(U) is weakly homotopy equivalent to \overline{K}(U_{Ket}) for each object U of Ket/X, and for every morphism f : U → V over X, the restriction map f^{∗} : \overline{K}_{Ket}(V) → \overline{K}_{Ket}(U) is homotopic to the one induced by the exact functor f^{∗} : \overline{Vect}(V_{Ket}) → \overline{Vect}(U_{Ket}). It is possible by a suitable rectification (for example, we can use the procedure in [Jar09]). Then by the Godement-Thomason construction (i.e. Definition 1.33 in [Tho85]), we can construct a fibrant spectrum ℍ^{·}(X_{Ket}, \overline{K}_{Ket}) and a natural map of fibrant spectra

\overline{K}(X_{Ket}) ⟶ ℍ^{·}(X_{Ket}, \overline{K}_{Ket}).

Finally, for each fibrant spectrum S, we take a fibrant spectrum S_{ℚ} and a map of spectra S → S_{ℚ} which induce isomorphisms π_{i}(S)_{ℚ} ≅ π_{i}(S_{ℚ}) on homotopy groups. Recall that this construction can be carried out functorially (for instance, it is enough to use the procedure in [BK72] and a fibrant replacement), so this notation makes sense also for a presheaf of fibrant spectra. We first state the “inversion formula”, which is one of the most important tools, not only for the proof of the structure theorem, also for its formulation. For a prime ideal 𝔮 of F, we denote (X^{𝔮})_{Ket} simply by X_{Ket}^{𝔮}, and a natural morphism of sites from X_{Ket} to X_{Ket}^{𝔮} by ε_{X}^{𝔮}. By a little abuse of notations, we also denote by ε_{X}^{𝔮} (resp. ε̄_{X}^{𝔮}) a morphism of ringed topoi from (X_{Ket}, 𝒪_{X}) to (X_{Ket}^{𝔮}, 𝒪_{X^{𝔮}}) (resp. (X_{Ket}, \overline{𝒪} _{X}) to (X_{Ket}^{𝔮}, \overline{𝒪} _{X^{𝔮}})). For instance, ε_{X∗}^{𝔮}(\overline{K}_{Ket})_{ℚ} is a presheaf of fibrant spectra that associates to U ∈ Ket/X^{𝔮} a spectrum weakly equivalent to \overline{K}((U ×_{X^{𝔮}}^{fs} X)_{Ket})_{ℚ}. We can also define ε_{X∗}^{𝔮}(\overline{K}′_{Ket})_{ℚ} similarly. Now we can construct a map of fibrant spectra

\overline{K}(X_{Ket}^{𝔮})_{ℚ} ⟶^{ε̄_{X}^{𝔮∗}} \overline{K}(X_{Ket})_{ℚ} ⟶ ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}_{Ket})_{ℚ}).

Theorem 3.14. (= Corollary 5.19) Let F and X be as in Convention 3.13. Then, for a prime ideal 𝔮 of F, this is a weak equivalence:

\overline{K}(X_{Ket}^{𝔮})_{ℚ} ⟶^{∼} ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}_{Ket})_{ℚ}).

As a result of the above theorem, we can define an “extraordinary map” as follows:

Definition 12. For a pair of prime ideals 𝔮′ ⊃ 𝔮 and an integer i ≥ 0, we define a map of fibrant spectra π_{i}(δ_{X}^{𝔮′,𝔮}) to be the composite of the following maps:

\overline{K}_{i}(X_{Ket}^{𝔮})_{ℚ} →^{ε̄_{X}^{𝔮∗}} \overline{K}_{i}(X_{Ket})_{ℚ} → π_{i}ℍ^{·}(X_{Ket}^{𝔮′^{∨}}, ε_{X∗}^{𝔮^{′∨}}(\overline{K}_{Ket})_{ℚ}) → \overline{K}_{i}(X_{Ket}^{𝔮′})_{ℚ}.

Here the third map is the inverse of the isomorphism between the i-th homotopy groups in Theorem 3.14. The notation π_{i}(δ_{X}^{𝔮′,𝔮}) is a little abusive because we will not consider the map of spectra which should be denoted by “δ_{X}^{𝔮′,𝔮}”, but it causes no confusion. We often omit the subscript X or the superscripts 𝔮, 𝔮′ if they are unnecessary.

For the next proposition, we introduce some additional notations. Observing Proposition 3.6 (2), for a pair of prime ideals 𝔮′ ⊃ 𝔮, we denote by ε_{X}^{𝔮,𝔮′} a natural morphism of sites from X_{Ket}^{𝔮} to X_{Ket}^{𝔮′}, and also denote by ε_{X}^{𝔮,𝔮′} (resp. \overline{ε}_{X}^{𝔮,𝔮′}) a morphism of ringed topoi from (X_{Ket}^{𝔮}, 𝒪_{X^{𝔮}}) to (X_{Ket}^{𝔮′}, 𝒪_{X^{𝔮′}}) (resp. from (X_{Ket}^{𝔮}, \overline{𝒪} _{X^{𝔮}}) to (X_{Ket}^{𝔮′}, \overline{𝒪} _{X^{𝔮′}})). Clearly they induce maps of fibrant spectra \overline{ε}_{X}^{𝔮,𝔮′∗} : \overline{K}(X_{Ket}^{𝔮′}) → \overline{K}(X_{Ket}^{𝔮}), ℍ^{·}(X_{Ket}^{𝔮′}, ε_{X∗}^{𝔮′}\overline{K}_{Ket}) ⟶ ℍ^{·}(X_{Ket}^{𝔮}, ε_{X∗}^{𝔮}\overline{K}_{Ket}), and moreover

ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}_{Ket})_{ℚ}) ⟶ ℍ^{·}(X_{Ket}^{𝔮^{′∨}}, ε_{X∗}^{𝔮^{′∨}}(\overline{K}_{Ket})_{ℚ})

(Note that we have 𝔮′^{∨} ⊂ 𝔮^{∨}). The following propositions easily follow from the construction:

Proposition 3.15. For a string of prime ideals 𝔮′′ ⊃ 𝔮′ ⊃ 𝔮 and an integer i ≥ 0, we have

1. The composite π_{i}(δ_{X}^{𝔮′,𝔮}) ∘ π_{i}(ε̄_{X}^{𝔮,𝔮′∗}) is the identity map.

2. The composite π_{i}(δ_{X}^{𝔮′′,𝔮′}) ∘ π_{i}(δ_{X}^{𝔮′,𝔮}) and π_{i}(δ_{X}^{𝔮′′,𝔮}) coincide.

3. If Y also satisfies the assumptions at the beginning of this section and f : X → Y is F -framed, then π_{i}(δ_{X}^{𝔮′,𝔮}) ∘ f^{∗} = f^{∗} ∘ π_{i}(δ_{Y}^{𝔮′,𝔮}) as maps from \overline{K}_{i}(Y_{Ket}^{𝔮})_{ℚ} to \overline{K}_{i}(X_{Ket}^{𝔮′})_{ℚ}.

Proposition 3.16. The homomorphism π_{0}(δ_{X}^{𝔮′,𝔮}) : \overline{K}_{0}(X_{Ket}^{𝔮})_{ℚ} → \overline{K}_{0}(X_{Ket}^{𝔮′})_{ℚ} is compatible with λ-ring structures.

Proof. First we note that the statement can be rewritten in the framework of the theory of simplicial presheaves. Then the proposition follows from the existence of the construction of product and λ-operations at the level of simplicial sets. More precisely, it is sufficient to use Gillet-Grayson’s G-construction in [GG87] (plus [GG03]), and to use [Gra89] for λ-operations, and [Jar09] (for example) for the product structure.

Next, we decompose Kummer etale K-groups “according to stratifications” by introducing an auxiliary category as below.

Definition 13. We define a category I_{F} as follows: The set of objects consists of the pairs (𝔭, 𝔮) of prime ideals of F satisfying 𝔭 ⊃ 𝔮. For a clarification, we describe these objects like (𝔭 ⊃ 𝔮). Each hom-set consists of at most one element, and we have a morphism (𝔭 ⊃ 𝔮) → (𝔭′ ⊃ 𝔮′) if and only if 𝔭 ⊃ 𝔭′ ⊃ 𝔮′ ⊃ 𝔮.

By Propositions 3.10 and 3.6 (2), for pairs of prime ideals 𝔭 ⊃ 𝔭′ and 𝔮′ ⊃ 𝔮, we have a morphism of sites

X[𝔭]_{Ket}^{𝔮} → X[𝔭′]_{Ket}^{𝔮′},

and a map of fibrant spectra

ℍ^{·}(X[𝔭′]_{Ket}^{𝔮′}, ε_{X[𝔭′]∗}^{𝔮′}\overline{K}_{Ket}) ⟶ ℍ^{·}(X[𝔭]_{Ket}^{𝔮}, ε_{X[𝔭]∗}^{𝔮}\overline{K}_{Ket}).

Taking their i-th homotopy groups (i ≥ 0), we can define a contravariant functor Π_{i} from I_{F} to the category of Abelian groups, by the rule

Π_{i}((𝔭 ⊃ 𝔮)) = π_{i}ℍ^{·}(X[𝔭]_{Ket}^{𝔮}, ε_{X[𝔭]∗}^{𝔮}\overline{K}_{Ket}).

Noting that \overline{K}_{i}(X_{Ket})_{ℚ} ⟶^{≅} π_{i}ℍ^{·}(X[∅]_{Ket}^{∅^{∨}}, ε_{X[∅]∗}^{∅^{∨}}(\overline{K}_{Ket})_{ℚ})(Theorem 3.14), we have a morphism of Abelian groups

\overline{K}_{i}(X_{Ket})_{ℚ} ⟶ lim_{𝔭⊃𝔮} π_{i}ℍ^{·}(X[𝔭]_{Ket}^{𝔮}, ε_{X[𝔭]∗}^{𝔮}(\overline{K}_{Ket})_{ℚ}),

where we use the notation “lim_{𝔭⊃𝔮}” to mean a limit with respect to the category I_{F}. The next theorem will be proven in Subsection 5.6.

Theorem 3.17. The above map is an isomorphism for each i ≥ 0.

Finally, we describe each piece more explicitly. As in [Hag03], we can define a Ket line bundle 𝒪_{X}(ξ) (more precisely, its isomorphism class) associated with an element ξ ∈ Γ(X, \overline{M}_{X}^{gp}) ⊗_{ℤ} ℚ′ to be its image by the homomorphism

Γ(X, \overline{M}_{X}^{gp}) ⊗_{ℤ} ℚ′ ≅ Γ(X_{Ket}, \overline{M}_{X_{Ket}}^{gp}) ⟶^{∂} H^{1}(X_{Ket}, 𝒪_{X_{Ket}}^{×}) ≅ Pic (X_{Ket}),

where ℚ′ = ℤ_{(p)} with p the characteristic of |X|. Here ∂ is induced by the short exact sequence of Abelian sheaves on X_{Ket}

0 → 𝒪_{X_{Ket}}^{×} → M_{X_{Ket}}^{gp} → \overline{M}_{X_{Ket}}^{gp} → 0.

In addition, we can complete this diagram as follows:

\xymatrix{Γ(X, \overline{M}_{X})_{ℤ} \ar[r]^{∂} \ar[d] & Pic (X_{Zar}) \ar[d] \\ Γ(X, \overline{M}_{X})_{ℚ′} \ar[r]^{∂} & Pic (X_{Ket}) \ar[r]^{ε̄^{red∗}} \ar[d] & \overline{Pic} (X_{Ket}) \ar[d] \\  & K_{0}(X_{Ket}) \ar[r]^{ε̄^{red∗}} & \overline{K}_{0}(X_{Ket}).}

Note that all maps are monoid homomorphisms (with K-groups regarded as monoids by the multiplication). For ξ ∈ Γ(X, \overline{M}_{X})_{ℚ′} we denote by \overline{𝒪} _{X}(ξ) the corresponding element in \overline{Pic} (X_{Ket}) or \overline{K}_{0}(X_{Ket}). Accordingly, using a frame F → Γ(X, \overline{M}_{X}), we can also define a pellicular Ket line bundle \overline{𝒪} _{X}(ξ) for ξ ∈ F_{ℚ′}. If a prime ideal 𝔮 of F is given, we can make the above procedure work for X^{𝔮} to construct the morphisms

(F \ 𝔮)_{ℚ′} ⟶ Γ(X_{Ket}^{𝔮}, \overline{M}_{X_{Ket}^{𝔮}}^{gp}) ⟶^{∂} Pic (X_{Ket}^{𝔮}).

Moreover, since the commutative diagram

\xymatrix{(F \ 𝔮)_{ℚ′} \ar[r] & Γ(X_{Ket}^{𝔮}, \overline{M}_{X_{Ket}^{𝔮}}^{gp}) \ar[r]^{∂} & Pic (X_{Ket}^{𝔮}) \\ (F \ 𝔮)_{ℤ} \ar[r] \ar@{^{(}->}[u] \ar@{^{(}->}[d] & Γ(X^{𝔮}, \overline{M}_{X^{𝔮}}^{gp}) \ar[u] \ar[d] \ar[r]^{∂} & Pic (X_{Zar}^{𝔮}) \ar[u] \ar[d]^{=} \\ F_{ℤ} \ar[r] & Γ(X, \overline{M}_{X}^{gp}) \ar[r]^{∂} & Pic (X_{Zar})}

induces a map

(F^{𝔮−div′})^{gp} ⟶ Pic (X_{Ket}^{𝔮}),

we can define objects 𝒪_{X_{Ket}^{𝔮}}(ξ) ∈ Pic (X_{Ket}^{𝔮}) and \overline{𝒪} _{X_{Ket}^{𝔮}}(ξ) ∈ \overline{Pic} (X_{Ket}^{𝔮}) for ξ ∈ (F^{𝔮−div′})^{gp}. Note that this defines a group endomorphism of \overline{K}_{i}(X_{Ket}^{𝔮}) for each i ≥ 0 via “− ⊗ \overline{𝒪} _{X_{Ket}^{𝔮}}(ξ)”. Notice also that, for prime ideals 𝔮 ⊂ 𝔮′ and an element ξ ∈ (F^{𝔮′−div′})^{gp}, we see (ε_{X}^{𝔮,𝔮′})^{∗}𝒪_{X^{𝔮′}} (ξ) ≅ 𝒪_{X^{𝔮}}(ξ) and (\overline{ε}_{X}^{𝔮,𝔮′})^{∗}\overline{𝒪} _{X^{𝔮′}} (ξ) ≅ \overline{𝒪} _{X^{𝔮}}(ξ) under the identification of (F^{𝔮′−div′})^{gp} with a subgroup of (F^{𝔮−div′})^{gp}.

Definition 14. Let 𝔮 be a prime ideal of F.

1. We denote by J(𝔮) the category such that (a) the set of objects is F (𝔮)_{ℚ},

(b) for two objects α and β, Hom (α, β) is the set consisting of one element β − α if β − α ∈ F (𝔮), and the empty set if not, and

(c) the composition is defined by using the addition in F (𝔮).

2. Let C be a category having countable coproducts and A ∈ Ob(C), and assume that we are given a monoid morphism L : F (𝔮) → Hom_{C}(A, A), where Hom_{C}(A, A) is regarded as a (not necessarily commutative) monoid by composition.

Then, a covariant functor G_{𝔮,A,L} from J(𝔮) to C is defined by

(a) G_{𝔮,A,L}(α) = A, independently of the object α, and

(b) For any pair of objects α and β such that β − α ∈ F (𝔮), G_{𝔮,A,L}(β − α) = L(β − α).

We set

A ⋊^{L} Λ[𝔮] = colim G_{𝔮,A,L}.

3. For a prime ideal 𝔭 of F, let C be the category of Abelian groups and A = K_{i}(X[𝔭]_{Zar}), and for γ ∈ F (𝔮) we define L(γ) to be the group morphism

K_{i}(X[𝔭]_{Zar}) ⟶ K_{i}(X[𝔭]_{Zar})

induced by the endofunctor “− ⊗ 𝒪_{X[𝔭]}(γ)” of Vect(X[𝔭]_{Zar}). Then we set

K_{i}(X[𝔭]_{Zar}) ⋊ Λ[𝔮] = A ⋊^{L} Λ[𝔮].

In practice, we use its variant K_{i}(X[𝔭]_{Zar}) ⋊ Λ′[𝔮] which is defined by replacing ℚ (in the definition of J(𝔮)) by ℚ′ = ℤ_{(p)}.

Remark 3.18. 1. This notation is indeed highly abusive, in the sense that K_{i}(X[𝔭]_{Zar}) ⋊ Λ[𝔮] is determined not only by K_{i}(X[𝔭]_{Zar}) and 𝔮, and actually the Abelian group Λ[𝔮] itself is not used in its definition. However, the author believes that it does not induce any risk of confusions. Anyway, the notation “⋊” symbolises the hope that we want to construct the “twisted” product of K(X[𝔭]_{Zar}) and Λ[𝔮] in a suitable sense.

2. As Abelian groups, we have an isomorphism

K_{i}(X[𝔭]_{Zar}) ⋊ Λ′[𝔮] ≅ K_{i}(X[𝔭]_{Zar}) ⊗_{ℤ} Λ′[𝔮].

As is already mentioned, with every ξ ∈ F (𝔮)_{ℚ′}, we associate a pellicular Ket line bundle \overline{𝒪} _{X[𝔭]_{Ket}^{𝔮^{∨}}}(ξ) via a map

F (𝔮)_{ℚ′} ↪ (F^{𝔮^{∨}−div′})^{gp} → Γ(X[𝔭]_{Ket}^{𝔮^{∨}}, \overline{M}_{X[𝔭]_{Ket}^{𝔮^{∨}}}^{gp}) ⟶^{∂} \overline{Pic}(X[𝔭]_{Ket}^{𝔮^{∨}}).

By using this, for α ∈ F (𝔮)_{ℚ′}, we can define a functor from Vect(X[𝔭]_{Zar}) to \overline{Vect}(X[𝔭]_{Ket}^{𝔮^{∨}}) which send each locally free sheaf of 𝒪_{X[𝔭]_{Zar}}-modules ℱ to a pellicular Kummer etale vector bundle ε̄^{∗}ℱ ⊗ \overline{𝒪} _{X[𝔭]_{Ket}^{𝔮^{∨}}}(−α) on X[𝔭]^{𝔮^{∨}}.

It induces a group homomorphism K_{i}(X[𝔭]_{Zar}) to \overline{K}_{i}(X[𝔭]_{Ket}^{𝔮^{∨}}), and a morphism K_{i}(X[𝔭]_{Zar}) ⋊ Λ′[𝔮] ⟶ \overline{K}_{i}(X[𝔭]_{Ket}^{𝔮^{∨}}) by the universality of a colimit.

Theorem 3.19. For prime ideals 𝔭 and 𝔮 of F satisfying 𝔭 ⊃ 𝔮 and an integer i ≥ 0, the homomorphism above is an isomorphism, that is, we have

K_{i}(X[𝔭]_{Zar}) ⋊ Λ′[𝔮] ⟶^{≅} \overline{K}_{i}(X[𝔭]_{Ket}^{𝔮^{∨}}).

For a simpler description of this isomorphism, we introduce the following notation.

Definition 15. For α ∈ F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ, take a representative a/n ∈ F (𝔮)_{ℚ′} for some a ∈ F (𝔮)_{ℤ} and n ∈ ℕ, and denote by x = [𝒪_{X}(a)] the image of a via a map F^{gp} → Pic (X_{Zar}) → K_{0}(X_{Zar}). Then we set

⟨⟨α⟩⟩ = [\overline{𝒪} _{X_{Ket}^{𝔮^{∨}}}(a/n)] ε̄^{∗}(exp(−(log x)/n)) ∈ \overline{K}_{0}(X_{Ket}^{𝔮^{∨}})_{ℚ},

where ε̄^{∗} is a natural map from K_{0}(X_{Zar})_{ℚ} to K_{0}(X_{Ket}^{𝔮^{∨}})_{ℚ}, and exp(T) and log(T) are formal power series in ℚ[[T]] and ℚ[[T − 1]], respectively.

We can easily check its well-definedness by noticing that x − 1 is nilpotent in K_{0}(X_{Zar}), and clearly we see that ⟨⟨−⟩⟩ is a monoid homomophism from F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ to \overline{K}_{0}(X_{Ket}^{𝔮^{∨}})_{ℚ} (The latter is regarded as a monoid by a multiplication). Note that these constructions also work for X[𝔭]. The following is only a rewrite of Theorem 3.19.

Corollary 3.20. Let notations and assumptions be as in Theorem 3.19. Then we have a group isomorphism

K_{i}(X[𝔭]_{Zar}) ⊗_{ℤ} Λ′[𝔮]_{ℚ} ⟶^{≅} \overline{K}_{i}(X[𝔭]_{Ket}^{𝔮^{∨}})_{ℚ},

which maps x ⊗ [α] to ε̄^{∗}(x)⟨⟨α⟩⟩ for x ∈ K_{i}(X[𝔭]_{Zar}) and α ∈ F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ.

This map enables us to make the λ-ring structure of the right hand side explicit. Here we will describe the action of Adams operations. For that we introduce that on Λ′[𝔮].

Definition 16. 1. We regard Λ′[𝔮] as a group ring over an Abelian group F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ. In addition, for each natural number m, we define Ψ_{Λ}^{m} to be a ring endomorphism on Λ′[𝔮] which sends [α] to [mα] for α ∈ F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ. We call them “Adams operations” on Λ′[𝔮].

2. On K_{0}(X[𝔭]_{Zar})⊗_{ℤ} Λ′[𝔮]_{ℚ}, we introduce a ring structure as a usual tensor product and the actions of “Adams operations” as Ψ_{Zar}^{m} ⊗Ψ_{Λ}^{m}, where Ψ_{Zar}^{m} on K_{0}(X[𝔭]_{Zar})_{ℚ} are usual Adams operations.

3. Let Ψ_{Ket}^{m} be Adams operations on K_{0}(X_{Ket})_{ℚ} constructed by the usual procedure from its λ-ring structure.

Remark that a priori we only see that the operations Ψ_{Ket}^{m} satisfy additivity, the usual property for line bundle elements, and the compatibility with pullback maps.

Corollary 3.21. In the case of i = 0, the isomorphism in Corollary 3.20 respects ring structures and Adams operations of both hand sides.

Proof. First we check the compatibility with Adams operations. We have only to check Ψ_{Ket}^{m}(ε̄^{∗}(x)⟨⟨α⟩⟩) = ε̄^{∗}(Ψ_{Zar}^{m}(x))Ψ_{Λ}^{m}(⟨⟨α⟩⟩) for x ∈ K_{0}(X[𝔭]_{Zar})_{ℚ} and α ∈ F (𝔮)^{gp} ⊗_{ℤ} ℚ′/ℤ. By the splitting principle (for K_{0}(X[𝔭]_{Zar})) and Corollary 3.20, we may assume that x is a line bundle element, and then the claim follows because ε̄^{∗}(x)⟨⟨α⟩⟩ can be written by a linear combination of line bundle elements. The preservation of product structures can be proven similarly.

The compatibility of these maps becomes clear by the theorem below.

Theorem 3.22. For prime ideals 𝔭 ⊃ 𝔭′ ⊃ 𝔮′ ⊃ 𝔮, we have a commutative diagram

\xymatrix{K_{i}(X[𝔭′]_{Zar}) ⊗_{ℤ} Λ′[𝔮′]_{ℚ} \ar[r] \ar[d] & π_{i}ℍ^{·}(X[𝔭′]_{Ket}^{𝔮′}, ε_{X[𝔭′]∗}^{𝔮′}(\overline{K}_{Ket})_{ℚ}) \ar[d] \\ K_{i}(X[𝔭]_{Zar}) ⊗_{ℤ} Λ′[𝔮]_{ℚ} \ar[r] & π_{i}ℍ^{·}(X[𝔭]_{Ket}^{𝔮}, ε_{X[𝔭]∗}^{𝔮}(\overline{K}_{Ket})_{ℚ}).}

Here

• the horizontal maps are the composites of isomorphisms given in Corollary 3.20 and Theorem 3.14,

• the right vertical one sheaf-theoretically defined, and

• the left vertical one induced by the inverse image functor with respect to X[𝔭] → X[𝔭′] and the monoid homomorphism F (𝔮′) → F (𝔮) given in Remark 3.5.

The proofs of Theorems 3.14, 3.17, 3.19 and 3.22 will be given in Section 5. Note that, when i = 0, the left vertical map is compatible with product and the actions of Adams operations defined above. Recalling that K(X_{Ket}) ≅ \overline{K}(X_{Ket}) in our case, we obtain the theorem and the corollary below, immediately from these theorems.

Theorem 3.23. For each i ≥ 0, we have an isomorphism

K_{i}(X_{Ket})_{ℚ} ⟶^{≅} lim_{𝔭⊃𝔮} K_{i}(X[𝔭]_{Zar}) ⊗_{ℤ} Λ′[𝔮]_{ℚ}.

Moreover, if i = 0, this isomorphism is compatible with Adams operations on both hand sides.

As a result, we can decompose K_{0}(X_{Ket})_{ℚ} via Adams operations to define what should be called “Kummer etale Chow groups”.

Definition 17. An element x in K_{0}(X_{Ket})_{ℚ} is called of weight i if there exists a finite set S of prime numbers and a natural number N ≥ 1 such that for every natural number m prime to all numbers in S, (Ψ_{Ket}^{m})^{N}(x) = m^{iN}x. We denote by K_{0}(X_{Ket})^{(i)} the subgroup of K_{0}(X_{Ket})_{ℚ} consisting of elements of weight i.

Corollary 3.24. We have an isomorphism

K_{0}(X_{Ket})_{ℚ} ≅ ⨁_{i=0}^{dim |X|} K_{0}(X_{Ket})^{(i)}.

Moreover, for each piece of the right hand side, we have an isomorphism of Abelian groups

K_{0}(X_{Ket})^{(i)} ≅ lim_{𝔭⊃𝔮} CH^{i}(X[𝔭]_{Zar}) ⊗ Λ′[𝔮]_{ℚ}.

In the rest of this paper we will prove the theorems stated above.

4 General theory of pellicular Kummer etale sheaves

4.1 Pellicular Kummer etale ringed topos

In this subsection, we develop a general theory of pellicular Ket ringed topos introduced in Subsection 3.2. First we prove a proposition on the Kummer etale topos.

Proposition 4.1. Let X be an fs log scheme and X_{red} its strictly closed subscheme with its underlying scheme |X|_{red}. Then the pullback map (in the category of fs log schemes) induces an equivalence of categories

(X_{Ket}) ≅ ((X_{red})_{Ket}).

Proof. First we prove that the functor Ket/X → Ket/X_{red} is fully faithful. This can be proven in the similar way to the classical case by using the following well-known or easily-proven facts; Let f : X → Y and g : Y → Z be morphisms of fs log schemes. Then

• If g and gf are Kummer etale, so is f.

• If g is Kummer etale and gf is an isomorphism, f is a strict open immersion.

• The schemes |Y ×_{X}^{fs} X_{red}| and |Y | have homeomorphic underlying topological spaces.

• If g is Kummer etale, f is strict and surjective, and gf is strict radicial, then g is an isomorphism.

Then the proposition follows from the next lemma and Théorème 4.1 in Exposé III of [SGA4].

Lemma 4.2. Let X be an fs log scheme, f : U → X_{red} a Kummer etale morphism, and u ∈ U. Then there exist an fs log scheme V Kummer etale over X and an etale morphism g : V ×_{X} X_{red} → U such that

\xymatrix{V ×_{X} X_{red} \ar[d]_{g} \ar[r] & V \ar[dd] \\ U \ar[d]_{f} \\ X_{red} \ar[r] & X}

is commutative and u ∈ Image (g).

Proof. We may assume that X has a chart X → Spec ℤ[P] with P fs and sharp such that P ≅ \overline{M}_{X,\overline{f(u)}}. Then, by Proposition 2.4, we can construct a diagram

\xymatrix{ & W_{2} \ar[ddl]_{f′} \ar[d]^{f′′} \\  & W_{1} \ar[r] \ar[d] & Spec ℤ[Q] \ar[d]^{Spec ℤ[h]} \\ U \ar[r]^{f} & X_{red} \ar[r] & Spec ℤ[P],}

where h : P → Q is of Kummer type with #Coker h^{gp} invertible on W_{2}, u ∈ Image (f′), and f′ and f′′ are classically etale. Here we may also assume that #Coker h^{gp} is invertible on X. If we set V_{1} = X ×_{Spec ℤ[P]}^{fs} Spec ℤ[Q], then W_{1} ≅ V_{1} ×_{X}^{fs} X_{red}. In addition, we can find V_{2} classically etale over V_{1} such that W_{2} ≅ W_{1} ×_{V_{1}} V_{2}. This V_{2} is a desired log scheme.

Next we investigate some properties of pellicular Ket ringed topos.

Proposition 4.3. Let X be an fs log scheme and ℱ a quasi-coherent sheaf of 𝒪_{X_{Zar}}-modules. Then the presheaf on X_{Ket} which associates to π : X′ → X an Abelian group Γ(X′, 𝒪_{X′}^{red} ⊗_{𝒪′_{X}} π^{∗}ℱ) is a sheaf.

Proof. Note first that the sheaf condition is satisfied for strictly etale morphisms since, if f : U → V is an etale morphism of schemes, then U_{red} → V_{red} is also etale and (U ×_{V} U)_{red} ≅ U_{red} ×_{V_{red}} U_{red}. By Corollary 2.5, it suffices to show the next lemma.

Lemma 4.4. Assume that we are given a reduced ℤ[1/n, ζ_{n}]-algebra A, an fs sharp monoid P and a monoid homomorphism P → A. Then, for any Amodule M,

0 ⟶ M ⟶ A_{n}^{red} ⊗_{A} M ⟶ (A_{n} ⊗_{A}^{fs} A_{n})^{red} ⊗_{A} M

is exact. Here,

A_{n} = A ⊗_{ℤ[P]} ℤ[P^{1/n}] and A_{n} ⊗_{A}^{fs} A_{n} = A ⊗_{ℤ[P]} ℤ[P^{1/n} ⊕_{P}^{sat} P^{1/n}],

and the middle and right arrows are induced by natural ring homomorphisms A → A_{n} and A_{n} ⇉ A_{n} ⊗_{A}^{fs} A_{n}, respectively.

Proof. We first recall that it is already known that

0 ⟶ M ⟶ A_{n} ⊗_{A} M ⟶ (A_{n} ⊗_{A}^{fs} A_{n}) ⊗_{A} M

is exact (Lemma 3.3 in [Hag03]). Set I_{n} = Hom (P^{1/n}/P, ℤ [ζ_{n}]^{×}) and let N_{n} be the nilradical of A_{n}. Note that A_{n} is acted by I_{n}. By the above fact we obtain that A ⟶^{≅} (A_{n})^{I_{n}} (Here, (−)^{I_{n}} is the I_{n}-invariant part), which implies (N_{n})^{I_{n}} = 0. By the exactness of the functor (−)^{I_{n}} (from the category of I_{n}-ℤ[1/n]-modules to that of Abelian groups), we also have A ⟶^{≅} (A_{n}^{red})^{I_{n}}. Since (A_{n} ⊗_{A}^{fs} A_{n})^{red} ≅ ∏_{I_{n}}(A_{n}^{red}), the lemma follows for M = A. For any A-module M, using a free resolution A^{⊕X} → A^{⊕Y} → M → 0 and the exactness of (−)^{I_{n}}, we see M ⟶^{≅} (A_{n}^{red} ⊗_{A} M)^{I_{n}}, which proves the lemma.

Corollary 4.5. Let X be an fs log scheme.

1. The presheaf \overline{𝒪} _{X} is indeed a sheaf.

2. For any quasi-coherent sheaf of 𝒪_{X_{Zar}}-modules ℱ and a Kummer etale morphism π : X′ → X, we have

Γ(X′, ε̄_{X}^{∗}ℱ) ⟶^{≅} Γ(X′, 𝒪_{X′}^{red} ⊗_{𝒪_{X′}} π^{∗}ℱ).

3. For any Zariski quasi-coherent sheaf ℱ on X, we have ε̄_{∗}ε̄^{∗}ℱ ≅ 𝒪_{X_{red}} ⊗ ℱ.

Assume moreover that |X| is reduced. Then

4. For a quasi-coherent sheaf of 𝒪_{X_{Zar}}-modules ℱ, the canonical morphism ℱ → ε̄_{∗}ε̄^{∗}ℱ is an isomorphism.

5. The functor ε̄^{∗} : Qcoh(X_{Zar}) → \overline{Mod}(X_{Ket}) is fully faithful.

The following proposition is easily verified and will be used often.

Proposition 4.6. Let X be an fs log scheme and i : X_{red} → X the natural strict morphism. Then 1. The morphism i induces the two functors i_{∗} and i^{∗} which give equivalences

of categories between \overline{Mod}(X_{Ket}) and \overline{Mod}((X_{red})_{Ket}). Similar results hold for \overline{Qcoh}, \overline{Coh}, and \overline{Vect}.

2. We have an isomorphism of functors

i_{∗}ε̄_{X_{red}}^{∗} ≅ ε̄_{X}^{∗}i_{∗}^{Zar}

from Qcoh((X_{red})_{Zar}) to \overline{Mod}(X_{Ket}). Here i_{∗}^{Zar} denotes the natural functor from Mod((X_{red})_{Zar}) to Mod(X_{Zar}).

Proof. Use Proposition 4.1, Lemma 4.2 and Corollary 4.5. Note also that we have an isomorphism of endofunctors i_{Zar}^{∗}i_{∗}^{Zar} ≅ Id of Mod((X_{red})_{Zar}), where i_{Zar}^{∗} denotes the natural functor from Mod(X_{Zar}) to Mod((X_{red})_{Zar}).

In the following we often identify \overline{Mod}(X_{Ket}) and \overline{Mod}((X_{red})_{Ket}) for simplicity.

Definition 18. An fs log scheme X is said to have the property (EX) if X satisfies the following condition: For every Kummer etale morphism X′ → X, there exist a Kummer etale covering X′′ → X′ of X′ such that the functor

ε̄_{X′′_{red}}^{∗} : Mod((X′′_{red})_{Zar}) → \overline{Mod}(X′′_{Ket})

is exact.

Clearly, for a Kummer etale morphism f : Y → X, if X has the property (EX), so does Y. On the other hand, if f is Kummer etale surjective and Y has the property (EX), so does X.

Remark 4.7. Note that the exactness of ε̄_{X}^{∗} implies that |X| is reduced.

The importance of this condition will be made clear by the proposition below.

Proposition 4.8. Let X be an fs log scheme.

1. The subcategory \overline{Vect}(X_{Ket}) has a structure of exact categories in the natural way.

2. We assume that |X| is Noetherian and X has the property (EX).

(a) For an object ℱ in \overline{Mod}(X_{Ket}), the followings are equivalent; i. ℱ is a pellicular Ket coherent sheaf. ii. There exists a Kummer etale covering {X′ → X} of X such that ℱ |_{X′} belongs to the essential image by ε̄_{X′}^{∗} of Coh(X′_{Zar}). iii. There exists a Kummer etale covering {X′ → X} of X such that ℱ |_{X′} belongs to the essential image by ε̄_{X′_{red}}^{∗} of Coh((X′_{red})_{Zar}), under the identification of \overline{Mod}(X′_{Ket}) with \overline{Mod}((X′_{red})_{Ket}) in Proposition 4.6.

In particular, the functor ε̄_{X}^{∗} : Mod(X_{Zar}) → \overline{Mod}(X_{Ket}) induces an functor ε̄_{X}^{∗} : Coh(X_{Zar}) → \overline{Coh}(X_{Ket}).

(b) The category \overline{Coh}(X_{Ket}) is Abelian, and the inclusion functor to \overline{Mod}(X_{Ket}) is exact.

3. The similar statements hold for quasi-coherent sheaves (for any fs log scheme X).

4. Assume moreover that the functor ε̄_{X}^{∗} : Mod(X_{Zar}) → \overline{Mod}(X_{Ket}) is exact. Then the functors ε̄_{X}^{∗} : Vect(X_{Zar}) → \overline{Vect}(X_{Ket}) and ε̄_{X}^{∗} : Qcoh(X_{Zar}) → \overline{Qcoh}(X_{Ket}), induced by the restriction, are exact. If X is Noetherian and has the property (EX), the similar statement holds also for \overline{Coh}(X_{Ket}).

Proof. We begin with the proof of (2-a). To prove (i)⇒(ii), assume ℱ ∈ \overline{Coh}(X_{Ket}). Then we have a Kummer etale covering U of X with U Noetherian and a morphism f : \overline{𝒪} _{U}^{m} → \overline{𝒪} _{U}^{n} such that ℱ |_{U} ≅ Coker f. Refining the covering U if necessary, we may assume that Γ(U, 𝒪_{U}) → Γ(U, 𝒪_{U}^{red}) is surjective. Then by Corollary 4.5 (5), we have a morphism f̃ : 𝒪_{U_{Zar}}^{m} → 𝒪_{U_{Zar}}^{n} on U_{Zar} which induces f by ε̄_{U}^{∗}. Then we obtain ε̄_{U}^{∗}(Coker f̃) ≅ Coker f. The implication (ii)⇒(iii) is trivial. For the implication (iii)⇒(i), it suffices to show that, for a Noetherian fs log scheme U such that ε̄_{U_{red}}^{∗} is exact, a coherent sheaf of 𝒪_{U_{red}}-modules ℱ̃ on (U_{red})_{Zar}, and a homomorphism f : \overline{𝒪} _{U}^{m} → ε̄_{U_{red}}^{∗} ℱ̃, the kernel Ker f is Ket locally finitely generated (Here we used the property (EX)). By Corollary 4.5 (5) and Remark 4.7 we have a homomorphism f̃: 𝒪_{(U_{red})_{Zar}}^{m} →

ℱ̃such that f = ε̄_{U_{red}}^{∗}(f̃). Since Ker f̃ is Zariski locally finitely generated, we see that ε̄_{U_{red}}^{∗}(Ker f̃) ≅ Ker f is also finitely generated by the exactness of ε̄_{U_{red}}^{∗}. Next we prove (2-b). We have only to check that if f : ℱ → ℱ′ is a homomorphism of pellicular Ket coherent sheaves, then Ker f and Coker f are also pellicular Ket coherent. Take a Ket covering U and Zariski coherent sheaves ℱ̃and ℱ̃′ on U_{red} such that ℱ |_{U} ≅ ε̄_{U_{red}}^{∗} ℱ̃, ℱ′|_{U} ≅ ε̄_{U_{red}}^{∗} ℱ̃′, and ε̄_{U_{red}}^{∗} is exact. By Corollary 4.5 (5), f comes from some f̃: ℱ̃→ ℱ̃′. Then Ker f̃ and Coker f̃ are coherent, which means that Ker f and Coker f are pellicular Ket coherent.

(3) is verified similarly and more easily. (1) and (4) are obvious.

Definition 19. A pellicular Ket quasi-coherent sheaf (a pellicular Ket vector bundle, respectively) on X is said to be classical if it belongs to the essential image of Qcoh((X_{red})_{Zar}) (Vect((X_{red})_{Zar}), respectively) by ε̄_{X_{red}}^{∗}. A pellicular Ket line bundle is defined similarly. If X is Noetherian and has the property (EX), the notion of classicality of pellicular Ket coherent sheaves is also defined similarly.

Remark 4.9. For pellicular Ket (quasi-)coherent sheaves, the notion of classicality can also be defined by using the essential image of Coh(X_{Zar}) (or Qcoh(X_{Zar})).

Later we will consider some sufficient conditions for fs log schemes to have the property (EX). Now we introduce a pellicular K′-theory spectrum.

Definition 20. For a Noetherian fs log scheme X satisfying (EX), we define a pellicular Ket K′-theory spectrum \overline{K}′(X_{Ket}) to be the spectrum constructed from the exact category \overline{Coh}(X_{Ket}).

The relation to equivariant sheaf theory is given by the next proposition.

Proposition 4.10. Let n be a positive integer, X a Noetherian fs log scheme over Spec ℤ[1/n] and X ⟶ Spec ℤ[P] a chart with P fs and sharp. Then we have a natural equivalence of categories between the one of pellicular Ket quasi-coherent sheaves which become classical on X̃_{n} and the one of Zariski quasi-coherent sheaves of 𝒪_{(X̃_{n})_{red}}-modules with an action of I_{n} ⋊ Γ_{n} (For the notations, see Subsection 2.1.3). Similarly for coherent sheaves, vector bundles and line bundles.

Proof. This follows from the argument in the usual sheaf theory via Proposition 2.1 and Corollary 4.5 (5).

Proposition 4.11. For a pellicular Ket quasi-coherent sheaf ℱ on an fs log scheme X, ε̄_{∗}ℱ is a quasi-coherent sheaf on X_{Zar}. The similar statement holds for pellicular Ket coherent sheaves and (Zariski) coherent sheaves if X is Noetherian and has the property (EX).

Proof. We can make the same argument as in Proposition 3.14 in [Hag03] work by using Corollary 4.5 (3) and Proposition 4.8.

Proposition 4.12. Let X be an fs log scheme and π : X′ → X a strictly etale surjective morphism and ℱ a pellicular Ket quasi-coherent sheaf. Then ℱ ≅ ε̄_{X}^{∗} ℱ̃for some ℱ̃∈ Qcoh(X_{Zar}) if and only if π^{∗}ℱ ≅ ε̄_{X′}^{∗} ℱ̃′ for some ℱ̃′ ∈ Qcoh(X′_{Zar}). The similar statement holds for pellicular Ket coherent sheaves if, in addition, X is Noetherian and has the property (EX). If the natural functor Vect(X_{Zar}) → Vect((X_{red})_{Zar}) is essentially surjective (e.g. |X| is semi-local or reduced), the similar statement holds for pellicular Ket vector (or line) bundles.

Proof. In the case where X (and X′) is reduced, it follows from Corollary 4.5 (5) and the usual descent. In general, use Proposition 4.6 (2).

Corollary 4.13. Let X be a Noetherian equi-characteristic fs log scheme and X → Spec ℤ[P] a chart with P fs and sharp. Then, for a pellicular Ket quasicoherent sheaf ℱ on X, there exists a positive integer n, invertible on X, and a quasi-coherent sheaf ℱ̃on (X_{n})_{Zar} such that ε̄_{X_{n}}^{∗} ℱ̃≅ ℱ |_{X_{n}}. Similarly for coherent sheaves if, in addition, X has the property (EX). If, moreover, the natural functor Vect(X_{Zar}) → Vect((X_{red})_{Zar}) is essentially surjective, the similar statements hold for vector bundles and line bundles.

Proof. Use Proposition 2.2 and Proposition 4.12 (cf. [Hag03] Corollary 4.10).

4.2 Regularly stratified log schemes

In this subsection, we consider some interesting sufficient conditions for the functor ε̄^{∗} to be exact.

Definition 21. Let X be a locally Noetherian fs log scheme and x a point of |X| such that \overline{M}_{X,x̄} is finitely generated free of rank r. Take a standard basis \{e_{1}, …, e_{r}\} of \overline{M}_{X,x̄} ≅ ℕ^{r} and choose their liftings ẽ_{i} ∈ M_{X,x̄}(1 ≤ i ≤ r).

1. We say that X is weakly regularly stratified at x if, after a suitable permutation of e_{i}’s, {α(ẽ_{1}), …, α(ẽ_{s})} becomes a regular sequence of 𝒪_{X,x̄} and that α(ẽ_{s+1}) = ⋯ = α(ẽ_{r}) = 0 for some s ≤ r.

2. We say that X is weakly quasi-regularly stratified at x if after a suitable permutation of e_{i}’s, each α(ẽ_{1}), …, α(ẽ_{s}) is a non zero-divisor in 𝒪_{X,x̄} and that α(ẽ_{s+1}) = ⋯ = α(ẽ_{r}) = 0 for some s ≤ r.

If, in the condition (1)(resp. (2)), we can take s = r, we say that X is regularly (resp. quasi-regularly) stratified at x. We also call X regularly stratified, weakly regularly stratified and so on, if X is so at every point.

Of course, these conditions are independent of the choice of liftings, and the (weakly) regular stratifiedness implies the (weakly) quasi-regular stratifiedness. The following properties are easily checked:

Proposition 4.14. 1. If X is (weakly) regularly stratified at x and η be a generalisation of x (i.e. x ∈ \overline{\{η\}}), then X is (weakly) regularly stratified also at η. Similar statements hold for (weakly) quasi-regular stratifiedness.

2. For a weakly quasi-regularly stratified fs log scheme X, the functor ε_{X}^{∗} : Mod(X_{Zar}) → Mod(X_{Ket}) is exact.

3. If X is (weakly) quasi-regularly stratified, so is X_{red}.

4. Let f : X → Y be a strict morphism of locally Noetherian fs log schemes such that |f| is flat. Then, if Y is (weakly) quasi-regularly stratified (or (weakly) regularly stratified), so is X. If |f| is faithfully flat, the converse also holds.

5. Assume that X is F -framed. If X is (weakly) (quasi-)regularly stratified, then so is X^{𝔮} for any prime ideal 𝔮 of F. If X is weakly regularly stratified, then so is X[𝔭]^{𝔮} for any prime ideals 𝔭 and 𝔮 of F.

Proof. The properties (1), (3) and (4) are immediately checked, and (2) is proven in Proposition 3.12 in [Hag03]. (5) is easily proven by using Proposition 3.7 and Lemma 3.9.

Example 1. 1. Let A be a Noetherian ring, r a natural number, and a_{1}, …, a_{r} elements of A. Consider a morphism of monoids ℕ^{r} → A which maps e_{i} to a_{i}, where {e_{i}} is the standard basis of ℕ^{r}. If {a_{1}, …, a_{r}} is a regular sequence, then the fs log scheme associated with this pre-log structure is regularly stratified. Similarly for the other notions.

2. Every regular and (weakly) log regular log scheme (see Subsection 3.2) is (weakly) regularly stratified.

To check the exactness of ε̄_{X}^{∗}, the next proposition is useful.

Proposition 4.15. Let A be a ring, n a natural number invertible in A and a ∈ A a non zero-divisor, then we have A_{red}[T]/(T^{n} −a) ≅ (A[T]/(T^{n} −a))_{red}. In particular, if A is reduced, then A[T]/(T^{n} − a) is reduced.

Proof. First we prove the latter part. We may assume that A is Noetherian. Then by considering localisations A_{𝔭_{i}} at the minimal primes 𝔭_{1}, …, 𝔭_{𝔯}, we have an embedding of A into a product ∏ A_{𝔭_{i}} of fields such that the image of a is non-zero in each component. So we can reduce to the case where A is a field, and in this case the second assertion is trivial. Now we can easily prove the first part by using an isomorphism (A_{1}⊗_{A_{3}}A_{2})_{red} ≅ ((A_{1})_{red} ⊗_{(A_{3})_{red}} (A_{2})_{red})_{red} for any ring A_{3} and any rings A_{1} and A_{2} over A_{3}.

Corollary 4.16. Let X be an fs log scheme, X → Spec ℤ[ℕ^{r}] its chart, n a natural number invertible in X and s a natural number such that s ≤ r. We set N′ = (ℕ^{s})^{1/n} ⊕ ℕ^{r−s}(⊃ ℕ^{r}) and X′ = X ×_{Spec ℤ[ℕ^{r}]}^{fs} Spec ℤ[N′].

1. If X is weakly quasi-regularly stratified, then X′_{red} is also weakly quasiregularly stratified, and |X′|_{red} is flat over |X|_{red}.

2. If X is quasi-regularly stratified and reduced, then so is X′. In particular, |X′| is flat over |X|.

Proof. Note first that we may assume that X be strictly local. We then have only to use the above proposition and the induction, noting that if a ∈ A is a non zero-divisor, then it is so also in A[T]/(f) for any monic polynomial f ∈ A[T].

Proposition 4.17. Let X be a weakly quasi-regularly stratified fs log scheme whose underlying scheme is reduced. Then, 1. for a log geometric point x(log) over x ∈ |X|, there exists a cofinal subset

{U_{λ}}_{λ∈Λ} in the category of all Kummer etale neighbourhoods of x(log) such that, for every λ ∈ Λ,

(a) (U_{λ})_{red} is weakly quasi-regularly stratified, and flat over X.

(b) ε̄_{(U_{λ})_{red}}^{∗} is exact.

2. The functor ε̄_{X}^{∗} is exact.

If, moreover, X is quasi-regularly stratified, then we can take a set {U_{λ}}_{λ∈Λ} so that each U_{λ} is quasi-regularly stratified and reduced.

Proof. First we prove the statement (2). Take an arbitrary point x ∈ |X|. We may assume that X is local whose closed point is x, and that we have a chart φ : X → Spec ℤ[ℕ^{r}] such that φ^{∗} : ℕ^{r} ≅ \overline{M}_{X,x̄}. Then, for n ≥ 1, prime to the characteristic of X, (X_{n})_{red} is flat over X by Corollary 4.16, and from this we obtain the exactness of ε̄_{X}^{∗} by Proposition 2.4. Next we prove (1). We may assume that we have a chart φ as above. Then, by Corollary 4.16, for any fs log scheme V etale over U_{n}, where U ⊂ |X| is a Zariski open neighbourhood of x, and n is invertible in U, we see that V_{red} is again a weakly quasi-regularly stratified and reduced, so that ε̄_{V_{red}}^{∗} is also exact. Now the first result also follows from Proposition 2.4. The last statement can be proven similarly.

Corollary 4.18. Let X be a weakly quasi-regularly stratified fs log scheme. Then ε̄_{X_{red}}^{∗} is exact and X_{red} has the property (EX).

Corollary 4.19. For a reduced quasi-regularly stratified fs log scheme X, we have an isomorphism 𝒪_{X_{Ket}} ≅ \overline{𝒪} _{X}.

Proposition 4.20. Let X and Y be Noetherian fs log schemes and f : X → Y a classically etale morphism. Assume that Y is weakly quasi-regularly stratified. Then we have an isomorphism

ε̄_{Y}^{∗} f_{∗} ≅ f_{∗}ε̄_{X}^{∗}

of functors from Qcoh(X_{Zar}) to \overline{Qcoh}(Y_{Ket}).

Proof. Let ℱ be an object in Qcoh(X_{Zar}). For any y ∈ |Y |, we have

(ε̄_{Y}^{∗} f_{∗}ℱ)_{y(log)}  ≅ colim_{V} (ε̄_{Y}^{∗} f_{∗}ℱ)(V),
(f_{∗}ε̄_{X}^{∗}ℱ)_{y(log)}  ≅ colim_{V} (f_{∗}ε̄_{X}^{∗}ℱ)(V).

Here we may assume that V runs through the essentially small category of Kummer etale neighbourhoods of y(log) whose maximal reduced closed subscheme is flat over Y, which is cofinal in the category of all Ket neighbourhoods (Proposition 4.17).

For such a V we set U = V ×_{Y}^{fs} X. Then, by the classical etaleness of f we obtain |U|^{red} ≅ (|V | ×_{|Y |} |X|)^{red} ≅ |V |^{red} ×_{|Y |} |X|. So, by Corollary 4.5 (2) and the flatness of V^{red} over Y, we have isomorphisms

Γ(V, ε̄_{Y}^{∗} f_{∗}ℱ) ≅ Γ(V^{red}, 𝒪_{V^{red}} ⊗_{𝒪_{Y}} f_{∗}ℱ)
≅ Γ(U^{red}, 𝒪_{U^{red}} ⊗_{𝒪_{X}} ℱ)
≅ Γ(U, ε̄_{X}^{∗}ℱ)
≅ Γ(V, f_{∗}ε̄_{X}^{∗}ℱ).

Now the proposition follows.

4.3 Some propositions

In this subsection we collect some propositions which play essential roles later. First we prove some results on pellicular Kummer etale sheaves in the case of local schemes.

Proposition 4.21. Let X be a Noetherian local equi-characteristic fs log scheme of characteristic exponent p, equipped with a chart X → Spec ℤ[P] with P fs and sharp. Then

1. For each i ≥ 0 we have an isomorphism

\overline{K}_{i}(X_{Ket}) ≅ colim_{n} K_{i}((X̃_{n})_{red}, I_{n} ⋊ Γ_{n}),

where the right hand side is a colimit of equivariant (Zariski) K-groups indexed by natural numbers prime to p. A similar statement holds for K′-groups if X is weakly quasi-regularly stratified (For the notations, see Subsection 2.1.3).

2. For any pellicular Ket coherent module ℱ on X, there exists a surjective morphism ⊕_{i}ℒ_{i} → ℱ from a finite direct sum of pellicular Ket line bundles.

Proof. They are only direct consequences to Proposition 4.10 and Corollary 4.13. Indeed, the statement (1) follows immediately from these propositions, and (2) follows from some simple arguments in equivariant module theory for semi-local rings.

Proposition 4.22. Let X be a Noetherian local equi-characteristic fs log scheme of characteristic exponent p. Assume that we are given an F -frame θ : F → Γ(X, \overline{M}_{X}), and let F_{0} ⊂ F be the (unique) direct summand of F such that θ : F_{0} ≅ \overline{M}_{X,x}, where x is the closed point of |X|. Then the frame θ induces a canonical isomorphism

F_{0}^{gp} ⊗ (ℚ/ℤ)′ ≅ Pic (X_{Ket}) ≅ \overline{Pic} (X_{Ket})

(For the definition of the isomorphism, see the argument after Theorem 3.17).

Before giving the proof of this proposition, we introduce a notation.

Definition 22. Let P be a finitely generated free monoid, n a positive integer, T = Spec ℤ[1/n][P] and T̃_{n} = Spec ℤ[1/n, ζ_{n}][P^{1/n}]. Recall that the group Ĩ_{n} = I_{n} ⋊ Γ_{n} acts on the scheme T̃_{n} (Subsection 2.1.3). Then, for any element α ∈ P^{1/n}, we have a map defined by

Ĩ_{n} = I_{n} ⋊ Γ_{n} → ℤ[1/n, ζ_{n}][P^{1/n}]^{×} ; (φ, γ) ↦ φ(α).

Since this satisfies the 1-cocycle condition, it induces an Ĩ_{n}-equivariant line bundle on (T̃_{n})_{Zar}, free of rank one as a line bundle. We denote by 𝒪(α) this line bundle (or its pullback with respect to some morphism to T̃_{n}).

The lemma below is checked by a direct calculation.

Lemma 4.23. Let F be a finitely generated free monoid, X a Noetherian equicharacteristic fs log scheme of characteristic exponent p, and X → Spec ℤ[F] a chart. We regard X as F -framed naturally by this chart. Then the diagram

\xymatrix{colim_{(p,n)=1}F^{1/n} \ar[r] \ar[d]_{≅} & colim_{(p,n)=1}Pic ((X̃_{n})_{Zar}, Ĩ_{n}) \ar[d] \\ F^{div′} \ar[r] & Pic (X_{Ket})}

is commutative, where

• the upper Picard group is the equivariant one,

• the upper horizontal morphism is induced from α ↦ 𝒪(α),

• the lower horizontal one as in the paragraphs after Theorem 3.17, and

• the vertical ones the natural ones (cf. Proposition 3.21 in [Hag03]).

Proof. (of Proposition 4.22) We may assume that F = F_{0}, and that there exists a chart X → Spec ℤ[F] compatible with the frame θ (Proposition 3.7). Then, since Pic ((X̃_{n})_{Zar}) is trivial, we have the following commutative diagram:

\xymatrix{colim_{n}F^{1/n}/F \ar[r] \ar[d]_{≅} & colim_{n}Pic (X̃_{n}, Ĩ_{n}) \ar[r] \ar[d] & colim_{n}Pic ((X̃_{n})_{red}, Ĩ_{n}) \ar[d] \\ F^{div′}/F \ar[r] & Pic (X_{Ket}) \ar[r] & \overline{Pic} (X_{Ket}),}

where the upper colimit is taken with respect to positive integers prime to p, the Picard groups are equivariant (Zariski) ones, and the morphisms are as in Lemma 4.23 or the natural ones (For the right vertical morphism, see Proposition 4.10). Now, as is easily seen by the argument in equivariant module theory, the upper horizontal morphisms are isomorphisms, and by Proposition 4.10 and Corollary 4.13, so are the vertical ones.

The proposition below will be used in the proof of the comparison of K- and K′-groups.

Proposition 4.24. Let X be an F -framed Noetherian equi-characteristic divisorial fs log scheme with the property (EX) and x ∈ |X|. Then,

1. For any pellicular Ket coherent sheaves of \overline{𝒪} _{X}-modules ℰ, ℱ and a morphism of sheaves f : ℰ_{x} → ℱ_{x} on (Spec 𝒪_{X,x} )_{Ket}, there exist a pellicular Ket line bundle ℒ and an s ∈ Γ(X, ℒ^{−1}) such that fs_{x} : (ℰ ⊗ℒ)_{x} → ℱ_{x} is extended to a global morphism ϕ : ℰ ⊗ ℒ → ℱ.

2. For any pellicular Ket coherent sheaf ℱ, there exist a pellicular Ket vector bundle ℰ and a surjective \overline{𝒪} _{X}-homomorphism ℰ → ℱ.

Proof. By Proposition 4.11, the claim (1) is proven similarly to [Hag03] Lemma 4.11, and (2) follows from (1), Proposition 4.21 (2) and Proposition 4.22 as in [Hag03] Proposition 7.2.

Later we will also need the following proposition, which can be proven easily.

Proposition 4.25. Let {X_{i}}_{i∈I} be a system of Noetherian fs log schemes with the property (EX), indexed by a filtered category I. Assume that any transition map π_{ij} : X_{j} → X_{i} is strict and |π_{ij}| is affine and etale. We denote by X a projective limit \underleftarrow{lim}_{i∈I} X_{i} in the category of fs log schemes and assume that X also has the property (EX). Then we have

colim_{i∈I} \overline{K}′_{q}((X_{i})_{Ket}) ⟶^{≅} \overline{K}′_{q}(X_{Ket}).

If every X_{i} and every π_{ij} is F -framed, then a similar statement holds for \overline{K}′(−_{Ket}^{𝔮}) for any prime 𝔮 of F.

5 The proof of the main theorem

5.1 Localisation sequence for pellicular Ket K-theory

Definition 23. Let X be a Noetherian fs log scheme, Y a closed subscheme of |X| and U its complement. We endow them with the induced log structures. We denote by i the natural closed immersion from Y to X, and by j the natural open immersion from U to X. Then we define a ring object \overline{𝒪} _{Y,X} in (Y_{Ket}) to be ε_{Y}^{−1}𝒪_{Y_{Zar}} ⊗_{ε_{Y}^{−1}i^{−1}𝒪_{X_{Zar}}} i^{−1}\overline{𝒪} _{X}.

The followings are easily proven.

Proposition 5.1. For a log scheme X′ Kummer etale over X, set Y′ = X′×_{X}^{fs} Y. Then we have a natural isomorphism \overline{𝒪} _{Y,X}|_{Y′} ≅ \overline{𝒪} _{Y′,X′}.

Proposition 5.2. 1. The homomorphism i^{−1}\overline{𝒪} _{X} → \overline{𝒪} _{Y,X} is surjective.

2. The kernel \overline{I}_{Y} of the above homomorphism i^{−1}\overline{𝒪} _{X} → \overline{𝒪} _{Y,X} is equal to i^{−1}((ε_{X}^{−1}I_{Y_{Zar}})\overline{𝒪} _{X}), where I_{Y_{Zar}} is the ideal sheaf of 𝒪_{X_{Zar}} defining Y and (ε_{X}^{−1}I_{Y_{Zar}})\overline{𝒪} _{X} is the ideal sheaf of \overline{𝒪} _{X} generated by the image of ε_{X}^{−1}I_{Y_{Zar}}.

3. If ε̄_{X}^{∗} is exact, then the homomorphism ε_{Y}^{−1}𝒪_{Y_{Zar}} → \overline{𝒪} _{Y,X} is flat.

4. If X has the property (EX), the sheaf \overline{𝒪} _{Y,X} is coherent as a \overline{𝒪} _{Y,X}-module, and so is i_{∗}\overline{𝒪} _{Y,X} as a \overline{𝒪} _{X}-module.

Proof. (1), (2) and (3) are obvious, and (4) follows from (2), Proposition 4.8 and the above proposition.

Remark 5.3. In general, \overline{𝒪} _{Y,X} is far from isomorphic to \overline{𝒪} _{Y} (indeed it is so very often).

Definition 24. 1. An \overline{𝒪} _{Y,X}-module ℱ is said to be coherent if it is coherent with respect to (Y_{Ket}, \overline{𝒪} _{Y,X}) in the sense of J.-P. Serre. We denote the full subcategory of Mod(Y_{Ket}, \overline{𝒪} _{Y,X}) consisting of coherent \overline{𝒪} _{Y,X}-modules by \overline{Coh}(Y_{Ket}; X).

2. For an X with the property (EX), we define \overline{K}′(Y_{Ket}; X) to be a K-theory spectrum constructed from this Abelian category.

The next proposition is easily proven by the dévissage theorem ([Qui73]).

Proposition 5.4. We have natural weak equivalences of spectra

\overline{K}′(Y_{Ket}; X) ≅ \overline{K}′((Y_{red})_{Ket}; X) ≅ \overline{K}′((Y_{red})_{Ket}; X_{red}).

Notice that we have a natural exact functor \overline{Coh}(Y_{Ket}; X) → \overline{Coh}(X_{Ket}). We denote by i_{∗} the induced homomorphism \overline{K}′_{q}(Y_{Ket}; X) → \overline{K}′_{q}(X_{Ket}).

Theorem 5.5. Let X be a Noetherian equi-characteristic weakly quasi-regularly stratified fs log scheme, Y a strictly closed subscheme and U its open complement, which we endow with the induced log structure. Then we have a canonical long exact sequence

⋯ →^{∂} \overline{K}′_{q}(Y_{Ket}; X) →^{i_{∗}} \overline{K}′_{q}(X_{Ket}) →^{j^{∗}} \overline{K}′_{q}(U_{Ket}) →^{∂} \overline{K}′_{q−1}(Y_{Ket}; X) →^{i_{∗}} ⋯

⋯ →^{∂} \overline{K}′_{0}(Y_{Ket}; X) →^{i_{∗}} \overline{K}′_{0}(X_{Ket}) →^{j^{∗}} \overline{K}′_{0}(U_{Ket}) → 0.

Proof. First we may assume that X is reduced because taking (−)_{red} makes the conditions and the conclusion unchanged (Proposition 4.14 (3) and the above proposition). Then we see that ε̄_{X}^{∗} is exact by Corollary 4.18. As usual we have only to show the following three lemmata (cf. the proof of Theorem 4.5 in [Hag03]).

Lemma 5.6. Let X be a Noetherian fs log scheme, Y a strictly closed subscheme and ℱ a pellicular Ket coherent sheaf of \overline{𝒪} _{X}-modules. Assume that \overline{M}_{X,x̄} is finitely generated free for each geometric point x̄ ∈ X. If we have ℱ |_{U} = 0, then there exists a natural number N such that ((ε_{X}^{−1}I_{Y_{Zar}})\overline{𝒪} _{X})^{N}ℱ = 0.

Proof. First note that ℱ_{x(log)} = 0 for any x ∉ |Y | by assumption. Take a Kummer etale covering f : X′ → X and a Zariski coherent 𝒪_{X′}-module ℱ̃such that ℱ |_{X′} ≅ ε̄_{X′}^{∗} ℱ̃. Here, by assumption on \overline{M}, we may assume |X′| is flat over |X|. Put Y′ = Y ×_{X} X′. Since we have ℱ_{x′(log)} ≅ \overline{𝒪} _{X′,x′(log)} ⊗_{𝒪_{X′,x′}} ℱ̃_{x′} for every x′ ∈ |X′|, the set {x′ ∈ |X′| |ℱ̃_{x′} ≠ 0} is contained in |Y′| by the faithful flatness of 𝒪_{X′,x′} → 𝒪_{X′,x′(log)}. Thus we have a natural number N such that I_{Y′_{Zar}}^{N} ℱ̃= 0. By the surjectivity of ε̄_{X′}^{∗}I_{Y′_{Zar}} → ((ε_{X}^{−1}I_{Y_{Zar}})\overline{𝒪} _{X})|_{X′}, the lemma now follows.

Lemma 5.7. Let X be a Noetherian equi-characteristic weakly quasi-regularly stratified fs log scheme, U a strictly open subscheme and j : U → X a natural open immersion. Then, for any pellicular Ket quasi-coherent module ℱ on U, j_{∗}ℱ is pellicular Ket quasi-coherent on X.

Proof. This is proven similarly to the proof of [Hag03] Proposition 4.6, by using Proposition 4.8 (3), Corollary 4.13 and Proposition 4.20.

Lemma 5.8. Let X be a Noetherian reduced weakly quasi-regularly stratified fs log scheme and ℱ a pellicular Ket quasi-coherent sheaf. Then we have

ℱ ≅ \underrightarrow{lim} ℱ_{i},

where ℱ_{i} run through all pellicular Ket coherent sheaves such that ℱ_{i} ⊂ ℱ.

Proof. (cf. [Hag03] Proposition 4.7.) Take a Kummer etale covering π : X′ → X and a Zariski quasi-coherent 𝒪_{X′_{red}}-module ℱ̃(not a 𝒪_{X′}-module) such that π^{∗}ℱ ≅ ε̄_{X′_{red}}^{∗} ℱ̃. By Proposition 4.17, we may assume that ε̄_{X′_{red}}^{∗} is also exact

and that X′ is Noetherian. The sheaf ℱ̃can be written as

ℱ̃≅ \underrightarrow{lim} ℱ̃_{i},

using all Zariski coherent sub-𝒪_{X′_{red}}-modules ℱ̃_{i} ⊂ ℱ̃. Then we have π^{∗}ℱ ≅ \underrightarrow{lim} ε̄_{X′_{red}}^{∗} ℱ̃_{i} and π_{∗}π^{∗}ℱ ≅ π_{∗} \underrightarrow{lim} ε̄_{X′_{red}}^{∗} ℱ̃_{i} ≅ \underrightarrow{lim} π_{∗}ε̄_{X′_{red}}^{∗} ℱ̃_{i} (Note that as X

and X′ are Noetherian and {ε̄_{X′_{red}}^{∗} ℱ̃_{i}} forms a direct system, inductive limit commutes with π_{∗}). If we define ℱ_{i} by the Cartesian diagram

\xymatrix{ℱ_{i} \ar[r] \ar@{^{(}->}[d] & π_{∗}ε̄_{X′_{red}}^{∗} ℱ̃_{i} \ar@{^{(}->}[d] \\ ℱ \ar[r] & π_{∗}π^{∗}ℱ ,}

then we have ℱ ≅ \underrightarrow{lim} ℱ_{i}. By the following commutative diagram

\xymatrix{π^{∗}ℱ_{i} \ar[r] \ar@{^{(}->}[drr] & π^{∗}π_{∗}ε̄_{X′_{red}}^{∗} ℱ̃_{i} \ar[r] & ε̄_{X′_{red}}^{∗} ℱ̃_{i} \ar[d] \\  &  & π^{∗}ℱ ,}

we see that π^{∗}ℱ_{i} is pellicular Ket coherent, as it is a pellicular Ket quasicoherent subsheaf of a pellicular Ket coherent sheaf ε̄_{X′_{red}}^{∗} ℱ̃_{i}, and X′ is Noetherian and has the property (EX) (Use Proposition 4.8).

As usual, we see that the presheaf of fibrant spectra ε_{X∗}\overline{K}′_{Ket} on Et/|X|(the category of schemes etale over |X|) defined by

“U ↦ \overline{K}′(U_{Ket})” (where U has the induced log structure)

satisfies “a Nisnevich Brown-Gersten property” (i.e. a cd-excision property in p.286 of [Jar97]) if X is a Noetherian equi-characteristic weakly quasi-regularly stratified fs log scheme. The more precise statement is as follows.

Corollary 5.9. Assume that we are given a Cartesian diagram of Noetherian equi-characteristic weakly quasi-regularly stratified fs log schemes

\xymatrix{V′ \ar[d]^{f′} \ar@{^{(}->}[r]^{j′} & U′ \ar[d]^{f} \\ V \ar@{^{(}->}[r]^{j} & U}

with vertical morphisms (classically) etale and horizontal ones strictly open immersions. We assume that the induced morphism f′′ : U′ \ V′ → U \ V is isomorphism if both sides are endowed with the induced reduced log scheme structure. Then the induced square of spectra

\xymatrix{\overline{K}′(U_{Ket}) \ar[d]^{f^{∗}} \ar[r]^{j^{∗}} & \overline{K}′(V_{Ket}) \ar[d]^{_{f′_{∗}}} \\ \overline{K}′(U′_{Ket}) \ar[r]^{_{j′_{∗}}} & \overline{K}′(V′_{Ket})}

is homotopy Cartesian. In particular, if X is a weakly quasi-regularly stratified fs log scheme whose underlying scheme is Noetherian, equi-characteristic, and of finite Krull dimension, we have a weak equivalence

\overline{K}′(X_{Ket}) ⟶^{≅} ℍ^{·}(X_{Nis}, ε_{X∗}\overline{K}′_{Ket}).

Similar statements hold for \overline{K}′(−_{Ket}^{𝔮}) and ℍ^{·}(X_{Nis}, ε_{X∗}\overline{K}′(−_{Ket}^{𝔮})) if U or X is endowed with an F -frame structure.

Proof. Set W = U \ V and W′ = U′ \ V′. Then we have f′′^{−1}\overline{𝒪} _{W,U} ≅ \overline{𝒪} _{W′,U′}. Now Corollary follows easily from the above theorem. The latter part follows from Corollary 7.68 in [Jar97]. The case for X_{Ket}^{𝔮} immediately follows from that for 𝔮 = ∅ by noting Proposition 3.6 (5).

5.2 Comparison of pellicular Ket K-theory and pellicular Ket K′-theory

Theorem 5.10. Let F and X be as in Convention 3.13. Then we have a natural weak equivalence

\overline{K}(X_{Ket}) ≅ \overline{K}′(X_{Ket}).

A similar proposition holds for X[𝔭]_{Ket}^{𝔮} for prime ideals 𝔭 and 𝔮.

Proof. The latter case is reduced to the former (by Proposition 3.12), the proof of which is completely parallel to the one of Proposition 7.1 in [Hag03]. Indeed, we have only to use Proposition 4.24 and the following lemma.

Lemma 5.11. Let X be a weakly log regular fs log scheme whose underlying scheme is Noetherian and regular. If we are given a chart X → Spec ℤ[P] with P fs and sharp, then, for any positive integer n, (X_{n})_{red} is again Noetherian, regular and weakly log regular. In particular, for any log scheme X′ Kummer etale over X, there exists a Kummer etale covering X′′ → X′ such that X′′_{red} is regular and weakly log regular.

Proof. The former is straightforward, and the latter follows from the former by Proposition 2.4.

Combining Theorem 5.10 and Corollary 5.9, we obtain the following corollary:

Corollary 5.12. Let F and X be as in Convention 3.13. Then we have a weak equivalence

\overline{K}(X_{Ket}) ⟶^{≅} ℍ^{·}(X_{Nis}, ε_{X∗}\overline{K}_{Ket}).

5.3 Inversion formula

We fix a finitely generated free monoid F also in this subsection. Here we prove Theorem 3.14.

Theorem 5.13. Let X be an F -framed weakly quasi-regularly stratified fs log scheme such that |X| is finite dimensional, Noetherian, separated and equicharacteristic of characteristic p. Then, for a prime ideal 𝔮 of F, we have a weak equivalence

\overline{K}′(X_{Ket}^{𝔮})_{ℚ} ⟶^{≅} ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ}).

Proof. First we show that we may assume that X is Henselian local. For a general X, consider the following up-to-homotopy commutative diagram

\xymatrix{\overline{K}′(X_{Ket}^{𝔮})_{ℚ} \ar[d] \ar[r] & ℍ^{·}(X_{Nis}, \overline{K}′(−_{Ket}^{𝔮})_{ℚ}) \ar[d] \\ ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ}) \ar[r] & ℍ^{·}(X_{Nis}, ℍ^{·}(−_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ})),}

where ℍ^{·}(−_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ}) and \overline{K}′(−_{Ket}^{𝔮})_{ℚ} are presheaves of fibrant spectra on X_{Nis} naturally defined. Here the upper horizontal morphism is a weak equivalence by Corollary 5.9, the lower horizontal one by the Cartan-Leray-type theorem proven formally (See Theorem 1. 56 in [Tho85]). Therefore, the weak equivalence of the left vertical morphism follows from that of the right vertical one. Thus the theorem is reduced to the Henselian local case by Proposition 4.25. From now on we will assume that X be Henselian local. Using the spectral sequence

H^{i}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{j,Ket})_{ℚ}) ⟹ π_{j−i}ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ}),

where ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{j,Ket})_{ℚ} is the sheaf on X_{Ket}^{𝔮^{∨}} associated with “U ↦ \overline{K}′_{j}((U ×_{X^{𝔮^{∨}}}^{fs} X)_{Ket})_{ℚ}”, and the vanishing of the higher group cohomology of profinite groups with ℚ-vector space coefficients, we have an isomorphism

π_{j}ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{Ket})_{ℚ}) ⟶^{≅} Γ(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{j,Ket})_{ℚ}).

Moreover, as is easily checked, the right hand side can be rewritten as follows:

Γ(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}′_{j,Ket})_{ℚ}) ⟶^{≅} (\underrightarrow{lim} ; X̃ \overline{K}′_{j}((X̃ ×_{X^{𝔮^{∨}}}^{fs} X)_{Ket})_{ℚ})^{π_{1}(X_{Ket}^{𝔮^{∨}})},

where X̃ runs through Kummer etale neighbourhoods of X^{𝔮^{∨}} (with respect to a fixed log geometric point over the closed point), and π_{1}(X_{Ket}^{𝔮^{∨}}) is the projective

limit of their automorphism groups over X^{𝔮^{∨}}. The notation (−)^{π_{1}(X_{Ket}^{𝔮^{∨}})} denotes the invariant part for the natural action. For a further reduction, we introduce some notations:

Definition 25. Let F be a finitely generated free monoid, and X an F -framed fs log scheme, and 𝔰 a prime ideal of F. Assume that we are given a chart X → Spec ℤ[F] which lifts the given frame. For a positive integer n invertible on X, we set

X̃_{n}^{𝔰} = X^{𝔰} ×_{Spec ℤ[1/n][F \𝔰]}^{fs} Spec ℤ[1/n, μ_{n}][(F \ 𝔰)^{1/n}],

I_{n}^{𝔰} = Hom ((F \ 𝔰)^{1/n}/(F \ 𝔰), ℤ[μ_{n}]^{×}),

and Ĩ_{n}^{𝔰} = I_{n}^{𝔰} ⋊ Gal(ℚ(μ_{n})/ℚ) (cf. Subsection 2.1.3). We regard X̃_{n}^{𝔰} as an fs log scheme by the chart X̃_{n}^{𝔰} → Spec ℤ[1/n, μ_{n}][(F \ 𝔰)^{1/n}]. Note that the log scheme X̃_{n}^{𝔰} has a natural group action of Ĩ_{n}^{𝔰}, and that, for prime ideals 𝔰 ⊂ 𝔯, the natural morphism X̃_{n}^{𝔰} → X̃_{n}^{𝔯} is compatible with the natural surjective homomorphism Ĩ_{n}^{𝔰} → Ĩ_{n}^{𝔯}.

The followings are easily checked:

Lemma 5.14. 1. The set

{X̃_{m}^{𝔮^{∨}} | m ∈ ℕ, (m, p) = 1}

is cofinal in the category of Kummer etale neighbourhoods of X^{𝔮^{∨}} (for a suitable choice of log geometric points over the closed points).

2. For a positive integer m, the set

{X̃_{mr}^{𝔮^{∨}} ×_{X^{𝔪}}^{fs} X̃_{n}^{𝔮} | r, n ∈ ℕ, (r, p) = (n, p) = 1}

is cofinal in the category of Kummer etale neighbourhoods of the log

scheme X̃_{m}^{𝔮^{∨}} ×_{X^{𝔮^{∨}}}^{fs} X = X̃_{m}^{𝔮^{∨}} ×_{X^{𝔪}}^{fs} X^{𝔮}, where 𝔪 is the maximal ideal of F (for a suitable choice of log geometric points over the closed points).

Therefore it suffices to prove that

\overline{K}′_{j}(X_{Ket}^{𝔮})_{ℚ} ≅ (\underrightarrow{lim}_{m} \overline{K}′_{j}((X̃_{m}^{𝔮^{∨}} ×_{X^{𝔮^{∨}}}^{fs} X)_{Ket})_{ℚ})^{π_{1}(X_{Ket}^{𝔮^{∨}})},

or by using Proposition 4.21 and the above lemma,

\underrightarrow{lim}_{n} K′_{j}(|X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮})_{ℚ} ≅ \underrightarrow{lim}_{m} ( \underrightarrow{lim}_{n,r} K′_{j}({ \atop |X_{mr}^{𝔮^{∨}}} ×_{X^{𝔪}}^{fs} X̃_{n}^{𝔮}|_{red}, { \atop I_{mr,m}^{𝔮^{∨}}} × Ĩ_{n}^{𝔮})_{ℚ})Ĩ_{m}^{𝔮^{∨}} ,

where { \atop I_{mr,m}^{𝔮^{∨}}} = Kernel (Ĩ_{mr}^{𝔮^{∨}} → Ĩ_{m}^{𝔮^{∨}}), and r, m and n runs positive integers prime to p. Thus, Theorem 5.13 (in the Henselian local case) is reduced to the following lemma (In (1), we only need the case 𝔯 = 𝔮^{∨}):

Lemma 5.15. Let X and F be as in Theorem 5.13, and assume that |X| is Henselian local, and that we are given a chart X → Spec ℤ[F] lifting the frame.

1. For each positive integers m and n prime to p, and prime ideals 𝔯 and 𝔮 of F such that 𝔯 ∧ 𝔮 = ∅, we have a natural isomorphism

K′_{j}(|X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮})_{ℚ} ≅ K′_{j}(|X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮})Ĩ_{m}^{𝔯} ; ℚ,

for every j ≥ 0.

2. For every positive integer n prime to p and a prime ideal 𝔮, we have a natural isomorphism

\underrightarrow{lim}_{m} K′_{j}({ \atop |X_{m}^{𝔮^{∨}}} ×_{X^{𝔪}}^{fs} X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮}) ≅ \underrightarrow{lim}_{m,r} K′_{j}({ \atop |X_{mr}^{𝔮^{∨}}} ×_{X^{𝔪}}^{fs} X̃_{n}^{𝔮}|_{red}, { \atop I_{mr,m}^{𝔮^{∨}}} × Ĩ_{n}^{𝔮}),

for every j ≥ 0.

To prove (2), we set K_{m,r} = K′_{j}({ \atop |X_{mr}^{𝔮^{∨}}} ×_{X^{𝔪}}^{fs} X̃_{n}^{𝔮}|_{red}, { \atop I_{mr,m}^{𝔮^{∨}}} × Ĩ_{n}^{𝔮}). Then we have a transition morphism K_{m,r} → K_{m′,r′} if m|m′ and mr|m′r′. In particular, for every (m, r), we have a morphism K_{m,r} → K_{mr,1}, which means the set {K_{m,1}} is cofinal. Hence (2) follows. The claim (1) is reduced to the lemma below by the descending induction on the number of generators of 𝔯.

Lemma 5.16. Let 𝔯′ = 𝔯 ∨ ⟨e⟩, where e is an element of the standard basis of F such that e ∉ 𝔯, and suppose that 𝔮 ∧ 𝔯′ = ∅. Then we have

K′_{j}(|X̃_{m}^{𝔯′} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮})_{ℚ} ≅ K′_{j}(|X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red}, Ĩ_{n}^{𝔮})_{ℚ}^{μ_{n}}

for every j ≥ 0, and m and n prime to p.

Finally, using the following two facts proven by Proposition 4.15, we can reduce this lemma to the next proposition.

• The diagram of schemes

\xymatrix{|X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}| \ar[r] \ar[d] & Spec R[e^{1/m}] \ar[d] \\ |X̃_{m}^{𝔯′} ×_{X^{𝔪}} X̃_{n}^{𝔮}| \ar[r]^{f} & Spec R[e]}

is Cartesian.

• We have one of the followings, depending on whether f^{∗}(e) is zero or a non-zero divisor.

1. |X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red} ≅ |X̃_{m}^{𝔯′} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red}.

2. The diagram of schemes

\xymatrix{|X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red} \ar[r] \ar[d] & |X̃_{m}^{𝔯} ×_{X^{𝔪}} X̃_{n}^{𝔮}| \ar[d] \\ |X̃_{m}^{𝔯′} ×_{X^{𝔪}} X̃_{n}^{𝔮}|_{red} \ar[r] & |X̃_{m}^{𝔯′} ×_{X^{𝔪}} X̃_{n}^{𝔮}|}

is Cartesian.

Now the proof of Theorem 5.13 is completed.

Proposition 5.17. Let G be a finite group, Z a Noetherian and equicharacteristic scheme of characteristic exponent p with the action of G, n a positive integer prime to p, and R = ℤ[1/n, μ_{n}]. Suppose that we are given a G-equivariant morphism Z → Spec R[T] with Spec R[T] trivially acted by G. Let Y = Z ×_{Spec R[T]} Spec R[T^{1/n}], π : Y → Z the natural map. Note that μ_{n} acts on Y via the natural action “T^{1/n} ↦ ζ · T^{1/n}” on R[T^{1/n}]. Assume, in addition, that π_{∗}𝒪_{Y_{red}} is free as a G-equivariant 𝒪_{Z_{red}}-module (Hence, Y_{red} is flat over Z_{red}). Then we have an isomorphism of equivariant K′-groups

K′_{j}(Z_{red}, G)_{ℚ} ⟶ K′_{j}(Y_{red}, G)_{ℚ}^{μ_{n}}

for j ≥ 0.

Proof. Consider the diagram

{0}_{R} ↪ 𝔸_{R}^{1} ↩ (𝔸^{1} \ {0})_{R}

and by taking the fiber product with respect to Y → Z → Spec R[T] construct the diagram

\xymatrix{Y_{0} \ar[r]^{i_{Y}} \ar[d] & Y \ar[d] & Y \ Y_{0} \ar[d] \ar[l]_{j_{Y}} \\ Z_{0} \ar[r]^{i_{Z}} & Z & Z \ Z_{0}. \ar[l]_{j_{Z}}}

Then we obtain a morphism between exact sequences ([Tho87])

\xymatrix{⋯ \ar[r] & K′_{j}(Y_{0}, G)_{ℚ}^{μ_{n}} \ar[r]^{i_{Y ∗}} & K′_{j}(Y, G)_{ℚ}^{μ_{n}} \ar[r]^{j_{Y}^{∗}} & K′_{j}(Y \ Y_{0}, G)_{ℚ}^{μ_{n}} \ar[r] & ⋯ \\ ⋯ \ar[r] & K′_{j}(Z_{0}, G)_{ℚ} \ar[u] \ar[r]^{i_{Z∗}} & K′_{j}(Z, G)_{ℚ} \ar[r]^{j_{Z}^{∗}} \ar[u] & K′_{j}(Z \ Z_{0}, G)_{ℚ} \ar[r] \ar[u] & · · · .}

The right vertical homomorphism is an isomorphism since Y \Y_{0} is Galois etale over Z \ Z_{0} with Galois group μ_{n}, and so is the left one as is seen by noting that μ_{n} acts on (Y_{0})_{red} ≅ (Z_{0})_{red} trivially, and using the next lemma:

Lemma 5.18. Let G be a finite group and W a Noetherian scheme with an action of G, and set W [ϵ]/(ϵ^{m}) = W ×_{Spec ℤ} Spec ℤ[ϵ]/(ϵ^{m}) and define f : W [ϵ]/(ϵ^{m}) → W to be the natural map . Then the composite

K′_{j}(W, G) ⟶^{f^{∗}} K′_{j}(W [ϵ]/(ϵ^{m}), G) ⟶^{f_{∗}} K′_{j}(W, G)

is the multiplication by m, and f_{∗} is an isomorphism for j ≥ 0. In particular, f^{∗} is an isomorphism up to torsion.

Thus we deduce that the middle vertical homomorphism is also an isomorphism. Since the composites

K′_{j}(Z, G)_{ℚ} → K′_{j}(Y, G)_{ℚ}^{μ_{n}} ≅ (K′_{j}(Y, G)_{ℚ})_{μ_{n}} → K′_{j}(Z, G)_{ℚ}

and

K′_{j}(Z_{red}, G)_{ℚ} → K′_{j}(Y_{red}, G)_{ℚ}^{μ_{n}} ≅ (K′_{j}(Y_{red}, G)_{ℚ})_{μ_{n}} → K′_{j}(Z_{red}, G)_{ℚ}

are isomorphisms by assumptions, Proposition easily follows.

Corollary 5.19. Let F and X be as in Convention 3.13. Then, for a prime ideal 𝔮 of F, we have a weak equivalence

\overline{K}(X_{Ket}^{𝔮})_{ℚ} ⟶^{≅} ℍ^{·}(X_{Ket}^{𝔮^{∨}}, ε_{X∗}^{𝔮^{∨}}(\overline{K}_{Ket})_{ℚ}).

Proof. It follows from Theorems 5.10 and 5.13.

5.4 Explicit description

In this subsection we prove Theorem 3.19.

Definition 26. Let F be a finitely generated free monoid, 𝔮 and 𝔯 its prime ideals, i a natural number, and Y an F -framed Noetherian equi-characteristic fs log scheme. Then we define a monoid morphism L : F (𝔮) → End_{Ab}(\overline{K}_{i}(Y_{Ket}^{𝔯})) by associating with each α ∈ F (𝔮) the group homomorphism

L(α) : \overline{K}_{i}(Y_{Ket}^{𝔯}) ⟶ \overline{K}_{i}(Y_{Ket}^{𝔯})

induced by the endofunctor “− ⊗ \overline{𝒪} _{Y} (α)” of \overline{Vect}(Y_{Ket}^{𝔯}). Now we set

\overline{K}_{i}(Y_{Ket}^{𝔯}) ⋊ Λ[𝔮] = \overline{K}_{i}(Y_{Ket}^{𝔯}) ⋊^{L} Λ[𝔮]

(See Definition in Subsection 3.3). Similarly, we can also define \overline{K}_{i}(Y_{Ket}^{𝔯}) ⋊ Λ′[𝔮] and \overline{K}(Y_{Ket}^{𝔯}) ⋊ Λ^{(′)}[𝔮].

Let 𝔯′ ⊃ 𝔯 be prime ideals such that 𝔮 ∧ 𝔯 = ∅. Then we have a natural map \overline{Pic} (X_{Ket}^{𝔮^{∨}}) → \overline{Pic} (X_{Ket}^{𝔯}). Thus we can construct a homomorphism \overline{K}_{i}(Y_{Ket}^{𝔯′}) ⋊ Λ[𝔮] → \overline{K}_{i}(Y_{Ket}^{𝔯}), as in Subsection 3.3 (See the argument before Theorem 3.19).

Proposition 5.20. Let F be a finitely generated free monoid and (Y, θ) an F - framed fs log scheme such that |Y | is Noetherian, local and equi-characteristic of characteristic exponent p. We denote by y its closed point. We assume that two primes 𝔯 and 𝔮 of F satisfy the following conditions:

1. 𝔯 ∧ 𝔮 = ∅.

2. For any element m ∈ θȳ(𝔮) ⊂ \overline{M}_{Y,ȳ}, all (any) of its liftings m̃ ∈ M_{Y,ȳ} are mapped to zero by the log-structure map α_{Y} : M_{Y} → 𝒪_{Y}.

Then the natural map

\overline{K}_{i}(Y_{Ket}^{𝔯∨𝔮}) ⋊ Λ′[𝔮] ⟶ \overline{K}_{i}(Y_{Ket}^{𝔯})

is an isomorphism for i ≥ 0.

Proof. Take and fix a chart Y → Spec ℤ[F] lifting the given frame (Proposition 3.7). By Proposition 4.21 and Lemma 4.23, it suffices to show that, for any natural number n prime to p and any integer i ≥ 0, the morphism below is an isomorphism

K_{i}(|{ \atop Y_{n}^{𝔯∨𝔮}}|_{red}, Ĩ_{n}^{𝔯∨𝔮}) ⊗ ℤ[F (𝔮)^{1/n}/F (𝔮)] → K_{i}(|Ỹ_{n}^{𝔯}|_{red}, Ĩ_{n}^{𝔯})

of (Zariski) equivariant K-groups (For the notation, see Definition before Lemma 5.14). Here this morphism maps [ℱ] ⊗ [α] to [π^{∗}ℱ ⊗ 𝒪(α)], where π is the natural morphism, and 𝒪(α) is the pull-back of an equivariant line bundle on the scheme Spec ℤ[μ_{n}][F (𝔮)^{1/n}] with the action of Hom (F (𝔮)^{1/n}/F (𝔮), ℤ[μ_{n}]^{×}) ⋊ Γ_{n} corresponding to α (See Definition after Proposition 4.22). Note that

Hom (F (𝔮)^{1/n}/F (𝔮), ℤ[μ_{n}]^{×}) × I_{n}^{𝔯∨𝔮} ≅ I_{n}^{𝔯}

by the assumption (1).

In addition, by the assumption (2), |{ \atop Y_{n}^{𝔯∨𝔮}}|_{red} ≅ |Ỹ_{n}^{𝔯}|_{red} and this has a trivial action of Hom (F (𝔮)^{1/n}/F (𝔮), ℤ[μ_{n}]^{×}), so the proposition can be deduced easily from the argument in equivariant module theory.

Corollary 5.21. Let F and Y be as in Proposition 5.20. Then for a prime 𝔮 of F satisfying the assumption (2), we have a weak equivalence

\overline{K}((Y_{red})_{Zar}) ⋊ Λ′[𝔮] ⟶ \overline{K}(Y_{Ket}^{𝔮^{∨}}).

Taking Corollary 5.12 into consideration, we obtain the following corollary:

Corollary 5.22. Let F and X be as in Convention 3.13, and 𝔮 a prime ideal of F. If, for any m ∈ 𝔮, θ(m) ∈ Γ(X, \overline{M}_{X}) is pseudo-zero, then we have a weak equivalence

\overline{K}((X_{red})_{Zar}) ⋊ Λ′[𝔮] ⟶ \overline{K}(X_{Ket}^{𝔮^{∨}}).

Since X[𝔭] clearly satisfies the condition in Corollary 5.22, we obtain Theorem 3.19.

5.5 Compatibility

Here we prove Theorem 3.22. First we prove the following compatibilities.

Proposition 5.23. Let F and X be as in Convention 3.13. Then 1. For prime ideals 𝔮 and 𝔯 of F, the diagram

\xymatrix{\overline{K}_{i}(X_{Ket}^{𝔮}) \ar[r]^{π_{i}(δ)} \ar[d]_{ε̄^{∗}} & \overline{K}_{i}(X_{Ket}^{𝔮∨𝔯}) \ar[d]^{ε̄^{∗}} \\ \overline{K}_{i}(X_{Ket}^{𝔮∧𝔯}) \ar[r]^{π_{i}(δ)} & \overline{K}_{i}(X_{Ket}^{𝔯})}

is commutative.

2. The morphism δ is compatible with the \overline{K}_{0}-module structure. More precisely, for prime ideals 𝔮 ⊂ 𝔯 of F, we have an equality

π_{i}(δ_{X}^{𝔯,𝔮})(x · y) = π_{i}(δ_{X}^{𝔯,𝔮})(x) · π_{0}(δ_{X}^{𝔯,𝔮})(y)

as elements of \overline{K}_{i}(X_{Ket}^{𝔯})_{ℚ}, for x ∈ \overline{K}_{i}(X_{Ket}^{𝔮})_{ℚ} and y ∈ \overline{K}_{0}(X_{Ket}^{𝔮})_{ℚ}.

Proof. (1) We set 𝔰 = 𝔮 ∧ 𝔯, 𝔱 = 𝔮 ∨ 𝔯, and 𝔲 = 𝔮 ∨ (𝔯^{∨})(= 𝔮 ∨ (𝔱^{∨})). Then we have (X^{𝔮})^{(𝔱\𝔮)^{∨}} = X^{𝔲}. Here recall that X^{𝔮} is F \ 𝔮-framed, and note that 𝔱 \ 𝔮 is a prime ideal of F \ 𝔮 (Note also that (𝔱 \ 𝔮)^{∨} denotes the dual in F \ 𝔮, that is, 𝔱^{∨} \ 𝔮). Similarly we have (X^{𝔰})^{(𝔯\𝔰)^{∨}} = X^{𝔲}. Therefore, by using the morphisms of sites

ε_{X^{𝔮}}^{(𝔱\𝔮)^{∨}} : X^{𝔮} → X^{𝔲} and ε_{X^{𝔰}}^{(𝔯\𝔰)^{∨}} : X^{𝔰} → X^{𝔲},

we can define presheaves of fibrant spectra ε_{∗}^{(𝔱\𝔮)^{∨}} \overline{K}_{Ket} and ε_{∗}^{(𝔯\𝔰)^{∨}} \overline{K}_{Ket} on Ket/X^{𝔲}. More concretely, we have

ε_{∗}^{(𝔱\𝔮)^{∨}} \overline{K}_{Ket}(U) = \overline{K}((U ×_{(X^{𝔮})^{(𝔱\𝔮)^{∨}}}^{fs} X^{𝔮})_{Ket}) = \overline{K}((U ×_{X^{𝔲}}^{fs} X^{𝔮})_{Ket})

for U ∈ Ket/(X^{𝔮})^{(𝔱\𝔮)^{∨}} and

ε_{∗}^{(𝔯\𝔰)^{∨}} \overline{K}_{Ket}(U) = \overline{K}((U ×_{(X^{𝔰})^{(𝔯\𝔰)^{∨}}}^{fs} X^{𝔰})_{Ket}) = \overline{K}((U ×_{X^{𝔲}}^{fs} X^{𝔰})_{Ket})

for U ∈ Ket/(X^{𝔰})^{(𝔯\𝔰)^{∨}}. Then we have a morphism of presheaves of spectra

ε_{∗}^{(𝔱\𝔮)^{∨}} \overline{K}_{Ket} → ε_{∗}^{(𝔯\𝔰)^{∨}} \overline{K}_{Ket},

which induces a commutative diagram

\xymatrix{\overline{K}(X_{Ket}^{𝔮}) \ar[r] \ar[d]_{ε̄^{∗}} & ℍ^{·}((X^{𝔮})_{Ket}^{(𝔱\𝔮)^{∨}}, ε_{∗}^{(𝔱\𝔮)^{∨}}(\overline{K}_{Ket})_{ℚ}) \ar[d]_{ε̄^{∗}} & \overline{K}((X^{𝔮})_{Ket}^{(𝔱\𝔮)})_{ℚ} \ar[d]_{ε̄^{∗}} \ar[l] \\ \overline{K}(X_{Ket}^{𝔰}) \ar[r] & ℍ^{·}((X^{𝔰})_{Ket}^{(𝔯\𝔰)^{∨}}, ε_{∗}^{(𝔯\𝔰)^{∨}}(\overline{K}_{Ket})_{ℚ}) & \overline{K}((X^{𝔰})_{Ket}^{(𝔯\𝔰)})_{ℚ}. \ar[l]}

The proposition now follows.

(2) can be proven in the similar way to Proposition 3.16.

Now, according to Proposition 3.15, Theorem 3.22 is reduced to the next proposition:

Proposition 5.24. Let F and X be as in Convention 3.13, and 𝔪 its maximal ideal. Take an element e of the basis of F and set 𝔫 = F \ e^{ℕ}. Then, by the composite of the natural maps

(e^{ℕ})_{ℚ′} → Pic (X_{Ket}^{𝔫}) → \overline{K}_{0}(X_{Ket}^{𝔫})_{ℚ} ⟶^{π_{0}(δ_{X}^{𝔪,𝔫})} \overline{K}_{0}(X_{Ket}^{𝔪})_{ℚ} ≅ K_{0}((X_{red})_{Zar})_{ℚ},

e^{α}(α ∈ ℚ′) is mapped to exp(α log[𝒪_{X}(e)]), where [𝒪_{X}(e)] ∈ K_{0}((X_{red})_{Zar}) is the image of e by the composite

e^{ℕ} ⊂ F → Γ(X_{red}, \overline{M}_{X_{red}}) → Pic ((X_{red})_{Zar}) → K_{0}((X_{red})_{Zar}).

Here exp and log are defined by formal power series.

Proof. We may assume that X is connected. Denote by a the image of e^{1/n} in K_{0}((X_{red})_{Zar})_{ℚ}. Since δ is compatible with the product structure, a^{n} must be equal to [𝒪_{X}(e)]. Now we can easily deduce the proposition, since the n-th power map on K_{0}((X_{red})_{Zar})_{ℚ} is an isomorphism, which follows from the fact that the map log : K_{0}((X_{red})_{Zar})_{ℚ} → K_{0}((X_{red})_{Zar})_{ℚ} is an isomorphism.

5.6 The end of the proof

The rest of this section is devoted to the proof of Theorem 3.17.

Theorem 5.25. Let F and X be as in Convention 3.13, 𝔭 and 𝔮 prime ideals of F, e an element of the basis of F, and ⟨e⟩ the prime ideal generated by e. We set 𝔭′ = 𝔭 ∨ ⟨e⟩ and 𝔮′ = 𝔮 ∨ ⟨e⟩. Then the natural commutative diagram

\xymatrix{ℍ^{·}(X[𝔭]_{Ket}^{𝔮′}, ε_{X[𝔭]∗}^{𝔮′}(\overline{K}_{Ket})_{ℚ}) \ar[r] \ar[d] & ℍ^{·}(X[𝔭]_{Ket}^{𝔮}, ε_{X[𝔭]∗}^{𝔮}(\overline{K}_{Ket})_{ℚ}) \ar[d] \\ ℍ^{·}(X[𝔭′]_{Ket}^{𝔮′}, ε_{X[𝔭′]∗}^{𝔮′}(\overline{K}_{Ket})_{ℚ}) \ar[r] & ℍ^{·}(X[𝔭′]_{Ket}^{𝔮}, ε_{X[𝔭′]∗}^{𝔮}(\overline{K}_{Ket})_{ℚ})}

is homotopy Cartesian.

Proof. By Corollary 5.9 and Proposition 4.25, we may assume that X is local. By using Corollary 5.19 and noting that π_{q}(δ) is surjective, we have only to show that the diagram

\xymatrix{\overline{K}_{q}(X[𝔭]_{Ket}^{𝔮^{∨}∧⟨e⟩^{∨}})_{ℚ} \ar[r]^{π_{q}(δ)} \ar[d]^{i^{∗}} & \overline{K}_{q}(X[𝔭]_{Ket}^{𝔮^{∨}})_{ℚ} \ar[d]^{i^{∗}} \\ \overline{K}_{q}(X[𝔭 ∨ ⟨e⟩]_{Ket}^{𝔮^{∨}∧⟨e⟩^{∨}})_{ℚ} \ar[r]^{π_{q}(δ)} & \overline{K}_{q}(X[𝔭 ∨ ⟨e⟩]_{Ket}^{𝔮^{∨}})_{ℚ}}

is bi-Cartesian (i.e. Cartesian and co-Cartesian), where i is the natural inclusion. Next we consider the diagram

\xymatrix{\overline{K}_{q}(X[𝔭]_{Ket}^{𝔮^{∨}})_{ℚ} \ar[d]^{i^{∗}} \ar[r]^{ε̄^{∗}} & \overline{K}_{q}(X[𝔭]_{Ket}^{𝔮^{∨}∧⟨e⟩^{∨}})_{ℚ} \ar[r]^{π_{q}(δ)} \ar[d]^{i^{∗}} & \overline{K}_{q}(X[𝔭]_{Ket}^{𝔮^{∨}})_{ℚ} \ar[d]^{i^{∗}} \\ \overline{K}_{q}(X[𝔭 ∨ ⟨e⟩]_{Ket}^{𝔮^{∨}})_{ℚ} \ar[r]^{ε̄^{∗}} & \overline{K}_{q}(X[𝔭 ∨ ⟨e⟩]_{Ket}^{𝔮^{∨}∧⟨e⟩^{∨}})_{ℚ} \ar[r]^{π_{q}(δ)} & \overline{K}_{q}(X[𝔭 ∨ ⟨e⟩]_{Ket}^{𝔮^{∨}})_{ℚ}.}

Since the outer rectangle is clearly bi-Cartesian (Proposition 3.15), it suffices to prove that so is the left square. Now we can reduce the theorem to the next lemma:

Lemma 5.26. Suppose that F and X be as in Convention 3.13, and in addition that X be local. Let 𝔯 ⊂ F be a prime and e ∈ F an element of the basis of F. Then the diagram

\xymatrix{\overline{K}_{q}(X_{Ket}^{𝔯∨⟨e⟩}) \ar[d]_{ε̄^{∗}} \ar[r]^{i^{∗}} & \overline{K}_{q}(X[⟨e⟩]_{Ket}^{𝔯∨⟨e⟩}) \ar[d]_{ε̄^{∗}} \\ \overline{K}_{q}(X_{Ket}^{𝔯}) \ar[r]^{i^{∗}} & \overline{K}_{q}(X[⟨e⟩]_{Ket}^{𝔯})}

is bi-Cartesian for every q ≥ 0.

Proposition 4.21 and the next proposition imply this lemma, thus the theorem is proven.

Proposition 5.27. Let n be a positive integer, R = ℤ[1/n, μ_{n}], G a finite group acted by Γ_{n} = Gal(ℚ(μ_{n})/ℚ), and Z a Noetherian equi-characteristic semi-local scheme of characteristic exponent p with the action of a finite group G ⋊ Γ_{n}. Suppose that p is prime to n, and that we are given a G ⋊ Γ_{n}-equivariant morphism Z → 𝔸_{R}^{1} = Spec R[e] with G ⋊ Γ_{n} naturally acting on 𝔸_{R}^{1} through Γ_{n}. We let the left square below be the fiber product of the right square with respect to Z → 𝔸_{R}^{1}:

\xymatrix{W_{n} \ar[r]^{i′} \ar[d]_{π′} & Z_{n} \ar[d]^{π} & \{0\} ×_{𝔸_{R}^{1}} (𝔸_{R}^{1})_{n} \ar[r] \ar[d] & (𝔸_{R}^{1})_{n} \ar[d] \\ W \ar[r]^{i} & Z & \{0\} \ar[r] & 𝔸_{R}^{1},}

where (𝔸_{R}^{1})_{n} = Spec R[e^{1/n}] and {0} is the closed subscheme consisting of the origin. Note that μ_{n} acts naturally on Z_{n} and W_{n}. Suppose that 1394 K. Hagihara

1. Z_{red}, W_{red} and (Z_{n})_{red} are regular (Note that W_{red} ↪ Z_{red} and (W_{n})_{red} ↪ (Z_{n})_{red} become regular immersions automatically),

2. π_{red} : (Z_{n})_{red} → Z_{red} is flat, and

3. the natural map W_{red} → W ×_{Z} Z_{red} is an isomorphism. Then the diagram consisting of (Zariski) equivariant K-groups

\xymatrix{K_{q}(Z_{red}, G ⋊ Γ_{n}) \ar[r]^{i^{∗}} \ar[d]_{π^{∗}} & K_{q}(W_{red}, G ⋊ Γ_{n}) \ar[d]_{π^{′∗}} \\ K_{q}((Z_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n}) \ar[r]^{i^{′∗}} & K_{q}((W_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n})}

is bi-Cartesian for every q ≥ 0. Proof. Since (Z_{n} \ W_{n})_{red} is Galois etale over (Z \ W)_{red} with Galois group μ_{n}, K′_{r}((Z \W)_{red}, G⋊Γ_{n}) ⟶^{π^{∗}} K′_{r}((Z_{n} \W_{n})_{red}, (G×μ_{n})⋊Γ_{n}) is an isomorphism for r = q, q + 1, so we see that the diagram

\xymatrix{K′_{q}(W_{red}, G ⋊ Γ_{n}) \ar[d]_{_{pr{∗ \atop 1}}} \ar[r]^{i_{∗}} & K′_{q}(Z_{red}, G ⋊ Γ_{n}) \ar[d]_{π^{∗}} \\ K′_{q}(V, (G × μ_{n}) ⋊ Γ_{n}) \ar[r]^{j_{∗}} & K′_{q}((Z_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n})}

is bi-Cartesian by Thomason’s localisation sequence ([Tho87]), where V = W_{red} ×_{Z_{red}} (Z_{n})_{red} and j is the natural closed immersion. Next, by considering the composite of morphisms

K′_{q}(W_{red}, G ⋊ Γ_{n})  ⟶^{i_{∗}}  K′_{q}(Z_{red}, G ⋊ Γ_{n})
(canonical)^{−1}
⟶

K_{q}(Z_{red}, G ⋊ Γ_{n})
⟶^{i^{∗}}  K_{q}(W_{red}, G ⋊ Γ_{n})

and its counterpart for (Z_{n})_{red}, we have a commutative diagram

\xymatrix{K′_{q}(W_{red}, G ⋊ Γ_{n}) \ar[r]^{i^{∗}∘(can)^{−1}∘i_{∗}} \ar[d]_{_{pr{∗ \atop 1}}} & K_{q}(W_{red}, G ⋊ Γ_{n}) \ar[d]^{π^{∗}} \\ K′_{q}(V, (G × μ_{n}) ⋊ Γ_{n}) \ar[r]^{i^{∗}∘(can)^{−1}∘j_{∗}} & K_{q}((W_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n}).}

It suffices to show that this diagram is bi-Cartesian.

Documenta Mathematica 21 (2016) 1345–1396

Now, since W_{red} ≅ (W_{n})_{red} and the action of μ_{n} on (W_{n})_{red} is trivial, we have isomorphisms

K′_{q}(W_{red}, G ⋊ Γ_{n}) ⊗_{ℤ} ℤ[ℤ/nℤ] ⟶^{≅} K′_{q}((W_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n})
⟶^{≅} K′_{q}(V, (G × μ_{n}) ⋊ Γ_{n}), and
K_{q}(W_{red}, G ⋊ Γ_{n}) ⊗_{ℤ} ℤ[ℤ/nℤ] ⟶^{≅} K_{q}((W_{n})_{red}, (G × μ_{n}) ⋊ Γ_{n}).

Here the first isomorphism and the third are defined by the rule x ⊗ [i] ↦ π^{′∗}x · [𝒪(e^{i/n})], where 𝒪(e^{i/n}) is the pull-back of μ_{n} ⋊ Γ_{n}-equivariant line bundle on (𝔸_{R}^{1})_{n} corresponding to e^{i/n} (See Definition after Proposition 4.22), with respect to (W_{n})_{red} → Spec R[e^{1/n}]/(e^{1/n}) ↪ Spec R[e^{1/n}]. In addition, noting that V (and W_{red}) is defined by the equation “e = 0” in (Z_{n})_{red} (and Z_{red}, respectively), we easily see that the above diagram can be rewritten as below, via these isomorphisms (The proofs of Proposition 1.6 and Lemma 1.7 in [Vis91] work also for our situation):

\xymatrix{K′_{q}(W_{red}, G ⋊ Γ_{n}) \ar[d] \ar[r]^{0} & K_{q}(W_{red}, G ⋊ Γ_{n}) \ar[d] \\ K′_{q}(W_{red}, G ⋊ Γ_{n}) ⊗_{ℤ} ℤ[ℤ/nℤ] \ar[r]^{can⊗f} & K_{q}(W_{red}, G ⋊ Γ_{n}) ⊗_{ℤ} ℤ[ℤ/nℤ],}

where, the right vertical morphism maps x to x ⊗ [0], the left vertical x to x ⊗ ∑_{i=0}^{n−1}[i], and f is the multiplication by [0] − [1]. Since this diagram is clearly bi-Cartesian, Proposition follows.

References

[BK72] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972. MR 0365573 (51 #1825)

[GG87] Henri Gillet and Daniel R. Grayson, The loop space of the Qconstruction, Illinois J. Math. 31 (1987), no. 4, 574–597. MR 909784 (89h:18012)

[GG03] , Erratum to: “The loop space of the Q-construction” [Illinois J. Math. 31 (1987), no. 4, 574–597; MR0909784 (89h:18012)], Illinois J. Math. 47 (2003), no. 3, 745–748. MR 2007234 (2004h:18012)

[Gra89] Daniel R. Grayson, Exterior power operations on higher K-theory, K-Theory 3 (1989), no. 3, 247–260. MR 1040401 (91h:19005)

[Hag03] Kei Hagihara, Structure theorem of Kummer étale K-group, K-Theory 29 (2003), no. 2, 75–99. MR 2 029 756

[Jar97] J. F. Jardine, Generalized étale cohomology theories, Progress in Mathematics, vol. 146, Birkhäuser Verlag, Basel, 1997. MR MR1437604 (98c:55013)

[Jar09] John F. Jardine, The K-theory presheaf of spectra, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, 2009, pp. 151–178. MR 2544389 (2010k:55012)

[Kat89] Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 99b:14020

[Kat94] , Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073– 1099. MR 95g:14056

[Nak92] Chikara Nakayama, Logarithmic étale cohomology, 1992, Master’s Thesis(in Japanese).

[Nak97] , Logarithmic étale cohomology, Math. Ann. 308 (1997), no. 3, 365–404. MR 98j:14022

[Qui73] Daniel Quillen, Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 49 #2895

[SGA4] Théorie des topos et cohomologie étale des schémas., Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963– 1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 269, 270, 305. MR 50 #7130

[Tho85] R. W. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR MR826102 (87k:14016)

[Tho87] , Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539– 563. MR MR921490 (89c:18016)

[Vis91] Angelo Vistoli, Higher equivariant K-theory for finite group actions, Duke Math. J. 63 (1991), no. 2, 399–419. MR MR1115114 (92d:19005)

[Wal85] Friedhelm Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796 (86m:18011)

Kei Hagihara

Department of Mathematics Keio University, Japan kei.hagihara@gmail.com