Exactness, integrality, and log modifications

Fumiharu Kato

1 Introduction

The exactness and the integrality of morphisms of log schemes, introduced by Kazuya Kato [3], have been widely accepted as practically useful concepts in log geometry. For instance, a log smooth morphism is underlain by a flat morphism if it is integral, while the underlying morphism of a log smooth morphism in general might be fairly pathological.

In this note we will prove that, under a few minor conditions, any morphism of fs log schemes can be modified by so-called log modifications to be an exact, and moreover an integral morphism. Our main theorem is the following (Theorem 3.13 and Theorem 3.16):

Theorem. Let f: X → Y be a morphism of locally noetherian fs log schemes. Suppose the underlying structure of f and Y are quasi-compact.

(1) There exists a log-ideal K of Y such that the base change of f by the log blow-up of Y along K is exact.

(2) There exists a commutative diagram

\begin{matrix}X̃ & ⟶ & X \\ f̃↓ & & ↓f \\ Ỹ & ⟶ & Y \end{matrix}

of fs log schemes such that the horizontal arrows are log modifications and f̃ is integral.

The log blow-up is also the concept which was already studied by Kazuya Kato in his unpublished works (cf. [4][5]). Roughly speaking, a log blow-up is the composite of a usual blow-up centered at an ideal which comes from an ideal of the log strtucture (log-ideal) followed by normalization. The log modification is a certain class of universally surjective log étale morphisms, which of course includes log blow-up’s.

Notice that the second part of the theorem above (especially for a log smooth morphism f) can be compared to a weak log-version of the famous flattening theorem by Raynaud and Gruson [9].

Notation and conventions. All sheaves are considered with respect to the étale topology. Log schemes are denoted simply by capital letters. For a log scheme X, we denote by X^{∘}, O_{X}, M_{X}, and \overline{M}_{X} the underlying scheme, the structure sheaf of X^{∘}, the log structure of X, and the characteristic (= M_{X}/O_{X}^{×}) of X, respectively. For a commutative and unitary monoid P the associated group (resp. the subgroup of invertible elements) is denoted by P^{gp} (resp. P^{×}).

2 Exact morphism and integral morphism

In this section we recall the definition of exactness and integrality, and basic properties of them, especially in relation to charts; we also recall the notion of neat and good charts which is useful in the next section.

Definition 2.1

(1) A homomorphism Q → P of monoids is said to be exact if the diagram

Q ⟶ P
↓  ↓
Q^{gp} ⟶ P^{gp}

is Cartesian in the category of sets.

(2) A morphism f: X → Y of log schemes is said to be exact at x ∈ X if the homomorphism \overline{M}_{Y,\overline{y}} → \overline{M}_{X,\overline{x}} is exact, where y = f(x). (Here and hereafter \overline{·} means the separable closure of points.) The morphism f is said to be exact if it is exact at any point of X.

If a homomorphism h: Q → P of monoids is exact and Q^{×} = 1, then h is injective. In particular, if a morphism f: X → Y is exact at x ∈ X, then \overline{M}_{Y,\overline{y}} → \overline{M}_{X,\overline{x}} is injective.

Let P be a monoid and A a ring. We denote by S(P)_{A} the log scheme underlain by the scheme Spec A[P] endowed with the log structure induced by P → A[P]. We write, in particular, S(P) = S(P)_{Z}. The log structure of S(P) is denoted by M_{P}. (Caution: Even if Q → P is exact, S(P) → S(Q) is not necessarily exact.)

Let X be a log scheme. A chart of X modeled on a monoid P is a strict morphism λ: X → S(P) of log schemes. The chart λ is said to be good at a point x ∈ X if P → \overline{M}_{X,\overline{x}} is an isomorphism. Good charts always exist at any point, provided X is fs.

Let f: X → Y be a morphism of log schemes. A chart of f modeled on a homomorphism h: Q → P of monoids is a commutative diagram

X ⟶ S(P)
f↓  ↓S(h)
Y ⟶ S(Q)

with the horizontal arrows strict. This chart of f is said to be neat at x ∈ X if h^{gp} is injective, P^{gp}/Q^{gp} → \overline{M}_{X/Y,\overline{x}} is an isomorphism, and P → \overline{M}_{X,\overline{x}} is exact. This chart of f is said to be good at x ∈ X if the charts X → S(P) and Y → S(Q) are good at x and f(x), respectively.

Lemma 2.2 (A. Ogus [8, Theorem 2.13]) Let f: X → Y be a morphism of fine log schemes and x ∈ X a point. Suppose there exists a chart Y → S(Q) of Y modeled on a fine monoid Q. Then f has fppf locally a neat chart at x modeled on a homomorphism of fine monoids extending the given chart Y → S(Q). Furthermore, if the order of the torsion part of \overline{M}_{X/Y,\overline{x}}^{gp} is invertible in the residue field k(\overline{x}), then the neat chart as above can be found étale locally.

Corollary 2.3 If f: X → Y is a morphism of fs log schemes exact at x ∈X, then f allows fppf locally a good chart at x.

Proof. Take étale locally a good chart Y → S(Q) at y = f(x) and extend it fppf locally to a neat chart (X → S(P), Y → S(Q), Q → P) at x. Since \overline{M}_{Y,\overline{y}} → \overline{M}_{X,\overline{x}} is injective, the chart X → S(P) is automatically good at x. □

Definition 2.4 A homomorphism h: Q → P of integral monoids is said to be integral if for any integral monoid Q′ and a homomorphism Q′ → Q, the push-out P ⊕_{Q} Q′ in the category of monoids is integral.

It is known that the integrality of h: Q → P is equivalent to the following condition: If a_{1}, a_{2} ∈ Q, b_{1}, b_{2} ∈ P and h(a_{1})b_{1} = h(a_{2})b_{2}, there exist a_{3}, a_{4} ∈ Q and b ∈ P such that b_{1} = h(a_{3})b, b_{2} = h(a_{4})b, and a_{1}a_{3} = a_{2}a_{4} (cf. [3, (4.1)]). It is also known that h: Q → P is integral and injective if and only if the map Z[h]: Z[Q] → Z[P] is flat.

Lemma 2.5

(1) Let h: Q → P be an integral homomorphism of integral monoids such that h^{−1}(P^{×}) = Q^{×}. Then h is exact.

(2) Consider the commutative diagram

P′ ⟶^{a} P
h′↑  ↑h
Q′ ⟶ Q

of integral monoids such that the horizontal arrow are surjective. Then:

(a) If h′ is integral and there exists a submonoid P′′ ⊆ P′ such that P′/P′′ ≅ P, then h is integral.

(b) If h is integral and the horizontal arrows are exact, then h′ is integral. (Here the quotient of a monoid M by a submonoid N is defined by the coset space M/ ∼ with the induced monoid structure, where a ∼ b if and only if there exist c, d ∈ N such that ac = bd.)

Proof. Straightforward. □

Definition 2.6 A morphism f: X → Y of integral log schemes is said to be integral at x ∈ X if \overline{M}_{Y,\overline{y}} → \overline{M}_{X,\overline{x}} is integral, where y = f(x). We say f is integral if it is integral at any point of X.

Note that the integrality of \overline{M}_{Y,\overline{y}} → \overline{M}_{X,\overline{x}} is equivalent to that of M_{Y,\overline{y}} → M_{X,\overline{x}}. By Lemma 2.5 (1), integral morphisms are exact. If h: Q → P is integral, then by Lemma 2.5 (2) (a), the morphism S(h): S(P) → S(Q) is integral.

In what follows, all push-outs are taken in the category of fs monoids, unless otherwise explicitely specified. Accordingly, all fiber products of fs log schemes are taken in the category fs log schemes, unless otherwise specified.

Let P ← Q → Q′ be homomorphisms of fs monoids and P^{gp} ← Q^{gp} → Q′^{gp} the induced morphisms, which gives rise to the push-out P^{gp} ⊕_{Q^{gp}} Q′^{gp} in the category of finitely generated abelian groups.

Lemma 2.7 The push-out of P ← Q → Q′ in the category of fs monoids is given by the smallest fs submonoid in P^{gp} ⊕_{Q^{gp}} Q′^{gp} containing those images of P and Q′, or what amounts to the same, the saturation of the submonoid in P^{gp} ⊕_{Q^{gp}} Q′^{gp} generated by those images of P and Q′.

Proof. It is enough to prove that the push-out in the category of fine monoids is the submonoid in P^{gp} ⊕_{Q^{gp}} Q′^{gp} generated by the images of P and Q′, which is easy to verify. □

Corollary 2.8 For a homomorphism Q → P of fs monoids such that Q^{gp} ≅ P^{gp}, the push-out of P ← Q → P is given by P.

We denote the push-out of P ← Q → Q′ by P_{Q′} if the monoid Q is evident in the context.

Lemma 2.9 For an exact homomorphism Q → P of fs monoids and a homomorphism Q → Q′ of fs monoids, the induced homomorphism Q′ → P_{Q′} is exact.

Proof. It is straightfoward to see that the induced homomorphism Q′ → P′ is exact, where P′ is the submonoid in P^{gp} ⊕_{Q^{gp}} Q′^{gp} generated by the images of P and Q′. The monoid P_{Q′} is the saturation of P′. Since Q′ is saturated, we see that Q′ → P_{Q′} is also exact. □

Corollary 2.10 Exactness is stable under base change.

3 Log blow-up and log modification

Definition 3.1 Let Q be a monoid. A subset K ⊆ Q (possibly empty) is said to be an ideal of Q if ax ∈ K for any a ∈ Q and x ∈ K.

Any ideal of a finitely generated monoid is finitely generated as an ideal. Products and intersections of ideals are obviously ideals. The correspondence K ↦ \overline{K}: = π(K) · \overline{Q} gives rise to the natural bijection between the set of all ideals in Q and the set of all ideals in \overline{Q}, where π: Q → \overline{Q} = Q/Q^{×} is the natural projection.

Notation 3.2 Let Q be an fs monoid. For a subset E ⊆ Q^{gp} we denote by Q⟨E⟩ the smallest fs submonoid in Q^{gp} which contains Q and E.

Definition 3.3 Let X be a log scheme. A log-ideal of X is a sheaf of ideals K of M_{X}.

Notation 3.4

(1) For an ideal K ⊆ Q we denote by K̃ the log-ideal of S(Q) generated by the image of K.

(2) Let f: X → Y be a morphism of log schemes and K a log-ideal of Y. Then we denote by f^{•}K the log-ideal of X generated by the image of f^{−1}K.

Clearly, for f: X → Y and g: Y → Z and a log-ideal K of Z, we have (g ∘f)^{•}K = f^{•}g^{•}K.

A log scheme X is called locally noetherian if the underlying scheme X^{∘} is locally noetherian.

Lemma 3.5 (A. Ogus [8, Proposition 2.6]) Let X be a locally noetherian fine log scheme and K a log-ideal. Then the following conditions are equivalent:

(1) For any x, y ∈ X such that x belongs to the topological closure of {y}, the natural map K\overline{x} → K\overline{y} is surjective.

(2) For any point x ∈ X and any étale neighborhood ι: U → X of x with chart λ: U → S(P) set K = β−1 ; \overline{x}(K\overline{x}), where β: P → M_{X}|_{U}. Then replacing U by a sufficiently small étale neighborhood of x, we have ι^{•}K = λ^{•}K̃.

Definition 3.6 Let X be a locally noetherian fine log scheme. Then a log-ideal K of X is said to be coherent if it satisfies the conditions in Lemma 3.5.

Definition 3.7 A coherent log-ideal K of X is said to be locally principal if it is étale locally generated by a single global section.

Definition 3.8 Let X be a locally noetherian fs log scheme and K a coherent logideal of X. Then the log blow-up ρ_{K}: B_{K}(X) → X of X along K is a morphism of locally noetherian fs log schemes satisfying the following conditions:

(1) The log ideal ρ_{K}^{•}K of B_{K}(X) is locally principal.

(2) For any morphism f: X′ → X of locally noetherian fs log schemes such that f^{•}K is locally principal, there exists a unique morphism f̃: X′ → B_{K}(X) such that f = ρ_{K} ∘ f̃.

The log blow-up along K is unique up to isomorphisms if it exists. We are going to construct ρ_{K}: B_{K}(X) → X. We should first start with the case X = S(P) for an fs monoid P.

The following construction is due to Kazuya Kato [5]: Let K ⊆ P be an ideal. Then the log blow-up B_{K}(P) → S(P) of S(P) along K is constructed as follows: Let I(K) be the ideal of Z[P] generated by K. Consider the natural morphism Proj ⊕_{n}I(K)^{n} → Spec Z[P]. Proj ⊕_{n}I(K)^{n} has the affine open covering

Proj ⊕_{n}I(K)^{n} = ⋃_{p∈K} Spec Z[P ⟨p^{−1}K⟩].

(It is enough for p to run over the generators of K.) The canonical log structures given by P ⟨p^{−1}K⟩ → Z[P ⟨p^{−1}K⟩] obviously glue to a fine log structure on Proj ⊕_{n}I(K)^{n}. The fs log scheme B_{K}(P) is defined by the saturation of this resulting log scheme. The morphisms S(P ⟨p^{−1}K⟩) → S(P) give rise to the desired morphism B_{K}(P) → S(P), which turns out to be the log blow-up in this case.

Let X be an fs log scheme underlain by a locally noetherian scheme and K a coherent log-ideal. Suppose X has a chart λ: X → S(P) modeled on an fs monoid P, and that the log-ideal K is generated by global sections. Take a point x ∈ X and set K = β−1 ; \overline{x}(K\overline{x}), where β: P → M_{X}. Since K is generated by global sections, the ideal K does not depend on the choice of x. In this situation we define the log blow-up B_{K}(X) → X by

B_{K}(X): = X ×_{S(P)} B_{K}(P) ⟶ X.

It can be checked without so much pain that this morphism satisfies the conditions in Definition 3.8. In particular, it does not depend on the choice of the chart λ.

In general we take an étale covering ⋃_{i∈I} U_{i} → X such that each U_{i} has allows a chart U_{i} → S(P_{i}) and K|_{U_{i}} is generated by global sections. Then from what we have seen above the local log blow-up’s B_{K|_{U_{i}}} (U_{i}) → U_{i} glue to a morphism of log schemes B_{K}(X) → X which is nothing but the log blow-up of X along the coherent log-ideal K. Note that log blow-up’s are log étale.

The following lemmas are easy to see by the universality of log blow-up’s:

Lemma 3.9 Let X be a locally noetherian fs log scheme and K a log-ideal of X. Then we have B_{K}(X) ×_{X} B_{K}(X) ≅ B_{K}(X).

Lemma 3.10 Let K and K′ be coherent log-ideals of X. Then we have a canonical isomorphisms

B_{K∗K′}(X) ≅ B_{ρ_{K}^{•}K′}(B_{K}(X)) ≅ B_{ρ_{K′}^{•}K}(B_{K′}(X))

over X, where K ∗ K′ denote the log-ideal of X generated by K and K′.

Example 3.11 It is worth while describing log blow-up’s in the toric case: Let Σ be a fan of N ≅ Z^{r} and T_{Σ} the assiciated toric variety over a ring A. For any cone σ ∈ Σ, the affine patch T_{σ} = Spec A[σ^{∨} ∩ M] (where M = Hom_{Z}(N, Z)) has the canonical log structure which gives rise to S(σ^{∨} ∩ M)_{A}. These patches glue to an fs log scheme which is also denoted by T_{Σ}. Let T = Spec A[M] be the torus included in T_{Σ} as a Zariski open set and i: T ↪ T_{Σ} the open immersion. Then T is exactly the locus on which the log structure is trivial. Furthermore, the log structure M_{T_{Σ}} on T_{Σ} is isomorphic to

O_{T_{Σ}} ∩ i_{∗}O_{T}^{×} ⟶ O_{T_{Σ}}.

Log blow-up’s of toric varieties coincides with those described in [6, I, §2]: For a log-ideal K of T_{Σ}, the ideal I(K) of O_{T_{Σ}} generated by K is invariant under the action of T. Conversely, to a T -invariant ideal of O_{T_{Σ}} we can associate the log ideal of T_{Σ} by taking the intersection with M_{T_{Σ}}. This correspondence gives rise to the bijection between the set of all log-ideals of T_{Σ} and that of all T -invariant ideals of O_{T_{Σ}}. If I is a T -invariant ideal of O_{T_{Σ}}, the log blow-up along I is given by a certain subdivision Σ̃ → Σ of the fan Σ; the fan Σ̃ is the subdivision consisting of cones on which the associated piecewise linear function ordI.

Any log blow-up of a toric variety therefore comes from subdivision of the fan. But a morphism of toric varieties by subdivision is not necessarily a log blow-up.

Lemma 3.12 Let Q → P be a homomorphism of fs monoids such that Q^{gp} ≅ P^{gp}. Then there exists an ideal K ⊆ Q and a strict open immersion S(P) ↪ B_{K}(Q) such that

S(P) = S(P)
↓  ↓
S(Q) ⟵ B_{K}(Q)

is a Cartesian diagram of fs log schemes.

Proof. Write T = S(P) and S = S(Q). We can set P = Q⟨\frac{a_{1}}{b} , …, \frac{a_{r}}{b} ⟩ in Q^{gp} ≅ P^{gp}, where b, a_{1}, …, a_{r} ∈ Q. Let K = (b, a_{1}, …, a_{r}) be the ideal of Q. Then the log blow-up B_{K}(Q) has the covering

B_{K}(Q) = ⋃_{i=0,...,r} V_{i},

where

V_{0} = S(Q⟨\frac{a_{1}}{b} , …, \frac{a_{r}}{b} ⟩),

which is isomorphic to T, and

V_{j} = S(Q⟨\frac{b}{a_{j}} , \frac{a_{1}}{a_{j}} , …, \frac{a_{r}}{a_{j}} ⟩)

for j > 0. By Lemma 2.7 we have T ×_{S} V_{0} ≅ T and T ×_{S} V_{j} ≅ S(P ⟨\frac{b}{a_{j}} ⟩) for j > 0. Since we can check T ×_{S} (V_{0} ∩ V_{j}) = T ×_{S} V_{j}, we have T ×_{S} B_{K}(Q) ≅ T. □

Theorem 3.13 Let f: X → Y be a morphism of locally noetherian fs log schemes. Suppose f^{∘} and Y^{∘} are quasi-compact. Then there exists a log-ideal K of Y such that the base change X_{B_{K}(Y)} → B_{K}(Y) by the log blow-up of Y along K is exact.

Proof. First we assume that \overline{M}_{X/Y}^{gp} is constant and that there exists a chart Y → S(Q). Then by Lemma 2.2 there exists an fppf covering {X_{i} → X} of X consisting of finitely many members such that each X_{i} → Y allows a chart (λ_{i}: X_{i} → S(P_{i}), μ: Y → S(Q), h_{i}: Q → P_{i}) such that h_{i} is injective and \overline{M}_{X/Y}^{gp} ≅ P_{i}^{gp}/Q^{gp}. Set Q_{i} = P_{i} ∩ Q^{gp}. Then Q → Q_{i} yields Q^{gp} ≅ Q_{i}^{gp}. Set Y_{i} = Y ×_{S(Q)} S(Q_{i}) for each i. By Lemma 3.12 we have an ideal K_{i} of Q such that S(Q_{i}) ↪ B_{K_{i}}(Q) gives the base change. Set K_{i} = μ^{•}K̃_{i} and consider the log blow-up B_{K_{i}}(Y) → Y. Then we have X_{i} ×_{Y} B_{K_{i}}(Y) ≅ X_{i}, since Y_{i} ×_{Y} B_{K_{i}}(Y) ≅ Y_{i}. Since Y_{i} → B_{K_{i}}(Y) is strict, we have \overline{M}_{X_{i}/ B_{K_{i}}(Y)}^{gp} ≅ \overline{M}_{X_{i}/Y}^{gp} , which is constant and is isomorphic to P_{i}^{gp}/Q_{i}^{gp}. Hence it follows from the exactness of Q_{i} → P_{i} that X_{i} → Y_{i} is exact, and hence X_{i} → B_{K_{i}}(Y) is exact. Let K be the log-ideal of Y generated by K_{i}’s. Since the exactness is stable under base changes, the base change X_{B_{K}(Y)} → B_{K}(Y) restricted on each X_{i} is exact, and hence the whole X_{B_{K}(Y)} → B_{K}(Y) is exact. Hence we have proved the theorem in this particular case.

In general we may still assume Y allows a chart Y → S(Q). The sheaf \overline{M}_{X/Y}^{gp} is constructible. We look at each locally closed set of X^{∘} on which \overline{M}_{X/Y}^{gp} is constant, and take a log blow-up to make f exact on this locus. Finally, we gather up the resulting log-ideal of Y and then take sufficiently large log blow-up. □

Definition 3.14 A morphism ρ: X̃ → X of locally noetherian fs log schemes is called a log modification if for any point x ∈ X there exists an étale neighborhood U → X of x, and a log blow-up Z → X̃_{U} such that the composite Z → X̃_{U} → U is also a log blow-up.

Note that if X is quasi-compact the morphism ρ: X̃ → X is a log modification if and only if there exists a log blow-up Z → X̃ such that the composite Z → X̃ → X is also a log blow-up.

The above definition of log modifications is motivated by the following facts:

• A morphism ρ: X̃ → X of quasi-projective varieties over a field is a modification (i.e. proper and birational morphism) if and only if there exists a blow-up Z → X̃ such that the composite Z → X is a blow-up; this follows easily from [1, Theorem 7.17] and Chow’s lemma.

• Any morphism of toric varieties coming from subdivision of fans are log modifications due to De Concini-Procesi’s Theorem (cf. [7, §1.7]).

Note that log modifications are log étale.

Lemma 3.15 (1) Log modifications are universally surjective. (A morphism X → Y of fs log

schemes is said to be universally surjective if for any morphism Y′ → Y of fs log schemes the underlying morphism of the base change X_{Y′} → Y′ is surjective.)

(2) If X → Y and Y → Z are log modifications, then the composite X → Z is a log modification.

(3) If X̃ → X is a log modification and Y → X is a morphism, then the base change X̃_{Y} → Y is a log modification.

Proof. First we prove (1). It is enough to prove that log blow-up’s are universally surjective. By the universality of log blow-up’s it suffices to show that any blow-up is surjective. By an easy observation we reduce to verify whether B_{K}(P) → S(P) is surjective for a monoid P and an ideal K. But this is trivial since the underlying morphism of B_{K}(P) → S(P) is a normalized blow-up in the usual sense. To prove (2) we may assume there exists a log blow-up B(Z) → Z which factors through Y by a log blow-up. We can take a log blow-up B(Y) → Y which factors through X by a log blow-up, since Y is easily seen to be quasi-compact. Take the log blow-up B(B(Z)) → B(Z) along the pull-back of the blown-up center of B(Y) → Y. Then we have the unique morphism B(B(Z)) → B(Y), and then (2) is proved. (3) is straightforward. □

Theorem 3.16 Let f: X → Y be a morphism of locally noetherian fs log schemes. Suppose f^{∘} and Y^{∘} are quasi-compact. Then there exists a commutative diagram

\begin{matrix}X̃ & ⟶ & X \\ f̃↓ & & ↓f \\ Ỹ & ⟶ & Y \end{matrix}

of fs log schemes such that the horizontal arrows are log modifications and f̃ is integral.

Proof. By Theorem 3.13 we may assume f is exact, since base changes by log blow-up’s are log modifications. Since we have assumed Y^{∘} is quasi-compact, we may assume Y allows a chart μ: Y → S(Q) which is good at a point. By Corollary 2.3 we can take fppf locally on X a chart (λ: X → S(P), μ: Y → S(Q), h: Q → P) of f which is good at a point. Then S(h): S(P) → S(Q) is a morphism of toric varieties which comes from a morphism of fans Σ′ → Σ (both of which are defined by single cones) underlain by φ: N′ → N.

For any ray ρ ∈ Σ′_{1} the image of ρ in N is either a ray or a point. If it is a ray, then take the primitive base n_{1} ∈ N of φ(ρ), and extend it to a Z-basis n_{1}, …, n_{r} of N. The r + 1-rays spanned by n_{1}, …, n_{r}, −(n_{1} + ⋯ + n_{r}) give the projective space P_{Z}^{r}, and the very ample divisor O(1) gives rise to the support function, which we denote by s_{ρ}. In case ρ is mapped to the point we set s_{ρ} = 0. Then

s = ∑_{ρ∈Σ′_{1}} s_{ρ}

gives a piecewise linear function on N ⊗_{Z} R which yields a log-ideal K on S(Q) (cf. [6, I, Theorem 9]). Set K′ = S(h)^{•}K. The induced morphism B_{K′}(P) → B_{K}(Q) is also an equivariant morphism of toric varieties which comes from Σ̃′ → Σ̃, where Σ̃ (resp. Σ̃′) is the subdivision of Σ (resp. Σ′) consisting of biggest cones on which s (resp. φ_{R}^{∗}s) is linear. Since any cones in Σ̃′ is mapped surjectively onto some cone in Σ̃, that the morphism of associated toric varieties B_{K′}(P) → B_{K}(Q) has constant fiber dimension. Taking more auxiliary log blow-up’s of B_{K}(Q) and B_{K′}(P), we may assume B_{K}(Q) is non-singular. In this case the morphism B_{K′}(P) → B_{K}(Q) is flat (since toric varieties are Cohen-Macaulay), or what amounts to the same, integral (as a morphism of fs log schemes) by [3, (4,1)].

We gather up these resulting local log-ideals on Y to make the log-ideal K. We set K′ = f^{•}K and then consider B_{K′}(X) → B_{K}(Y). Then by what we have seen above it is fppf locally integral, and hence is integral. □

Acknowledgments. This work owes much to conversations with Professor Richard Pink. The author expresses gratitude to him. The author thanks Professor Masanori Ishida and Professor Takeshi Kajiwara for discussion on the proof of Theorem 3.16. Thanks are also due to Kazuhiro Fujiwara for valuable discussions. The author thanks Institut Henri Poincaré and Max-Planck-Institut für Mathematik Bonn for the nice hospitality.

References

[1] Hartshorne, R.: Algebraic Geometry, Graduate Texts 49, Springer-Verlag, Heidelberg (1977).

[2] Illusie, L.: Logarithmic spaces (according to K. Kato), in Barsotti Symposium in Algebraic Geometry (V. Cristante and W. Messing, Eds.). Perspectives in Math. 15, 183–203. Academic Press, 1994.

[3] Kato, K.: Logarithmic structures of Fontaine–Illusie, in Algebraic Analysis, Geometry and Number Theory (J.-I. Igusa, ed.). Johns Hopkins Univ., 1988, 191–224.

[4] Kato, K.: Logarithmic structures of Fontaine–Illusie II — Logarithmic flat topology, preprint.

[5] Kato, K.: Logarithmic degeneration and Dieudonne theory, preprint.

[6] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I, Lecture Note in Math. 339, Springer-Verlag, Berlin, Heidelberg, New York (1973).

[7] Oda, T.: Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties. Ergebnisse der Math. (3) 15. Springer-Verlag, Berlin-New York, 1988.

[8] Ogus, A.: Lectures on Logarithmic De Rham Cohomology, preprint.

[9] Raynaud, M., Gruson. L: Critères de platitude et de projectivité. Techniques de ”platification” d’un module, Invent. Math. 13 (1971), 1–89.

Graduate School of Mathematics, Kyushu University, Hakozaki Higashiku, Fukuoka 812-8581, Japan.

Current Address: Max-Planck-Institut für Mathematik, P.O.Box: 7280, D-53072, Bonn, Germany.