Math. J. Okayama Univ. 59 (2017), 1–19

CATEGORICAL CHARACTERIZATION OF STRICTMORPHISMS OF FS LOG SCHEMESYuichiro Hoshi and Chikara Nakayama
Abstract. In the present paper, we study a categorical characterization of strict morphisms of fs log schemes. In particular, we prove thatstrictness of morphisms of fs log schemes is preserved by an arbitraryequivalence of categories between suitable categories of fs log schemes.
The main result of the present paper leads us to a relatively simple alternative proof of a result on a categorical representation of fs log schemes
proved by S. Mochizuki.

Introduction
Let S be an fs log scheme whose underlying scheme is locally noetherian.
Then, by considering noetherian fs log schemes of finite type over S, we
obtain a category Schlog (S) [cf. §0, Log Schemes]. In the present paper, we
discuss a categorical characterization of strict morphisms in this category
Schlog (S). Our main result is as follows [cf. Theorem 3.7]:
Theorem. Let S and T be fs log schemes whose underlying schemes are
locally noetherian,∼φ : Schlog (S) −→ Schlog (T )an equivalence of categories, and f a morphism in Schlog (S). Then it
holds that f is strict if and only if φ(f ) is strict.
The content of Theorem is in fact a formal consequence of a result of S.
Mochizuki [i.e., [Mzk2], Theorem A]. Moreover, Mochizuki also proved thata result concerning a categorical representation of fs log schemes follows from
Theorem, together with some arguments [cf. Remark 3.7.1]. In particular,
the proof of the main theorem of the present paper may be regarded as analternative proof of the categorical representation of fs log schemes alreadyproved by Mochizuki.
Here, let us discuss the differences between the arguments of [Mzk2] andthe present paper. In the present paper, by establishing [cf. Proposition 2.5]
a categorical characterization of fs log points [i.e., fs log schemes whose
underlying schemes are isomorphic to the spectra of fields], we obtain asimple proof of Theorem. In particular,
Mathematics Subject Classification. Primary 14A20; Secondary 14A15.
Key words and phrases. fs log scheme, strict morphism, fs log point.
1

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Y. HOSHI AND C. NAKAYAMA(i) the proof of the present paper may be regarded as a relatively simple
alternative proof of Theorem, as well as the categorical representation of fs
log schemes, i.e., [Mzk2], Theorem A.
Next, as explained in [Mzk2], Introduction, the theory of [Mzk2] arose as
an attempt to correct errors contained in the theory of [Mzk1] that was
supposed to lead us to Theorem. By comparing [Mzk2] with the present
paper from this point of view, one may find that(ii) the argument of [Mzk1] [that contains errors] is closer to the argumentof the present paper than the argument of [Mzk2].
On the other hand, let us observe that, in the proof of the present paper,
one cannot avoid the use of the following two existence results:• the existence of certain non-separated schemes in the category Schlog (S)[cf. the proof of Lemma 2.4],• the existence of closed points of noetherian schemes [cf., e.g., the proofof Proposition 2.5].
By contrast, there is no need to use the above two existence results in thearguments of [Mzk2]. As a result,(iii) the arguments of the present paper only work under relatively restricted conditions, whereas the arguments of [Mzk2] can be applied in moregeneral situations, e.g., even if one replaces the category Schlog (S) [of noetherian fs log schemes of finite type over S] by, for instance, the categoryof [not necessarily noetherian] separated fs log schemes locally of finite typeover S.
In the proof of Theorem, we prove, by applying Hilbert’s Theorem 90, asufficient condition [cf. Proposition 1.3, Remark 1.3.1] for an fs log point tobe quasi-split [cf. Definition 1.2, (ii)]. In Appendix of the present paper, we
also discuss, by considering twisted versions of Hilbert’s Theorem 90, further
such sufficient conditions [cf. Theorem A.5]. Note that the proof of Theorem
does not depend on these further sufficient conditions obtained in Appendix.

0. Notations and Conventions
Monoids: We shall refer to a commutative semigroup with the unit element
as a monoid. Let M be a monoid. Then we shall write M × ⊆ M for thedefsubmonoid consisting of invertible elements of M , M = M/M × , and M gp
for the groupification of M . Moreover,• we shall say that M is sharp if M × has only the unit element;

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

3• we shall say that M is integral if the natural homomorphism M → M gp
is injective [which thus implies that M may be regarded as a submonoid of
M gp ];• we shall say that M is saturated if M is integral, and, moreover, foreach x ∈ M gp , it holds that x ∈ M if the submonoid of M gp generated by xintersects nontrivially M ⊆ M gp ;• we shall say that M is fs if M is finitely generated and saturated.

Let h : M → N be a homomorphism of fs monoids. Then we shall write→ N gp for the homomorphism between the groupifications induced by h. Moreover,
hgp : M gp

• we shall say that h is local if M × = h−1 (N × );• we shall say that h is exact if M = (hgp )−1 (N ) [in M gp ].

Log Schemes: A basic reference for the notion of log schemes is [Kato].
◦

Let X be an fs log scheme. Then we shall write X for the underlying◦

scheme of X, OX for the structure sheaf of X [regarded as an étale sheaf],◦

MX for the [étale] sheaf of monoids on X which defines the log structure ofdef×. Moreover,X, and MX = MX /OX• we shall say that X is of log rank n [where n is an integer] if thegroupification [which is necessarily a free module] of the stalk of MX at any◦

geometric point of X is of rank n;◦• we shall say that X is an fs log point if X is isomorphic to the spectrumof a field;• we shall say that X is a trivial log point if X is an fs log point and oflog rank 0;• we shall say that X is a standard log point if X is an fs log point andof log rank 1.
◦

Let f : X → Y be a morphism of log schemes. Then we shall write◦◦

f : X → Y for the underlying morphism of schemes of f . Moreover,◦• we shall say that f is strict if the natural homomorphism f −1 MY →MX is an isomorphism;

4

Y. HOSHI AND C. NAKAYAMA• we shall say that f is exact if the homomorphism [of fs monoids] obtained◦by considering the stalk of the homomorphism f −1 MY → MX at any◦

geometric point of X is exact.

◦

Let S be an fs log scheme whose underlying scheme S is locally noetherian.
Then we shall write
Schlog (S)for the category defined as follows: An object of Schlog (S) is a morphism oflog schemes X → S, where X is an fs log scheme whose underlying scheme
is noetherian, whose underlying morphism of schemes is of finite type. A
morphism in Schlog (S) [from an object X → S to an object Y → S] is amorphism of log schemes X → Y lying over S [whose underlying morphismof schemes is necessarily of finite type]. To simplify the exposition, we shalloften refer to the domain X of an arrow X → S which is an object of
Schlog (S) as an “object of Schlog (S)”.
We shall say that a morphism in Schlog (S) is an fs (respectively, a trivial;
a standard) log point if the domain of the morphism is an fs (respectively, atrivial; a standard) log point.

1. Characterization of Trivial and Standard Log Points
In the present §1, we give a categorical characterization of trivial andstandard log points [cf. Proposition 1.7 below]. In the present §1, let S bean fs log scheme whose underlying scheme is locally noetherian.
First, let us prove some facts on sharp fs monoids:defLemma 1.1. Let M be a sharp fs monoid. Write V = M gp ⊗Z Q anddefr = dimQ (V ). Then the following hold:(i) For each x ∈ M gp r{0}, there exists a local homomorphism h : M →N such that hgp (x) 6= 0.

(ii) Let L ⊆ V be a nonzero Q-subspace. Then there exist r local homomorphisms h1 , . . . , hr : M → N which satisfy the following two conditions:L(1)
The homomorphism hgp : M gp →L ri=1 Z induced by ther[necessarily local] homomorphism
h: M →
i=1 N given by mappingLrr
x ∈ M to (hi (x))i=1 ∈ i=1 N is injective.
(2) For every 1 ≤ i ≤ r, L is not contained in the kernel of the
Q-linear homomorphism hQ
i : V → Q induced by hi .

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

5(iii) Suppose that a finite group G acts on M . Then there exists a
homomorphism h : M → N which is local and G-equivariant [with respect
to the trivial action of G on N].
(iv) In the situation of (iii), suppose that r ≥ 2. Then there exists a
submonoid P ⊆ M gp such that M ( P , and, moreover, P is G-stable,
sharp, and fs.
Proof. Assertion (i) follows from [Mzk1], Lemma 2.5, (iii). Next, we verify
assertion (ii). It follows immediately from assertion (i) that there exist r
local homomorphisms h1 , . . . , hr : M → N which satisfy condition (1). Thus,
there exists 1 ≤ i0 ≤ r such that L 6⊆ Ker(hQi0 ). Then one verifies easily
that, by replacing hi [where 1 ≤ i ≤ r] by hi + hi0 (respectively, hi ) ifQ
L ⊆ Ker(hQi ) (respectively, L 6⊆ Ker(hi )), we obtain r homomorphismsof the desired type. This completes the
P proof of assertion (ii). Assertion(iii) follows [by considering the sumg∈G h ◦ g : M → N for some local
homomorphism h : M → N] from assertion (i).
Finally, we verify assertion (iv). Let h : M → N be a G-equivariant local
homomorphism [cf. assertion (iii)]. Let x ∈ M gp r M be such that hgp (x) ∈Nr{0}. [Note that one may verify existence of such an x as follows: Assume
that (hgp )−1 (N r {0}) ⊆ M . Let a ∈ M r M × . Then since h(a) ∈ N r {0}by the fact that h is local, it holds that Ker(hgp ) + a ⊆ (hgp )−1 (N r {0}) ⊆M , which thus implies that Ker(hgp ) + a ⊆ h−1 ({h(a)}). On the other
hand, since M is finitely generated, and h is local, one verifies easily thath−1 ({h(a)}) is finite. In particular, Ker(hgp ) + a, hence also Ker(hgp ), is
finite, in contradiction to our assumption that r ≥ 2.] Write P ⊆ M gp forthe saturation of the submonoid N ⊆ M gp generated by M ⊆ M gp and theG-orbit of x ∈ M gp . Then it is immediate that M ( P , and that P is Gstable and fs. Moreover, since h is G-equivariant, it follows from our choiceof x that the image of every nontrivial element of N via hgp is contained in
N r {0}, which thus implies that N , hence also P , is sharp. This completes
the proof of assertion (iv), hence also of Lemma 1.1.
◦Definition 1.2. Let X be an fs log point. Thus, X is isomorphic tothe spectrum of a field k. Let ksep be a separable closure of k. Writedef◦

Gk = Gal(ksep /k), x → X for the geometric point determined by the sepadef

rable closure ksep , and M = MX,x . [Thus, the Gk -monoid M is naturally
isomorphic to the Gk -monoid obtained by forming the stalk MX,x ; moreover, M is sharp and fs.](i) We shall say that X is split if the action of Gk on M is trivial.

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Y. HOSHI AND C. NAKAYAMA(ii) We shall say that X is quasi-split if the Gk -equivariant surjection M  M has a Gk -equivariant splitting [which thus determines a Gk ∼× ×M →
equivariant isomorphism ksep
M — i.e., an isomorphism of sheaves∼
×
OX × MX → MX ].

Note that one verifies easily that the issue of whether or not X is split(respectively, quasi-split) does not depend on the choice of ksep .
Proposition 1.3. Let X be an fs log point. Suppose that X is split. Then
X is quasi-split. In particular, a standard log point is quasi-split.
Proof. Since the monoid N has no nontrivial automorphism, the final assertion follows from the first assertion. Let us verify the first assertion.
Since we are in the situation of Definition 1.2, we shall apply the notation
× → M gp →of Definition 1.2. Then we have an exact sequence 1 → ksep
M gp → 1 of Gk -modules. Thus, since M gp is a free module, to verify that
the Gk -equivariant surjection M gp  M gp , hence also M  M , has a Gk × )) =equivariant splitting, it suffices to verify that H 1 (Gk , HomZ (M gp , ksep{0}. On the other hand, since the action of Gk on M is trivial, this follows
from Hilbert’s Theorem 90. This completes the proof of Proposition 1.3. 
Remark 1.3.1. Proposition 1.3 gives us a sufficient condition for an fs log
point to be quasi-split. Now let us observe that Proposition 1.3 essentiallyfollows from Hilbert’s Theorem 90. In §A, we discuss, by considering twisted
versions of Hilbert’s Theorem 90, further such sufficient conditions.
Lemma 1.4. Let X be an object of Schlog (S). Then the following hold:(i) Suppose that X is an fs log point. Since we are in the situation ofDefinition 1.2, we shall apply the notation of Definition 1.2. Let φ : M → N
be a local homomorphism. Then there exists a finite separable extension
K of k which satisfies the following condition: If we write X 0 for the objectof Schlog (S) obtained by equipping the spectrum of K with the log structure
induced by the log structure of X, then there exists a standard log point
Y → X 0 such that the induced homomorphism between the stalks of “M”
coincides with φ.
(ii)◦

Let x ∈ X be a closed point. Then there exists a standard log◦

point f : Y → X such that the image of f coincides with {x}.

Proof. First, we verify assertion (i). We may assume without loss of generality, by replacing k by a suitable finite separable extension of k, that X is
split, hence also quasi-split [cf. Proposition 1.3]. Thus, assertion (i) follows∼
× ×M →
from the Gk -equivariant isomorphism ksep
M of Definition 1.2, (ii).

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

7

Next, we verify assertion (ii). We may assume without loss of generality,
by replacing X by the log scheme obtained by equipping the spectrum of the◦residue field of X at x with the log structure induced by the log structureof X, that X is an fs log point. Thus, by assertion (i), to verify assertion(ii), it suffices to verify existence of a local homomorphism M → N. On theother hand, this follows from Lemma 1.1, (i). This completes the proof ofassertion (ii), hence also of Lemma 1.4.
Definition 1.5. We shall say that an object X of Schlog (S) is minimal if
X is non-initial, and, moreover, every monomorphism in Schlog (S) from a
non-initial object to X is an isomorphism [cf. [Mzk1], Proposition 2.4]. We
shall say that a morphism in Schlog (S) is a minimal log point if the domainof the morphism is minimal.
Lemma 1.6. Let X be an object of Schlog (S). Then it holds that X is
minimal if and only if X is either a trivial log point or a standard log
point.
Proof. This is the content of [Mzk2], Proposition 1.6, (ii). Note that sufficiency also follows immediately from the surjectivity portion of necessityof [Mzk1], Proposition 2.3 [cf. also [Mzk2], Appendix]. Here, we give a
proof of necessity from the point of view of the present paper for the reader’sconvenience as follows:◦◦
Suppose that X is minimal. Since X is noetherian, X has a closed point.
Thus, one verifies immediately, by considering the strict closed immersion
into X determined by this closed point, that X is an fs log point. Now
since we are in the situation of Definition 1.2, we shall apply the notationof Definition 1.2.
Assume that the free module M gp is of rank ≥ 2. Then it follows from
Lemma 1.1, (iv), that there exists a Gk -stable submonoid P ⊆ M gp suchthat M ( P , and, moreover, P is sharp and fs. Thus, we have a Gk -stabledef
submonoid N = (M gp  M gp )−1 (P ) of M gp such that M ( N , and,moreover, the natural homomorphism M gp → N gp is an isomorphism.
Next, let us observe that since P is sharp, by mapping each element of
N r M to 0 ∈ ksep , we obtain a Gk -equivariant extension N → ksep of thehomomorphism M → ksep of monoids [where we regard ksep as a monoid bymultiplication] which defines the log structure of X. Moreover, one verifies
easily that this homomorphism N → ksep of monoids determines an fs log◦

structure on X. Write Y for the resulting [non-initial] fs log scheme. Thensince M ( N , and the natural homomorphism M gp → N gp is an isomorphism, the morphism Y → X [in Schlog (S)] induced by the natural inclusion
M ,→ N is a monomorphism but not an isomorphism. In particular, we8

Y. HOSHI AND C. NAKAYAMA

conclude that X is not minimal, in contradiction to our assumption that X
is minimal. This completes the proof of necessity of Lemma 1.6.
Proposition 1.7. Let X be an object of Schlog (S). Then the following hold:(i) The following two conditions are equivalent:(1) X is a trivial log point.
(2) X is minimal, and, moreover, there exists a minimal log point
f : Y → X such that Y has an endomorphism over X [relative to f ] which
is not an isomorphism.
(ii) The following two conditions are equivalent:(3) X is a standard log point.
(4) X is minimal but not a trivial log point.
Proof. These assertions follow immediately from Lemma 1.6.



2. Characterization of Fs Log Points
In the present §2, we give a categorical characterization of fs log points [cf.
Proposition 2.5 below]. In the present §2, let S be an fs log scheme whose
underlying scheme is locally noetherian and X an object of Schlog (S).
Definition 2.1.
(i) We shall say that a finite set {fi : Yi → X}i∈I consisting of standard
log points
F whose codomains are X is an epimorphic family of X if the morphism i∈I Yi → X from the coproduct of the Yi ’s to X determined by thefi ’s is an epimorphism in Schlog (S).
(ii) We shall say that a collection consisting of standard log points whose
codomains are X is an indispensable collection for X if every epimorphic
family of X has an element which belongs to the collection.
Definition 2.2. Suppose that X is an fs log point. Since we are in the situation of Definition 1.2, we shall apply the notation of Definition 1.2. Write
def
V = M gp ⊗Z Q. Let f : Y → X be a standard log point. Then, by consider◦◦ing a geometric point of Y which lifts x → X, we obtain a [necessarily local]
homomorphism M → N. We shall write L(f ) ⊆ V for the kernel of the
Q-linear homomorphism V → Q induced by this homomorphism M → N.
[Note that one verifies easily that L(f ) does not depend on the choice of the◦geometric point of Y .]

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

9

Lemma 2.3. Suppose that X is an fs log point. Let {fi : Yi → X}i∈I be
a nonempty finiteTset consisting of standard log points whose codomains are
X. Suppose that i L(fi ) = {0}. Then the finite set {fi : Yi → X}i∈I is an
epimorphic family of X.
◦◦◦

Proof. Let x → X be a geometric point of X and, for i ∈ I, y i → Y i a◦◦Tgeometric point of Y i which
lifts
x
→
X.
Then
since
i L(fi ) = {0}, theQ
inducedbythef’shomomorphism MX,x → i∈I MYF
i is injective, which
i ,y ithus implies that the morphism i∈I Yi → X is an epimorphism. This
completes the proof of Lemma 2.3.
Lemma 2.4. Let f : Y → X be an epimorphism in Schlog (S). Then every
◦

◦

◦

◦

closed point of X is contained in the image of f : Y → X.
◦◦

Proof. Suppose that the image of f does not contain a closed point x ∈ X.
Write Z for the object of Schlog (S) obtained by glueing two copies of X
along the open log subscheme X r {x} [via the identity automorphism of
X r {x}]. Then we have two distinct natural open immersions X ,→ Z
whose restrictions to X r {x} coincide, which thus implies that f is not an
epimorphism in Schlog (S). This completes the proof of Lemma 2.4.
Proposition 2.5. The following two conditions are equivalent:(1) X is an fs log point.
(2) Every indispensable collection for X has a finite subset which forms
an epimorphic family of X.
Proof. First, we verify the implication (1) ⇒ (2). Suppose that condition(1) is satisfied. Since we are in the situation of Definition 1.2, we shall apply
def

the notation of Definition 1.2. Write V = M gp ⊗Z Q.
Assume that there exists an indispensable collection A for X such that A
does not have any finite subset which forms an epimorphic family of X. [Notethat since there exists a nonempty epimorphic family of X by Lemma 1.1,(ii), Lemma 1.4, (i), and Lemma 2.3, it holds that A 6= ∅.] Then it follows
def T
immediately from Lemma 2.3 that L = f L(f ) 6= {0} — where f ranges
over the members of A. Thus, one verifies immediately from Lemma 1.1, (ii),that there exists a finite set {gj }j∈J consisting of standard log points whose
codomains are X such that L 6⊆ L(gj ) [cf. condition (2) of Lemma 1.1, (ii)]— which thus
T implies that gj does not belong to A for every j ∈ J — and,moreover, j L(gj ) = {0} [cf. condition (1) of Lemma 1.1, (ii)] — which
thus implies [cf. Lemma 2.3] that this finite set {gj }j∈J is an epimorphic
family. In particular, since A is indispensable, we obtain a contradiction.
This completes the proof of the implication (1) ⇒ (2).

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Y. HOSHI AND C. NAKAYAMA

Next, we verify the implication (2) ⇒ (1). Suppose that condition (2) is◦◦◦satisfied. Let x ∈ X be a closed point of X. [Note that since X is noetherian,◦a closed point of X always exists.] Write A for the collection consisting of
the standard log points whose codomains are X and images coincide with◦{x} ⊆ X. [Note that it follows from Lemma 1.4, (ii), that A 6= ∅.] Then it
follows from Lemma 2.4 that A is indispensable. In particular, since somefinite subset of A forms an epimorphic family of X [cf. condition (2)], again
by Lemma 2.4, we conclude that x is the unique closed point of X, which◦

thus implies that X is isomorphic to the spectrum of a noetherian local ringR.
Let π ∈ R r R× . Write A1X for the fs log scheme over X obtained byequipping Spec(R[t]) — where t is an indeterminate — with the log structure induced by the log structure of X. Then we have two strict closed
immersions f0 , fπ : X ,→ A1X over X determined by the R-linear homomorphisms R[t] → R given by mapping t ∈ R[t] to 0, π ∈ R, respectively.
Now let us observe that one verifies immediately that, for an fs log point◦

f : Y → X, if the image of f is {x}, then it holds that f0 ◦ f = fπ ◦ f . Thus,
since some finite subset of A forms an epimorphic family of X as verified
above, we conclude that f0 = fπ , hence also π = 0, which thus implies that
R is a field. This completes the proof of the implication (2) ⇒ (1), hence
also of Proposition 2.5.

3. Characterization of Strict Morphisms
In the present §3, we prove the main theorem of the present paper [cf.
Theorem 3.7 below]. In the present §3, let S be an fs log scheme whose
underlying scheme is locally noetherian, X and Y objects of Schlog (S), andf : X → Y a morphism in Schlog (S).
Lemma 3.1. Suppose that X is an fs log point, and that f is a monomorphism. Then it holds that f is strict if and only if f is a terminal object
among the fs log points Z → Y which satisfy that X ×Y Z is non-initial.
Proof. This is [Mzk1], Corollary 2.13. However, since the proof contains anerror [cf. [Mzk2], Appendix], we give a proof as follows: Let us first observe
that, to verify Lemma 3.1, we may assume without loss of generality, by
replacing Y by the log scheme obtained by equipping the spectrum of the◦◦residue field of Y at the image of f with the log structure induced by thelog structure of Y , that Y is an fs log point. Now, to verify necessity,
suppose that f is strict. Then since f is a strict monomorphism, one verifies
easily that f is an isomorphism. Thus, necessity is immediate. Next, weCATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

11

verify sufficiency. Since the identity automorphism of Y is an fs log point
which satisfies that X ×Y Y = X is non-initial, by our condition, f has asplitting over Y [i.e., a morphism s : Y → X over Y such that the compositefs

Y → X → Y is the identity automorphism of Y ]. Thus, f is an isomorphism.
This completes the proof of sufficiency, hence also of Lemma 3.1.
Lemma 3.2. It holds that f is strict if and only if, for every commutative
diagram in Schlog (S)Z −−−−→ X



f
y
y
W −−−−→ Y— where the horizontal arrows are fs log points, monomorphisms, andstrict — it holds that the left-hand vertical arrow is strict.

Proof. This is [Mzk1], Corollary 2.14. However, since the proof contains anerror [cf. [Mzk2], Appendix], we give a proof as follows: Necessity may be◦easily verified. Next, we verify sufficiency. Let x be a closed point of X.
◦◦[Note that since X is noetherian, a closed point of X always exists.] Write
Z → X for the strict morphism whose underlying morphism of schemes is◦given by the natural morphism from the spectrum of the residue field of X
at x and W → Y for the strict morphism whose underlying morphism ofschemes is given by the natural morphism from the spectrum of the residue◦◦

field of Y at f (x). Then one verifies easily that Z → X, hence also W → Y ,is a morphism in Schlog (S). Moreover, it holds that Z → X and W → Y aremonomorphisms. Thus, by our condition, the natural morphism Z → W is◦◦strict, which thus implies that f −1 MY → MX is an isomorphism at x ∈ X.
This implies sufficiency.
Lemma 3.3. Suppose that both X and Y are fs log points. Then thefollowing hold:(i) It holds that f is exact if and only if, for every fs log point Z → Y ,the fiber product X ×Y Z is non-initial.
(ii) It holds that f is strict if and only if the following condition issatisfied: f is exact, and, moreover, for every minimal log point Z → Y ,the second projection X ×Y Z → Z is strict.
Proof. Assertion (i) is essentially [Naka], (A.1), Proposition [cf. also theproof of [Naka], (A.1), Proposition]. Next, we verify assertion (ii). Necessity
may be easily verified. We verify sufficiency. Write M for the stalk of MX

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Y. HOSHI AND C. NAKAYAMA◦◦

at a geometric point of X, N for the stalk of MY at the geometric point of Y◦

determined by the geometric point of X, and φ : N → M for the [necessarily
exact — cf. our condition] local homomorphism induced by f .
Let ψ : N → N be a local homomorphism [cf. Lemma 1.1, (i)]. Write P for
the quotient by the torsion elements of the saturation of the pushout [in theφψ

category of monoids] of M ← N → N. Then, by our condition in the endof the statement of assertion (ii) [cf. also Lemma 1.6], together with [Naka],
Proposition (2.1.1), the natural homomorphism N → P is an isomorphism.
Thus, it follows that rankZ (Coker(φgp : N gp → M gp )) = rankZ (Coker(Z =Ngp → P gp )) = 0.
Assume that f is not strict, i.e., that φ is not an isomorphism. Then,since φ is exact, and Coker(φgp ) is of rank 0, it holds that Coker(φgp ) has anontrivial torsion. In particular, there exists a homomorphism π : N gp → Z
which does not factor through φgp . Next, observe that since N is finitely
generated, there exists a positive integer n such that the homomorphism
N gp → Z given by mapping x ∈ N gp to π(x) + n · ψ gp (x) ∈ Z maps N ⊆ N gp
to N ⊆ Z, and the resulting homomorphism ψ0 : N → N is local. Then it
follows from our choice of π [together with the fact that ψ factors throughφ — cf. the above discussion concerning P ] that ψ0 does not factor throughφ. Thus, by means of ψ0 , together with Lemma 1.4, (i), one may construct
a minimal log point Z → Y such that the second projection X ×Y Z → Z
is not strict, in contradiction to our condition. This completes the proof ofassertion (ii), hence also of Lemma 3.3.
Lemma 3.4. Suppose that Y is minimal. Then it holds that f is strict if
and only if every fs log point Z → X factors through a minimal log pointf

W → X such that the composite W → X → Y is strict.

Proof. First, we verify necessity. Suppose that f is strict. Let Z → X◦

be an fs log point. Write W for the log scheme obtained by equipping Z
with the log structure induced by the log structure of X. Thus, we have afactorization Z → W → X, where W → X is strict. Since f is strict andf

Y is minimal, the composite W → X → Y is strict and W is minimal [cf.
Lemma 1.6]. This completes the proof of necessity.
◦

Next, we verify sufficiency. Suppose that f is not strict. Then since X◦

is noetherian, one verifies easily that there exists a closed point x of X such◦◦that the homomorphism f −1 MY → MX is not an isomorphism at x ∈ X.
Thus, to verify sufficiency, we may assume without loss of generality, by
replacing X by the log scheme obtained by equipping [the reduced closed

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

13

subscheme determined by] {x} with the log structure induced by the log
structure of X, that X is an fs log point.
Assume that the identity automorphism X → X [which is in fact an fs
log point] factors through a minimal log point W → X which satisfies thatfthe composite W → X → Y is strict.

f

If Y is a trivial log point, then since the composite W → X → Y is strict,
it follows that W is a trivial log point, which thus implies that X is a trivial
log point. In particular, f is strict, in contradiction to our assumption thatf is not strict.
Thus, it follows from Lemma 1.6 that we may assume without loss ofgenerality that Y is a standard log point. Then since the composite W →f

X → Y is strict, it follows that W is a standard log point. Thus, it follows
immediately, by considering our factorization X → W → X of the identity
automorphism of X, that X is a standard log point, and, moreover, X →W , hence also the composite X → W → X → Y [i.e., f ], is strict, incontradiction to our assumption that f is not strict. This completes theproof of sufficiency, hence also of Lemma 3.4.
Lemma 3.5. Suppose that both X and Y are minimal. Then it holds thatf is strict if and only if the following condition is satisfied: There exists afactorization X → Z → Y of f such that Z is connected and either oflog rank 0 or of log rank 1, X → Z is a monomorphism, and Z → Ys
has a splitting [i.e., a morphism s : Y → Z such that Y → Z → Y is theidentity automorphism].
Proof. This follows from [Mzk2], Proposition 2.4.

Lemma 3.6. It holds that X is of log rank 0 (respectively, 1) if and only ifevery fs log point Z → X factors through a trivial (respectively, standard)log point W → X.
Proof. Necessity follows by considering a suitable strict monomorphism
W → X. Next, we verify sufficiency. One verifies easily that X is oflog rank 0 if every fs log point Z → X factors through a trivial log point
W → X. Thus, suppose that every fs log point Z → X factors through astandard log point W → X. Then it follows immediately from our condition◦

gp

that the module MX,x is of rank 1 for every geometric point x → X whose◦◦

image is closed in X. Write U ⊆ X for the maximal [necessarily open — cf.
the well-known constructibility of MX ] subset on which MX is trivial. If
U 6= ∅, then since U has a closed point [by the fact that U is noetherian],
we obtain a contradiction by our condition. This completes the proof of
sufficiency, hence also of Lemma 3.6.


14

Y. HOSHI AND C. NAKAYAMA

Theorem 3.7. Let S and T be fs log schemes whose underlying schemes
are locally noetherian,∼φ : Schlog (S) −→ Schlog (T )an equivalence of categories, and f a morphism in Schlog (S). Then it
holds that f is strict if and only if φ(f ) is strict.
Proof. Let X be an object of Schlog (S) and g a morphism in Schlog (S). Let
us first observe that it follows from Proposition 1.7, Proposition 2.5 that(1) it holds that X is a trivial (respectively, a standard; an fs) log point
if and only if φ(X) is a trivial (respectively, a standard; an fs) log point.
Moreover, it follows from Lemma 3.1, together with assertion (1), that(2) if g [hence also φ(g) — cf. (1)] is an fs log point and a monomorphism,
then it holds that g is strict if and only if φ(g) is strict.
Thus, it follows from Lemma 3.2 that, to verify Theorem 3.7, it is enoughto verify the following assertion (3):(3) If the domain and codomain of g [hence also of φ(g) — cf. (1)] are fs
log points, then it holds that g is strict if and only if φ(g) is strict.
Next, let us observe that, to verify assertion (3), it follows from
Lemma 3.3, together with assertion (1), that it suffices to verify thefollowing assertion (4):(4) If the codomain of g [hence also of φ(g)] is minimal, then it holds
that g is strict if and only if φ(g) is strict.
Next, to verify assertion (4), it follows from Lemma 3.4, together withassertion (1), that it is sufficient to verify the following assertion (5):(5) If the domain and codomain of g [hence also of φ(g)] are minimal,
then it holds that g is strict if and only if φ(g) is strict.
Thus, to verify Theorem 3.7, it follows from Lemma 3.5 that it suffices toverify the following assertion (6):(6) It holds that X is either of log rank 0 or of log rank 1 if and only if
φ(X) is either of log rank 0 or of log rank 1.
On the other hand, assertion (6) follows from Lemma 3.6, together withassertion (1). This completes the proof of Theorem 3.7.

Remark 3.7.1. The content of Theorem 3.7 is in fact a formal consequenceof a result of S. Mochizuki [i.e., [Mzk2], Theorem A]. Moreover, Mochizuki
also proved that a result concerning a categorical representation of fs log

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

15

schemes [i.e., [Mzk1], Theorem B; [Mzk2], Theorem A] follows from Theorem 3.7, together with some discussions [cf. the portion of [Mzk2] from thediscussion preceding [Mzk2], Proposition 3.7, to the end of [Mzk2], §3 —also [Mzk2], Appendix].
Appendix A. Twisted Versions of Hilbert’s Theorem 90
In §1, we gave a sufficient condition for an fs log point to be quasi-split [cf.
Proposition 1.3, Remark 1.3.1]. In the present §A, we discuss, by considering
twisted versions of Hilbert’s Theorem 90, further such sufficient conditions.
In the present §A, let k be a field and ksep a separable closure of k. Writedef

Gk = Gal(ksep /k). Let M be a sharp fs monoid equipped with a continuous
action of Gk [with respect to the discrete topology on M ].
Proposition A.1. Let X be an fs log point. Then the following three
conditions are equivalent:(1) X is quasi-split.
(2) There exist an fs log point Y and a morphism f : Y → X such that◦

Y is quasi-split, and, moreover, f is an isomorphism.

(3) There exist a standard log point Y and a morphism f : Y → X◦

such that f is an isomorphism.

Proof. The implication (3) ⇒ (2) follows from Proposition 1.3. The implication (1) ⇒ (3) follows from Lemma 1.1, (iii).
Finally, to verify the implication (2) ⇒ (1), suppose that condition (2) is◦◦◦satisfied. By means of the isomorphism f , let us identify X with Y . Since Y
is quasi-split, it follows from Definition 1.2, (ii), that the natural inclusion
×OY× ,→ MY has a splitting MY  OY× (= OX). Thus, by considering
the composite of the homomorphism MX → MY induced by f and the××above splitting MY  OX
, we obtain a splitting MX  OX
of the natural×
inclusion OX ,→ MX . In particular, X is quasi-split, i.e., condition (1) issatisfied. This completes the proof of the implication (2) ⇒ (1), hence alsoof Proposition A.1.

Definition A.2. We shall say that the pair (k, M ) is quasi-split if the following condition is satisfied: For every fs log scheme X whose underlying
scheme is the spectrum of k, if there exists a Gk -equivariant isomorphism of◦

M with the stalk of MX at the geometric point x → X determined by theseparable closure ksep , then X is quasi-split.

16

Y. HOSHI AND C. NAKAYAMA

Proposition A.3. It holds that the pair (k, M ) is quasi-split if and× )) = {0} [where the action of G
only if H 1 (Gk , HomZ (M gp , ksep
k on
def

× ) is given by g · φ = g ◦ φ ◦ g −1 ].
HomZ (M gp , ksep

Proof. One can prove sufficiency in the same way as the proof of Proposition 1.3. Next, we verify necessity. Suppose that (k, M ) is quasi-split. Let× → E → M gp → 1 be an exact sequence of G -modules corre1 → ksep
k
1
gp
×
1
× )).
sponding to an element of ExtGk (M , ksep ) = H (Gk , HomZ (M gp , ksep
defWrite N = (E  M gp )−1 (M ) ⊆ E. [Thus, N is isomorphic, as an ab× × M .] Then since M is sharp [which thus implies thatstract monoid, to ksep
× ], by mapping each element of N r k × to 0 ∈ k
N × = ksep
sep , we obtainsep
a Gk -equivariant homomorphism N → ksep of monoids [where we regard
ksep as a monoid by multiplication] which is an extension of the natural in× ,→ kclusion ksep
sep . Moreover, one verifies easily that this homomorphism
N → ksep of monoids determines an fs log structure on Spec(k). On theother hand, since (k, M ) is quasi-split, the resulting fs log scheme is quasisplit, which thus implies that the above exact sequence of Gk -modules has a
Gk -equivariant splitting. This completes the proof of Proposition A.3. defLemma A.4. Let K be a finite Galois extension of k. Write G =Gal(K/k). Let H ⊆ N ⊆ G be subgroups such that N is normal in G.
Let us define an action of G on the module Map(G/N, (K H )× ) [consistingdefof maps of sets G/N → (K H )× ] by g · φ = φ ◦ g −1 ; moreover, let us also
define an action of G on the module Map(G/H, K × ) [consisting of maps ofdef

sets G/H → K × ] by g · φ = g ◦ φ ◦ g −1 . Then the homomorphism
Map(G/N, (K H )× ) −→ Map(G/H, K × )φ
7→ (gH 7→ gφ(gN ))determines a G-equivariant isomorphism∼

Map(G/N, (K H )× ) −→ Map(G/H, K × )Nof G-modules. In particular, the G/N -module Map(G/H, K × )N is a coinduced module.
Proof. This follows from a straightforward computation.



Theorem A.5. If one of the following three conditions is satisfied, then thepair (k, M ) is quasi-split.
(1) The action of Gk on M is trivial.
(2) The Brauer group of every finite separable extension of k is zero.

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

17∼(3) There exists a [not necessarily Gk -equivariant] isomorphism M →⊕n
N of monoids for some positive integer n.
Proof. Theorem A.5 in the case where condition (1) is satisfied follows formally from Proposition 1.3. Theorem A.5 in the case where condition (2) issatisfied follows from Proposition A.3 and [Serre], Chapter X, Proposition
11, as well as [Serre], Chapter X, Corollary to Proposition 11.
Finally, we verify Theorem A.5 in the case where condition (3) is satisfied. Suppose that condition (3) is satisfied. Let us first observe that one
verifies easily that each automorphism of the monoid N⊕n arises from somepermutation of the n factors. Thus, it follows from Proposition A.3 that, tocomplete the verification of Theorem A.5, it suffices to verify that(†): for a finite set S and a finite Galois extension K ofdef

k whose Galois group G = Gal(K/k) acts on S, it holds
that H 1 (G, Map(S, K × )) = {0} [where the action of G on
def

Map(S, K × ) is given by g · φ = g ◦ φ ◦ g −1 ].

Next, let us observe that we may assume without loss of generality, by
replacing G by a p-Sylow subgroup of G [where p is a prime number], that
G in (†) is a [nontrivial] p-group. Next, let us observe that we may assume
without loss of generality, by replacing S by the G-orbit of an element ofS, that S in (†) is the G-set G/H for a subgroup H ⊆ G. Observe that if
H = G, then (†) follows from Hilbert’s Theorem 90; thus, we may assume
without loss of generality that H 6= G.
Let N ⊆ G be a normal subgroup such that H ⊆ N , and, moreover,[G : N ] = p. [Note that one may verify existence of such a normal subgroup
N as follows: Since G is a nontrivial p-group, there exists a subgroup Z ⊆ Gsuch that Z is of order p and contained in the center of G. If H · Z = G,
which thus implies that H × Z = G, then one may take H itself as “N ”.
If H · Z 6= G, then, by induction on the cardinality of “G”, one obtains anormal subgroup of G/Z of index p which contains the image of H. Thus,by considering the inverse image in G of this normal subgroup of G/Z, one
obtains a subgroup of the desired type.] Thus, by induction on the cardinalityof G, to verify (†), it suffices to verify that
H 1 (G/N, Map(G/H, K × )N ) = {0}.
On the other hand, this follows from Lemma A.4. This completes the proofof Theorem A.5.

Remark A.5.1. Examples of “k” which satisfies condition (2) in the statement of Theorem A.5 are given in the discussion entitled “Examples of

18

Y. HOSHI AND C. NAKAYAMA

Fields with Zero Brauer Group” in [Serre], p.162. For instance, a quasialgebraically closed field [i.e., a field which has property C1 ] — e.g., a finite
field — satisfies condition (2) in the statement of Theorem A.5.
Remark A.5.2. An example of an fs log point which is not quasi-split isgiven as follows: Write M for the monoid obtained by taking the quotientof N⊕3 by the relation (a, a, 0) ∼ (0, 0, 2a), where a ∈ N. Let us define andefaction of the Galois group Gal(C/R) on C× × M by σ(z, [a, b, c]) = ((−1)c ·σ(z), [b, a, c]) — where we write σ ∈ Gal(C/R) for the unique nontrivial
element and “[−]” (∈ M ) for the image of “(−)” (∈ N⊕3 ) in M — as well
as a Gal(C/R)-equivariant homomorphism C× × M → C of monoids by(z, [a, b, c]) 7→ z if (a, b, c) = (0, 0, 0) (respectively, 7→ 0 if (a, b, c) 6= (0, 0, 0)).
Then one verifies immediately that this Gal(C/R)-equivariant homomorphism C× × M → C determines an fs log structure on Spec(R). Next, let us
verify that the resulting fs log point is not quasi-split, i.e., that there is no
Gal(C/R)-equivariant splitting M gp → C× × M gp of the natural surjection
C× ×M gp  M gp . To this end, assume that there is a Gal(C/R)-equivariant
splitting φ : M gp → C× × M gp . Write s, t, u ∈ M gp for the respective imdef
def
ages of [1, 0, 0], [0, 1, 0], [0, 0, 1] in M gp , (zs , s) = φ(s), (zt , t) = φ(t), anddef(zu , u) = φ(u). Then since φ is Gal(C/R)-equivariant, it holds that
• (σ(zs ), t) = σ(φ(s)) = φ(σ(s)) = (zt , t), i.e., σ(zs ) = zt , and that

• (−σ(zu ), u) = σ(φ(u)) = φ(σ(u)) = (zu , u), i.e., −σ(zu ) = zu .

Moreover, since φ is a homomorphism, it holds that• s + t = 2u implies zs · zt = zu2 .

Thus, we conclude that|zs |2 = zs · σ(zs ) = zs · zt = zu2 = zu · (−σ(zu )) = −|zu |2 ,which thus implies that zs = 0, in contradiction to our assumption thatzs ∈ C× . This completes the proof of the fact that our fs log point is notquasi-split.

Acknowledgement
The first author was supported by JSPS KAKENHI Grant Number
15K04780. The second author was supported by JSPS KAKENHI Grant
Numbers 22540011, 23340008, 16K05093. The authors would like to thankShinichi Mochizuki and the referee for helpful comments. The second
author thanks J. C. for suggesting this paper.

CATEGORICAL CHARACTERIZATION OF STRICT MORPHISMS

19

References[Kato] K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry,and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press,
Baltimore, MD, 1989.
[Mzk1] S. Mochizuki, Categorical representation of locally noetherian log schemes, Adv.
Math. 188 (2004), no. 1, 222–246.
[Mzk2] S. Mochizuki, Monomorphisms in categories of log schemes, Kodai Math. J. 38(2015), no. 2, 365–429.
[Naka] C. Nakayama, Logarithmic étale cohomology, Math. Ann. 308 (1997), no. 3, 365–
404.
[Serre] J.-P. Serre, Local fields, Translated from the French by Marvin Jay Greenberg.
Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Heidelberg-Berlin,
1979.
Yuichiro Hoshi
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japane-mail address: yuichiro@kurims.kyoto-u.ac.jp
Chikara Nakayama
Department of Economics
Hitotsubashi University
Tokyo, 186-8601 Japane-mail address: c.nakayama@r.hit-u.ac.jp(Received February 18, 2016 )(Accepted May 13, 2016 )