1. Introduction
Zink proved that a p-divisible group over a regular
$\mathbb {F}_p$
-scheme with constant Newton polygon is isogenous to a p-divisible group admitting a slope filtration [Reference Zink21], and Oort and Zink generalized this result to the case of a normal base [Reference Oort and Zink17]. First, we briefly recall the result of Oort–Zink. Let S be a locally Noetherian normal
$\mathbb {F}_p$
-scheme. Let G be a p-divisible group over S with constant Newton polygon. Then there exists a completely slope divisible p-divisible group H and an isogeny
$G\to H$
. Here, a p-divisible group H over S is called completely slope divisible if H admits a filtration of p-divisible subgroups
$$\begin{align*}0=H_0\subset H_1 \subset \dots \subset H_m=H, \end{align*}$$
and if there are integers
$$\begin{align*}s\geq r_1> r_2 >\dots >r_m\geq 0 \end{align*}$$
such that the following conditions are satisfied:
-
1. The quasi-isogeny
$p^{-r_i}F^s\colon H_i\to H_{i}^{(p^s)}$
is an isogeny for
$i=1, \dots ,m$
. -
2. The p-divisible group
$H_{i}/H_{i-1}$
is isoclinic of slope
$r_i/s$
for
$i=1, \dots ,m$
.
The above theorem plays an important role in the theory of foliations in the moduli space of abelian varieties in characteristic p (see [Reference Oort16]).
In this article, we shall prove a log version of this theorem. Log p-divisible groups are degenerating objects of p-divisible groups in the framework of logarithmic geometry. In [Reference Kajiwara, Kato and Nakayama5], [Reference Kajiwara, Kato and Nakayama7], Kajiwara–Kato–Nakayama introduced the notion of log abelian varieties, which are degenerating objects of abelian varieties in the framework of logarithmic geometry, and realized the toroidal compactification of the moduli space of abelian varieties constructed by Faltings–Chai as the fine moduli of log abelian varieties. In the same way as the non-log case, the p-torsion part of a log abelian variety is a log p-divisible group. We expect that the study in this article is helpful to study the structure of the toroidal compactification of the moduli space of abelian varieties in characteristic p.
The following is one of the main results of this article.
Theorem 1.1 (See Corollary 3.7)
Let S be a normal fs log scheme over
$\mathbb {F}_{p}$
. Then, for a log p-divisible group G over S with constant Newton polygon, there exist a completely slope divisible log p-divisible group H and an isogeny
$G\to H$
.
This result can be deduced from the corresponding result of Oort–Zink by using the fact that a completely slope divisible log p-divisible group is written as an extension of a classical étale p-divisible group by a classical completely slope divisible formal Lie group (see Proposition 3.4).
Next, we consider a stronger statement. Oort–Zink proved the above result by extending an isogeny generically defined. We prove that we can make the same procedure for log p-divisible groups.
Theorem 1.2 (See Theorem 5.4)
Let S be a log regular fs log scheme over
$\mathbb {F}_p$
and U be a dense open subscheme of S. Let G be a log p-divisible group over S. Suppose that we are given a completely slope divisible log p-divisible group
$H_{U}$
over U and an isogeny
$G|_{U}\to H_{U}$
. Then there exist a completely slope divisible log p-divisible group H over S and an isogeny
$G\to H$
which restricts to the given isogeny
$G|_{U}\to H_{U}$
.
When
$\mathring {S}$
is regular, this theorem follows from a result on the extendability of complete slope divisible log p-divisible groups (see Proposition 5.3). In general cases, we reduce the problem to the above case by using resolutions of toric singularities in [Reference Nizioł12].
Note that we need the assumption of log regularity. Theorem 1.2 does not hold under the assumption of Theorem 1.1 (for a counterexample, see Remark 4.6).
The outline of this article is as follows. In Section 2, we review the definitions of Kummer log flat topology, log vector bundles, log finite group schemes, and log p-divisible groups, and we prove some fundamental properties of them. In Section 3, we introduce the notion of completely slope divisible log p-divisible groups and Newton polygons for log p-divisible groups. In Section 4, we review the definition and properties of log regular fs log scheme, and we study log finite group schemes and log vector bundles on log regular fs log schemes for the proof of Theorem 5.4. In Section 5, we shall prove Theorem 5.4.
Notation and convention:
-
1. Throughout this article, p is a prime number.
-
2. All monoids are commutative. For a monoid M, let
, and let
$M^{\text {gp}}$
be an abelian group associated with M. We say that M is sharp if
$M^{\times }=1$
. -
3. For an integer
$n\geq 1$
, a monoid M equipped with a monoid map
$\times n\colon M\to M$
is denoted by
$M^{1/n}$
. -
4. For a log scheme S, the structure sheaf of monoids is denoted by
$\mathcal {M}_{S}$
. The underlying scheme of S is denoted by
$\mathring {S}$
. -
5. Let
$\mathcal {P}$
be a property of schemes, including quasi-compact, quasi-separated, finite, and locally Noetherian. We say that a log scheme S satisfies
$\mathcal {P}$
if the underlying scheme
$\mathring {S}$
satisfies
$\mathcal {P}$
. -
6. Let
$\mathcal {P}$
be a property of morphisms of schemes. We say that a morphism of log schemes
$T\to S$
satisfies
$\mathcal {P}$
if the morphism of the underlying schemes
$\mathring {T}\to \mathring {S}$
satisfies
$\mathcal {P}$
. -
7. For a monoid M, the log scheme
$\mathrm {Spec}\mathbb {Z}[M]$
equipped with the log structure associated with a natural monoid map
$M\to \mathcal {O}_{\mathrm {Spec}\mathbb {Z}[M]}$
is denoted by
$\mathbb {A}_{M}$
. -
8. A chart
$P\to \mathcal {M}_{S}$
is called neat at
$s\in S$
if
$P\to \overline {\mathcal {M}_{S, \bar {s}}}$
is an isomorphism. For an fs log scheme S and a point
$s\in S$
, there exists an fs chart
$P\to \mathcal {M}_{S}$
which is neat at s by [Reference Ogus14, Chapter II, Proposition 2.3.7]. -
9. Unless otherwise specified, fiber products of saturated log schemes are taken in the category of saturated log schemes. For morphisms of saturated log schemes
$S_{1}\to S_{3}$
and
$S_{2}\to S_{3}$
, the saturated fiber product of
$S_{1}$
and
$S_{2}$
over
$S_{3}$
is simply denoted by
$S_{1}\times _{S_{3}} S_{2}$
.
, and let 
2. Preliminaries on log schemes and log p-divisible groups
Throughout this section, let S denote an fs log scheme.
2.1. Kummer log flat topologies
Let
$(\mathrm {fs}/S)$
denote the category of fs log schemes over S. Note that the category of schemes over
$\mathring {S}$
is regarded as a full subcategory of
$(\mathrm {fs}/S)$
by equipping schemes over
$\mathring {S}$
with the pullback log structure of
$\mathcal {M}_{S}$
.
In this article, we use the Kummer log flat topology introduced by Kato (see [Reference Kato10, Section 2]). Recall that, for
$U\in (\mathrm {fs}/S)$
, a family of morphisms
$\{f_{i}\colon U_{i}\to U\}$
is called a Kummer log flat cover if the following conditions are satisfied:
-
1. Each
$f_i$
is log flat, of Kummer type, and locally of finite presentation. -
2. The family is set-theoretically surjective, that is,
$\mathring {U}=\cup f_{i}(\mathring {U_{i}})$
.
The Kummer log flat topology on
$(\mathrm {fs}/S)$
is the Grothendieck topology given by Kummer log flat covers on
$(\mathrm {fs}/S)$
. The resulting site is denoted by
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
.
Proposition 2.1. For an fs log scheme T over S, the representable presheaf by T on
$(\mathrm {fs}/S)$
is a sheaf.
Proof. See [Reference Kato10, Theorem 3.1].
The following lemma is a useful property of Kummer morphisms.
Lemma 2.2. Let
$f\colon T\to S$
be a Kummer morphism of fs log schemes. Suppose that
$\mathring {T}$
is quasi-compact, and that we are given an fs chart
$P\to \mathcal {M}_{S}$
. Then there exists an integer
$n\geq 1$
such that, if we put
$S'=S\times _{\mathbb {A}_{P}} \mathbb {A}_{P^{1/n}}$
, the natural morphism
$T\times _{S} S'\to S'$
is strict.
Proof. See the proof of [Reference Kato10, Proposition 2.7(2)].
2.2. Log vector bundles
Let
$\mathcal {O}_{S_{\mathrm {kfl}}}$
be a sheaf of rings on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
given by
$T\mapsto \Gamma (\mathring {T}, \mathcal {O}_{\mathring {T}})$
. The fact that
$\mathcal {O}_{S_{\mathrm {kfl}}}$
is a sheaf on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
is proved in [Reference Kato10, Section 3.4]. For a quasi-coherent sheaf
$\mathcal {F}$
on
$\mathring {S}$
, we consider the
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-module
$\alpha \mathcal {F}$
given by
$T\mapsto \Gamma (\mathring {T},f^{*}\mathcal {F})$
, where
$f\colon \mathring {T}\to \mathring {S}$
is the structure morphism (cf. [Reference Nizioł13, Proposition 2.19]). This gives a fully faithful functor
$\alpha $
from the category of quasi-coherent
$\mathcal {O}_{\mathring {S}}$
-modules to the category of
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-modules. The functor
$\alpha $
is right exact and preserves exact sequences of locally free
$\mathcal {O}_{X}$
-modules.
Definition 2.3 (Log vector bundles)
For a scheme X, we let
$\mathcal {LF}(X)$
denote the category of vector bundles on X (i.e., locally free
$\mathcal {O}_{X}$
-modules of finite rank).
A log vector bundle on S is a locally free sheaf of finite rank on the ringed site
$((\mathrm {fs}/S)_{\mathrm {kfl}},\mathcal {O}_{S_{\mathrm {kfl}}})$
. In other words, an
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-module
$\mathcal {E}$
is called a log vector bundle if there exists a Kummer log flat cover
$\{U_i\to S\}$
such that each restriction
$\mathcal {E}|_{U_i}$
is isomorphic to
$\alpha \mathcal {F}_{i}$
for a vector bundle
$\mathcal {F}_{i}$
on
$\mathring {U_{i}}$
. A log vector bundle
$\mathcal {E}$
is called classical if
$\mathcal {E}$
is isomorphic to
$\alpha \mathcal {F}$
for a vector bundle
$\mathcal {F}$
on
$\mathring {S}$
. The category of log vector bundles on S is denoted by
$\mathcal {LLF}(S)$
. The functor
$\alpha $
induces a fully faithful functor
$$\begin{align*}\mathcal{LF}(\mathring{S})\to \mathcal{LLF}(S), \end{align*}$$
which is also denoted by
$\alpha $
. Obviously, the functor
$\alpha $
gives an equivalence from the category
$\mathcal {LF}(\mathring {S})$
to the category of classical log vector bundles on S, and we identify these categories.
Lemma 2.4. Let S be a quasi-compact fs log scheme with an fs chart
$P\to \mathcal {M}_{S}$
. Let
$\mathcal {E}$
be a log vector bundle on S. Then there exists an integer
$n\geq 1$
such that, if we set 
, the pullback of
$\mathcal {E}$
by
$S'\to S$
is classical.
Proof. Take a Kummer log flat cover
$T\to S$
such that
$\mathcal {E}|_{T}$
is classical. We may assume that
$\mathring {T}$
is quasi-compact. By Lemma 2.2, there exists an integer
$n\geq 1$
such that, if we set
the natural morphism
$T'\to S'$
is a strict fppf cover. As
$\mathcal {E}|_{T}$
is classical,
$\mathcal {E}|_{T'}$
is also classical. By fppf descent,
$\mathcal {E}|_{S'}$
is also classical.
Proposition 2.5. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of fs log schemes satisfying the condition
$(\ast )$
at the beginning of Appendix A. We put 
.
-
1. Suppose that the cofiltered category I has a final object
$0$
. Let
$\mathcal {E}_{0}$
and
$\mathcal {F}_{0}$
be objects of
$\mathcal {LLF}(S_{0})$
. We put , , , and . Then the natural map is an isomorphism.
$$\begin{align*}\displaystyle \varinjlim_{i\in I}\mathrm{Hom}(\mathcal{E}_{i}, \mathcal{F}_{i})\to \mathrm{Hom}(\mathcal{E},\mathcal{F}) \end{align*}$$
-
2. For any
$\mathcal {E}\in \mathcal {LLF}(S)$
, there exist
$i\in I$
and
$\mathcal {E}_{i}\in \mathcal {LLF}(S_{i})$
with an isomorphism
$\mathcal {E}_{i}|_{S}\cong \mathcal {E}$
.
, 
, 
, and 
. Then the natural map 
Proof. (1) By working Kummer log flat locally on
$S_{0}$
, this statement follows from the limit argument for usual vector bundles on schemes.
(2) Take a qcqs Kummer log flat cover
$T\to S$
such that
$\mathcal {E}|_{T}$
is classical. By Propositions A.3 (2) and A.5 (4),
$T\to S$
comes from a Kummer log flat cover
$T_{i}\to S_{i}$
and
$\mathcal {E}|_{T}$
comes from a vector bundle
$\mathcal {E}_{T_{i}}$
over
$\mathring {T_{i}}$
for some
$i\in I$
. By replacing i with a bigger object of I, we may assume that the descent datum of
$\mathcal {E}|_{T}$
over
$T\times _{S} T$
comes from the descent datum of
$\mathcal {E}_{T_{i}}$
over
$T_{i}\times _{S_{i}} T_{i}$
, which defines an object
$\mathcal {E}_{i}$
of
$\mathcal {LLF}(S_{i})$
with an isomorphism
$\mathcal {E}_{i}|_{S}\cong \mathcal {E}$
.
Proposition 2.6 (cf. [Reference Kato10, Theorem 6.2])
Let
$\mathcal {E}\in \mathcal {LLF}(S)$
. Suppose that the pullback of
$\mathcal {E}$
to a point
$s\in S$
is classical. Then there exists an open neighborhood
$U\subset S$
of s such that
$\mathcal {E}|_{U}$
is classical. In particular, if the pullback of
$\mathcal {E}$
to every point
$s\in S$
is classical, then
$\mathcal {E}$
is classical.
Proof. By the limit argument (Proposition 2.5), we may assume that
$\mathring {S}$
is the spectrum of a Noetherian strict local ring. In this case, the assertion follows from [Reference Kato10, Theorem 6.2].
2.3. Log finite group schemes
In this section, we review the definition of log finite group schemes introduced by Kato in [Reference Kato11] and prove some fundamental properties. For a scheme X, the category of finite locally free group schemes over X is denoted by
$(\mathrm {fin}/X)$
.
Definition 2.7 [Reference Kato11, Section 1]
(1) Let
$(\mathrm {fin}/S)_{\text {c}} $
be the category of sheaves of abelian groups on
$(\mathrm {fs}/S)_{\mathrm {kfl}} $
represented by finite locally free group schemes over
$\mathring {S}$
which are considered as strict log schemes over S. Obviously, the category
$(\mathrm {fin}/S)_{\text {c}}$
is naturally equivalent to the category
$(\mathrm {fin}/\mathring {S})$
, and we identify these categories.
(2) Let
$(\mathrm {fin}/S)_{\text {f}} $
be the category of sheaves G of abelian groups on
$(\mathrm {fs}/S)_{\mathrm {kfl}} $
such that there exists a Kummer log flat cover
$\{U_{i}\to S \} $
with
$G|_{U_{i}}\in (\mathrm {fin}/U_{i})_{\text {c}} $
. For
$G\in (\mathrm {fin}/S)_{\text {f}}$
, the sheaf 
on
$(\mathrm {fs}/S)_{\mathrm {kfl}} $
is called the Cartier dual of G. When G belongs to
$(\mathrm {fin}/S)_{\text {c}}$
, the sheaf
$G^{*}$
is representable by the Cartier dual of G in the usual sense. By working Kummer log flat locally on S, we see that
$G^{*}$
belongs to
$(\mathrm {fin}/S)_{\text {f}}$
for
$G\in (\mathrm {fin}/S)_{\text {f}}$
.
(3) Let
$(\mathrm {fin}/S)_{\text {d}} $
be the full subcategory of
$(\mathrm {fin}/S)_{\text {f}} $
consisting of objects G such that both G and
$G^{*}$
are representable by finite Kummer log flat fs log schemes of finite presentation over S. Objects in
$(\mathrm {fin}/S)_{\text {d}} $
are called log finite group schemes over S.
We have the inclusion relation of categories
$$\begin{align*}(\mathrm{fin}/S)_{\text{c}}\subset (\mathrm{fin}/S)_{\text{d}}\subset (\mathrm{fin}/S)_{\text{f}}. \end{align*}$$
When an object
$G\in (\mathrm {fin}/S)_{\text {f}}$
belongs to
$(\mathrm {fin}/S)_{\text {c}}$
, we say that G is classical.
Definition 2.8. For
$G\in (\mathrm {fin}/S)_{\text {f}}$
, we say that G is of order n if there exists a Kummer log flat cover
$\{U_{i}\to S\} $
such that, for each i,
$G|_{U_{i}}$
is a finite locally free group scheme of order n over
$\mathring {U_{i}}$
.
Example 2.9. If the log structure of S is trivial, the category
$(\mathrm {fin}/S)_{\text {f}} $
is equal to
$(\mathrm {fin}/S)_{\text {c}} $
. This follows from the fact that, for any fs log scheme T with a Kummer morphism
$T\to S$
, the log structure of T is also trivial.
Example 2.10. Let
$n\geq 1$
be an integer. Every log abelian variety A over S (for its definition, see [Reference Kajiwara, Kato and Nakayama5, Definition 4.1]) is a sheaf on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
by [Reference Zhao20, Theorem 2.1(1)]. The sheaf
$A[n]:=\mathrm {Ker}(\times n\colon A\to A)$
is a log finite group scheme of order
$n^{2\dim (A)} $
over S by [Reference Kajiwara, Kato and Nakayama6, Proposition 18.1(2)], Proposition 2.22 below, and [Reference Zhao20, Proposition 3.4(3)].
Example 2.11. Let S be an fs log scheme. Let
$\mathbb {G}_{m, \log }$
denote a sheaf on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
given by
$\mathbb {G}_{m, \log }(T)=\Gamma (\mathring {T}, \mathcal {M}_{T}^{\text {gp}})$
for
$T\in (\mathrm {fs}/S)$
. The fact that
$\mathbb {G}_{m, \log }$
is a sheaf is proved in [Reference Kato10, Theorem 3.2]. For an integer
$n\geq 1$
, there is a Kummer sequence [Reference Kato10, Proposition 4.2]
$$\begin{align*}0 \to \mu_{n} \to \mathbb{G}_{m, \log} \stackrel{(-)^{n}}{\to} \mathbb{G}_{m, \log} \to 0. \end{align*}$$
From this sequence, we get a connecting homomorphism
$$\begin{align*}\mathbb{G}_{m, \log}(S)=\Gamma(\mathring{S}, \mathcal{M}_{S}^{\text{gp}})\to H^{1}((\mathrm{fs}/S)_{\mathrm{kfl}}, \mu_{n})=\mathrm{Ext}^{1}_{(\mathrm{fs}/S)_{\mathrm{kfl}}}(\mathbb{Z}/n\mathbb{Z}, \mu_{n}) \end{align*}$$
which associates to any
$q\in \Gamma (S, \mathcal {M}_{S}^{\text {gp}})$
an extension of sheaves on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
$$\begin{align*}0\to \mu_{n}\to G_{q}\to \mathbb{Z}/n\mathbb{Z}\to 0. \end{align*}$$
The sheaf
$G_{q}$
is a log finite group scheme over S by [Reference Kato11, Proposition 2.3].
Lemma 2.12. The category
$(\mathrm {fin}/S)_{\mathrm {f}}$
satisfies Kummer log flat descent with respect to S.
Proof. For a site
$\mathscr {C}$
, write
$\mathrm {Shv}(\mathscr {C})$
for the category of abelian groups on
$\mathscr {C}$
. Obviously, the category
$\mathrm {Shv}((\mathrm {fs}/S)_{\mathrm {kfl}})$
satisfies Kummer log flat descent with respect to S. For
$G\in \mathrm {Shv}((\mathrm {fs}/S)_{\mathrm {kfl}})$
, the property that G belongs to
$(\mathrm {fin}/S)_{\mathrm {f}}$
is Kummer log flat locally on S. Therefore, the category
$(\mathrm {fin}/S)_{\mathrm {f}}$
also satisfies Kummer log flat descent with respect to S.
Lemma 2.13. Let S be a quasi-compact fs log scheme with an fs chart
$P\to \mathcal {M}_{S}$
. Let
$G\in (\mathrm {fin}/S)_{\text {f}}$
. Then there exists an integer
$n\geq 1$
such that, if we set 
, the pullback of G by
$S'\to S$
is classical.
Proof. This can be proved in the same way as Lemma 2.4.
Proposition 2.14. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of fs log schemes satisfying the condition
$(\ast )$
at the beginning of Appendix A. We put 
.
(1) Suppose that the cofiltered category I has a final object
$0$
. Let
$G_{0}$
and
$H_{0}$
be objects of
$(\mathrm {fin}/S_{0})_{\text {f}}$
. We put
$G_{i}:=G_{0}|_{S_{i}}$
,
$H_{i}:=H_{0}|_{S_{i}}$
,
$G:=G_{0}|_{S}$
, and
$H:=H_{0}|_{S}$
. Then the natural map
$$\begin{align*}\displaystyle \varinjlim_{i\in I}\mathrm{Hom}(G_{i}, H_{i})\to \mathrm{Hom}(G, H) \end{align*}$$
is an isomorphism.
(2) For any
$G\in (\mathrm {fin}/S)_{\text {f}}$
, there exist
$i\in I$
and
$G_{i}\in (\mathrm {fin}/S_{i})_{\text {f}}$
with an isomorphism
$G_{i}|_{S}\cong G$
.
(3) For any
$G\in (\mathrm {fin}/S)_{\text {d}}$
, there exist
$i\in I$
and
$G_{i}\in (\mathrm {fin}/S_{i})_{\text {d}}$
with an isomorphism
$G_{i}|_{S}\cong G$
.
Proof. (1) By working Kummer log flat locally on
$S_{0}$
, this statement follows from the limit argument for usual finite locally free group schemes over schemes.
(2) This follows from the same argument as Proposition 2.5 (2).
(3) This follows from (1) and the limit argument for finite Kummer log flat fs log schemes of finite presentation (Propositions A.3 and A.5 (4)).
Remark 2.15. The theory of log finite group schemes is introduced and developed in [Reference Kato11] under the assumption that the base log scheme is Noetherian. Since an fs affine Noetherian log scheme can be written as a limit of a cofiltered system of fs affine log schemes satisfying the condition (
$\ast $
) at the beginning of Appendix A by Lemma A.2, Proposition 2.14 enables us to generalize several results in [Reference Kato11] to the non-Noetherian case.
Let
$G\in (\mathrm {fin}/S)_{\text {f}}$
. We let
$\mathcal {A}_{G}$
denote the sheaf of morphisms
$\mathcal {M}or(G,\mathcal {O}_{S_{\mathrm {kfl}}})$
on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
. The
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-algebra structure on
$\mathcal {O}_{S_{\mathrm {kfl}}}$
makes
$\mathcal {A}_{G}$
an
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-algebra, and the abelian group structure on G defines a commutative Hopf
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-algebra structure on
$\mathcal {A}_{G}$
. If
$G\in (\mathrm {fin}/S)_{\text {c}}$
,
$\mathcal {A}_{G}$
coincides with the usual coordinate ring of G. Hence, by working Kummer log flat locally on S, we see that the underlying
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-module of
$\mathcal {A}_{G}$
belongs to
$\mathcal {LLF}(S)$
. Therefore,
$\mathcal {A}_{G}$
is a commutative Hopf object of the unitary symmetric monoidal category
$\mathcal {LLF}(S)$
.
Conversely, for a commutative Hopf object
$\mathcal {A}$
of
$\mathcal {LLF}(S)$
, we let
$G_{\mathcal {A}}$
denote the sheaf of homomorphisms of
$\mathcal {O}_{S_{\mathrm {kfl}}}$
-algebras
$\mathcal {H}om(\mathcal {A},\mathcal {O}_{S_{\mathrm {kfl}}})$
on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
. The commutative Hopf structure on
$\mathcal {A}$
defines an abelian group structure
$G_{\mathcal {A}}$
. By working Kummer log flat locally on S, we see that
$G_{\mathcal {A}}$
is an object of
$(\mathrm {fin}/S)_{\mathrm {f}}$
.
Proposition 2.16 (cf. [Reference Kato11, Proposition 2.15])
The assignments
$G\mapsto \mathcal {A}_{G}$
and
$\mathcal {A}\mapsto G_{\mathcal {A}}$
give a contravariant equivalence between the category
$(\mathrm {fin}/S)_{\text {f}}$
and the category of commutative Hopf objects of the unitary symmetric monoidal category
$\mathcal {LLF}(S)$
. Moreover, an object
$G\in (\mathrm {fin}/S)_{\text {f}}$
is classical if and only if
$\mathcal {A}_{G}$
is classical.
Proof. The former assertion follows from the definition, and the latter assertion follows the former one.
Proposition 2.17. Let
$H\subset G$
be objects of
$(\mathrm {fin}/S)_{\text {f}}$
. Suppose that G is classical. Then H and
$G/H$
are also classical.
Proof. By working Kummer log flat locally on S, we see that the inclusion morphism
$H\hookrightarrow G$
induces a surjection
$\mathcal {A}_{G}\twoheadrightarrow \mathcal {A}_{H}$
. Since
$\mathcal {A}_{G}$
is classical,
$\mathcal {A}_{H}$
is also classical by [Reference Nizioł13, Proposition 3.29]. Hence, Proposition 2.16 implies that H is classical. Therefore,
$G/H$
is also classical.
Construction 2.18 (Connected-étale sequences, cf. [Reference Kato11, (2.6)])
Let S be an fs log scheme such that
$\mathring {S}=\mathrm {Spec}R$
for a Henselian local ring R, and let
$G\in (\mathrm {fin}/S)_{\text {f}}$
. We shall construct a unique exact sequence of objects of
$(\mathrm {fin}/S)_{\text {f}}$
$$ \begin{align} 0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0 \end{align} $$
such that, for every finite Kummer log flat cover
$T\to S$
with
$G|_{T}$
being classical, the pullback of the sequence (2.1) is the connected-étale sequence for
$G|_{T}$
. This sequence is called the connected-étale sequence for G. The uniqueness follows from the uniqueness of connected-étale sequences in non-log cases.
First, we assume that R is strict local. Take a chart
$P\to \mathcal {M}_{S}$
. By Lemma 2.13, there exists an integer
$n\geq 1$
such that the pullback
$G|_{T}$
is classical, where we set 
. Since the underlying schemes of T,
$T\times _{S} T$
, and
$T\times _{S} T\times _{S} T$
are the spectrums of finite products of strict local rings, the connected-étale sequence for
$G|_{T}$
descends to the sequence (2.1) via Kummer log flat descent (Lemma 2.12) by the uniqueness of connected-étale sequences in non-log cases.
In general, what we proved in the previous paragraph and the limit argument (Proposition 2.14) allow us to take a strict finite étale cover
$S'\to S$
such that
$G|_{S'}$
admits a connected-étale sequence. By the uniqueness of connected-étale sequences, this descends to the desired sequence (2.1).
Lemma 2.19. Let S be an fs log scheme such that
$\mathring {S}$
is the spectrum of a Henselian local ring. Let
$H\subset G$
be objects in
$(\mathrm {fin}/S)_{\text {f}}$
. Then there are natural isomorphisms of objects in
$(\mathrm {fin}/S)_{\text {f}}$
$$ \begin{align*} H^{0}\cong H\cap G^{0}, \ H^{\acute{\rm e}\mathrm{t}}\cong \mathrm{Im}(H\to G^{\acute{\rm e}\mathrm{t}}), \ (G/H)^{0}\cong G^{0}/H^{0}, \ (G/H)^{\acute{\rm e}\mathrm{t}}\cong G^{\acute{\rm e}\mathrm{t}}/H^{\acute{\rm e}\mathrm{t}}. \end{align*} $$
Proof. The analogous statements for non-log finite locally free group schemes are well-known, to which one can reduce the original statement by Kummer log flat descent.
Proposition 2.20 (cf. [Reference Kato11, Proposition 2.7(3)])
Let S be an fs log scheme such that
$\mathring {S}=\mathrm {Spec} R$
for a Henselian local ring R with residue characteristic
$p>0$
. Let G be an object of
$(\mathrm {fin}/S)_{\text {f}}$
killed by some power of p. Then the object G belongs to
$(\mathrm {fin}/S)_{\text {d}}$
if and only if both of
$G^{0}$
and
$G^{\acute{\rm e}\mathrm{t}}$
are classical. Moreover, in this case,
$G^{0}$
(resp.
$G^{\acute{\rm e}\mathrm{t}}$
) is a connected finite locally free group scheme (resp. a finite étale group scheme) over
$\mathring {S}$
.
Proof. This is proved in [Reference Kato11, Proposition 2.7(3)] in the case, where R is Noetherian. In general, one can reduce the problem to this case by the limit argument (Proposition 2.14).
Lemma 2.21. Let G be a log finite group scheme over S. Suppose that p is locally nilpotent on
$\mathring {S}$
and that the order of
$(G|_{s})^{\acute{\rm e}\mathrm{t}}$
is constant for every
$s\in S$
. Then there exists a unique exact sequence
$$\begin{align*}0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0, \end{align*}$$
where
$G^{0}$
is a finite locally free radiciel group scheme over
$\mathring {S}$
and
$G^{\acute{\rm e}\mathrm{t}}$
is a finite étale group scheme over
$\mathring {S}$
.
Proof. Since the exact sequence as in the statement is unique if it exists, we may work strict étale locally on S. Hence, by the limit argument (Proposition 2.14), we may assume that
$\mathring {S}$
is the spectrum of a strict local ring. Consider the connected-étale exact sequence
$$\begin{align*}0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0, \end{align*}$$
where
$G^{0}$
is a finite locally free group scheme over
$\mathring {S}$
and
$G^{\acute{\rm e}\mathrm{t}}$
is a finite étale group scheme over
$\mathring {S}$
by Proposition 2.20. For every point
$t\in S$
, we have the following diagram:
By the assumption, the right vertical homomorphism is a surjection between finite étale group schemes of the same order, which is an isomorphism. Hence, the left vertical map is also an isomorphism. This proves that
$G^{0}$
is radiciel over
$\mathring {S}$
.
Proposition 2.22 (cf. [Reference Kato11, Proposition 2.16])
Let
$G\in (\mathrm {fin}/S)_{\text {f}}$
. Let
$\star \in \{\mathrm {d,c}\}$
be a subscript. Suppose that the pullback of G to a point
$s\in S$
belongs to
$(\mathrm {fin}/s)_{\star }$
. Then there exists an open neighborhood
$U\subset S$
of s such that
$G|_{U}$
belongs to
$(\mathrm {fin}/U)_{\star }$
. In particular, if the pullback of G to every point
$s\in S$
belongs to
$(\mathrm {fin}/s)_{\star }$
, then G also belongs to
$(\mathrm {fin}/S)_{\star }$
.
Proof. The limit argument (Proposition 2.14) allows us to assume that S is locally Noetherian, in which case the assertion is proved in [Reference Kato11, Proposition 2.16].
Lemma 2.23. Let G be a log finite group scheme over S of order p. Suppose that p is locally nilpotent over
$\mathring {S}$
. Then G is classical.
Proof. Due to Proposition 2.22, we may assume that the
$\mathring {S}$
is the spectrum of a field. Since the order of G is equal to the product of the orders of
$G^{0}$
and
$G^{\acute{\rm e}\mathrm{t}}$
, the log finite group scheme G is isomorphic to either
$G^{0}$
or
$G^{\acute{\rm e}\mathrm{t}}$
. Hence, the assertion follows from [Reference Kato11, Proposition 2.7(3)].
Lemma 2.24. Let S be an fs log scheme such that
$\mathring {S}=\mathrm {Spec} R$
for a reduced Henselian local ring R with residue characteristic
$p>0$
. Let G be a log finite group scheme over S killed by some power of p, and let
$H_{1}, H_{2}$
be log finite subgroup schemes of G. Suppose that
$H_{1}^{0}$
is contained in
$H_{2}^{0}$
,
$H_{1}^{\acute{\rm e}\mathrm{t}}$
is contained in
$H_{2}^{\acute{\rm e}\mathrm{t}}$
, and
$G^{0}$
is radiciel over S. Then
$H_{1}$
is contained in
$H_{2}$
.
Proof. By Lemma 2.19, we have the following commutative diagram of log finite group schemes:
By the above diagram, the canonical morphism
$H_{1}\to G/H_{2}$
factors as
$H_{1}\to H_{1}^{\acute{\rm e}\mathrm{t}}\to G^{0}/H_{2}^{0}\to G/H_{2}$
. Since
$H_{1}^{\acute{\rm e}\mathrm{t}}$
is reduced and
$G^{0}/H_{2}^{0}$
is radiciel over
$\mathring {S}$
, we get
$$\begin{align*}\mathrm{Hom}_{S}(H_{1}^{\acute{\rm e}\mathrm{t}}, G^{0}/H_{2}^{0})=0. \end{align*}$$
Therefore,
$H_{1}$
is contained in
$H_{2}$
.
Corollary 2.25. Let
$H\subset G$
be objects of
$(\mathrm {fin}/S)_{\text {f}}$
. Suppose that G belongs to
$(\mathrm {fin}/S)_{\text {d}}$
. Then H and
$G/H$
also belong to
$(\mathrm {fin}/S)_{\text {d}}$
.
Proof. Proposition 2.22 allows us to assume that
$\mathring {S}=\mathrm {Spec} k$
for a field k of characteristic
$p\geq 0$
. By [Reference Kato11, Proposition 2.1], it is enough to treat the case, where
$p>0$
and G is killed by some power of p.
Note that
$G^{0}$
and
$G^{\acute{\rm e}\mathrm{t}}$
are classical by Proposition 2.20. There are inclusion relations
$H^{0}\subset G^{0}$
and
$H^{\acute{\rm e}\mathrm{t}}\subset G^{\acute{\rm e}\mathrm{t}}$
by Lemma 2.19. Hence, Proposition 2.17 implies that
$H^{0}$
and
$H^{\acute{\rm e}\mathrm{t}}$
are also classical. By using Proposition 2.20 again, we see that H belongs to
$(\mathrm {fin}/S)_{\text {d}}$
. By applying this result to the subobject
$(G/H)^{*}\subset G^{*}$
, we conclude that
$G/H$
also belongs to
$(\mathrm {fin}/S)_{\text {d}}$
.
2.4. Log p-divisible groups
Definition 2.26 [Reference Kato11, Section 4]
Let
$(\mathrm {BT}/S)_{\text {f}}$
be the category of sheaves of abelian groups G on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
satisfying the following conditions:
-
1. The equation
$\displaystyle G=\bigcup _{n\geq 1} G[p^n]$
holds, where
$G[p^n]:=\mathrm {Ker}(\times p^n\colon G\to G)$
. -
2. The morphism
$\times p\colon G\to G$
is surjective. -
3. For each
$n\geq 1$
, the sheaf
$G[p^n]$
belongs to
$(\mathrm {fin}/S)_{\text {f}} $
.
Furthermore, for a subscript
$\star \in \{\mathrm {d,c}\}$
, we define a category
$(\mathrm {BT}/S)_{\star }$
as the full subcategory of
$(\mathrm {BT}/S)_{\text {f}}$
consisting of objects G such that
$G[p^{n}]$
belongs to
$(\mathrm {fin}/S)_{\star }$
for each
$n\geq 1$
. We say that
$G\in (\mathrm {BT}/S)_{\mathrm {f}}$
is classical if G belongs to
$(\mathrm {BT}/S)_{\mathrm {c}}$
. The category
$(\mathrm {BT}/S)_{\text {c}}$
is naturally equivalent to the category of p-divisible groups over
$\mathring {S}$
. Objects in
$(\mathrm {BT}/S)_{\text {d}}$
are called log p-divisible groups over S.
Definition 2.27. For
$G\in (\mathrm {BT}/S)_{\text {f}}$
, we say that G is of height h if
$G[p^n]$
is of order
$p^{nh}$
for each
$n\geq 1$
(equivalently,
$G[p]$
is of order
$p^h$
).
Example 2.28. Let
$n\geq 1$
. For any log abelian variety A over S with
$\mathrm {dim}(A)=d$
, the object 
is a log p-divisible group of height
$2d$
over S because
$A[p^n]$
is a log finite group scheme of order
$p^{2nd}$
by Example 2.10 and
$\times p\colon A\to A$
is surjective by [Reference Kajiwara, Kato and Nakayama6, Section 18.6].
Lemma 2.29. Let S be an fs log scheme such that
$\mathring {S}$
is the spectrum of a Henselian local ring. Let
$G\in (\mathrm {BT}/S)_{\text {f}}$
. Then
are objects of
$(\mathrm {BT}/S)_{\text {f}}$
, and there exists an exact sequence
$$\begin{align*}0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0. \end{align*}$$
Proof. Let
$\star \in \{0,{\acute{\rm e}\mathrm{t}}\}$
be a subscript and
$n,m\geq 1$
be integers. By Lemma 2.19, the homomorphism
$\times p^{n}\colon G[p^{n+m}]^{\star }\to G[p^{n+m}]^{\star }$
factors as
$$\begin{align*}G[p^{n+m}]^{\star}\to G[p^{m}]^{\star}\hookrightarrow G[p^{n+m}]^{\star}. \end{align*}$$
We also simply write
$\times p^{n}$
for the homomorphism
$G[p^{n+m}]^{\star }\to G[p^{m}]^{\star }$
defined above. Then, by applying Lemma 2.19 again, we see that the sequence
$$\begin{align*}0\to G[p^{n}]^{\star}\hookrightarrow G[p^{n+m}]^{\star}\stackrel{\times p^{n}}{\to} G[p^{m}]^{\star}\to 0 \end{align*}$$
is exact, which implies that
$G^{\star }$
is an object of
$(\mathrm {BT}/S)_{\text {f}}$
. The remaining assertion is clear.
Lemma 2.30. Let G be a log p-divisible group over S. Suppose that p is locally nilpotent on S and that the height of
$(G|_{s})^{\acute{\rm e}\mathrm{t}}$
is constant for every
$s\in S$
. Then there exists a unique exact sequence
$$\begin{align*}0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0, \end{align*}$$
where
$G^{0}$
is a formal Lie group over
$\mathring {S}$
and
$G^{\acute{\rm e}\mathrm{t}}$
is an étale p-divisible group over
$\mathring {S}$
.
Proof. This immediately follows from Lemma 2.21.
Definition 2.31. Let
$G_{1}$
and
$G_{2}$
be log p-divisible groups over S. A homomorphism
$f\colon G_{1}\to G_{2} $
is called an isogeny if f is surjective and
$\mathrm {Ker}(f)$
belongs to
$(\mathrm {fin}/S)_{\mathrm {f}}$
.
Let
$\mathrm {QIsog}(G_{1},G_{2})$
be the subset of
$\mathrm {Hom}(G_{1},G_{2})[1/p]$
consisting of elements which can be written as
$g/p^{n}$
for an integer
$n\geq 1$
and an isogeny
$g\colon G_{1}\to G_{2}$
. A quasi-isogeny from
$G_{1}$
to
$G_{2}$
is an element of the global section of the sheaf on the Zariski site of S associated with the presheaf given by
$U\mapsto \mathrm {QIsog}(G_{1}|_{U},G_{2}|_{U})$
for an open subset
$U\subset S$
. Note that, when S is quasi-compact, the set of quasi-isogenies from
$G_{1}$
to
$G_{2}$
is naturally in bijection with the set
$\mathrm {QIsog}(G_{1},G_{2})$
.
Lemma 2.32. Let G be a log p-divisible group and H be a log finite subgroup scheme of G. Then the sheaf
$G/H$
is a log p-divisible group, and the canonical map
$G\to G/H$
is an isogeny. Moreover, this gives a one-to-one correspondence between log finite subgroup schemes of G and isogenies from G.
Proof. By working Zariski locally on S, we may assume that H is killed by
$p^{d}$
for an integer
$d\geq 1$
. For an integer
$n\geq d$
, applying the snake lemma to the diagram
gives an exact sequence
$$\begin{align*}0\to G[p^{n}]/H\to (G/H)[p^{n}]\to H\to 0. \end{align*}$$
Hence, it follows from [Reference Kato11, Proposition 2.3] that
$(G/H)[p^{n}]$
is a log finite group scheme, and so
$G/H$
is a log p-divisible group.
Conversely, for an isogeny
$f\colon G_{1}\to G_{2}$
, the object
$\mathrm {Ker}(f)\in (\mathrm {fin}/S)_{\mathrm {f}}$
is a log finite group scheme by Corollary 2.25. This proves the assertion.
Proposition 2.33. Let
$f\colon G_{1}\to G_{2}$
be a surjection of log p-divisible groups over a locally Noetherian fs log scheme S. Suppose that, Zariski locally,
$\mathrm {Ker}(f)$
is killed by
$p^{d}$
for an integer
$d\geq 1$
. Then f is an isogeny.
Proof. It is enough to prove that
$\mathrm {Ker}(f)$
is a log finite group scheme over S. By working Zariski locally on S, we may assume that
$\mathrm {Ker}(f)\subset G_{1}[p^{d}]$
for an integer
$d\geq 1$
. The sheaf
$\mathrm {Ker}(f)$
is representable by an fs log scheme of finite presentation over S as it coincides with the kernel of the homomorphism of log finite group schemes
$G_{1}[p^{d}]\to G_{2}[p^{d}]$
induced from f. Hence, by the limit argument (Propositions 2.14 and A.3), we may assume that
$\mathring {S}$
is the spectrum of a strict local ring.
Since
$\mathrm {Ker}(f)$
is killed by
$p^{d}$
and f is surjective, there exists a unique homomorphism
$g\colon G_{2}\to G_{1}$
such that both
$fg$
and
$gf$
are equal to the multiplication by
$p^{d}$
. The homomorphism f uniquely induces homomorphisms
$f^{0}\colon G_{1}^{0}\to G_{2}^{0}$
and
$f^{\acute{\rm e}\mathrm{t}}\colon G_{1}^{\acute{\rm e}\mathrm{t}}\to G_{2}^{\acute{\rm e}\mathrm{t}}$
fitting into the following diagram:
Similarly, the homomorphism g uniquely induces homomorphisms
$g^{0}\colon G_{2}^{0}\to G_{1}^{0}$
and
$g^{\acute{\rm e}\mathrm{t}}\colon G_{2}^{\acute{\rm e}\mathrm{t}}\to G_{1}^{\acute{\rm e}\mathrm{t}}$
. By the uniqueness of the induced morphism on the connected part and the étale part, all of the homomorphisms
$f^{0}g^{0}$
,
$f^{\acute{\rm e}\mathrm{t}}g^{\acute{\rm e}\mathrm{t}}$
,
$g^{0}f^{0}$
, and
$g^{\acute{\rm e}\mathrm{t}}f^{\acute{\rm e}\mathrm{t}}$
are the multiplication by
$p^{d}$
. It follows from [Reference Chai, Conrad and Oort2, Proposition 3.3.8] that all of the homomorphisms
$f^{0}$
,
$f^{\acute{\rm e}\mathrm{t}}$
,
$g^{0}$
, and
$g^{\acute{\rm e}\mathrm{t}}$
are isogenies. By applying the snake lemma to the above diagram, we get an exact sequence
$$\begin{align*}0\to \mathrm{Ker}(f^{0})\to \mathrm{Ker}(f)\to \mathrm{Ker}(f^{\acute{\rm e}\mathrm{t}})\to 0. \end{align*}$$
Since
$\mathrm {Ker}(f^{0})$
and
$\mathrm {Ker}(f^{\acute{\rm e}\mathrm{t}})$
are finite locally free group schemes over
$\mathring {S}$
, the sheaf
$\mathrm {Ker}(f)$
is a log finite group scheme over S by [Reference Kato11, Proposition 2.3].
3. Completely slope divisible log p-divisible groups
Throughout this section, let S be an fs log scheme over
$\mathbb {F}_{p}$
.
Definition 3.1. Let G be a sheaf of abelian groups on
$(\mathrm {fs}/S)_{\mathrm {kfl}}$
(the examples of primary interest are objects of
$(\mathrm {fin}/S)_{\mathrm {f}}$
or
$(\mathrm {BT}/S)_{\mathrm {f}}$
). Let
$G^{(p^{n})}$
denote the pullback of G by the nth power Frobenius morphism on S.
For an fs log scheme T over S, we write
$T^{(F)}$
for the fs log scheme T equipped with the structure map
$T\to S\stackrel {F}{\to } S$
, where F is the Frobenius morphism on S. The Frobenius morphism on T is a map
$T^{(F)}\to T$
over S, and so we have an induced homomorphism
$G(T)\to G(T^{(F)})\cong G^{(p)}(T)$
. This defines a homomorphism
$F\colon G\to G^{(p)}$
, called the Frobenius homomorphism. When G belongs to
$(\mathrm {fin}/S)_{\mathrm {c}}$
or
$(\mathrm {BT}/S)_{\mathrm {c}}$
, the homomorphism F is nothing but the usual Frobenius homomorphism. For an integer
$n\geq 1$
, the composite of homomorphisms
$$\begin{align*}G\stackrel{F}{\to} G^{(p)}\stackrel{F^{(p)}}{\to} G^{(p^{2})}\stackrel{F^{(p^{2})}}{\to} \dots \stackrel{F^{(p^{n-1})}}{\to} G^{(p^{n})} \end{align*}$$
is simply denoted by
$F^{n}$
, where
$F^{(p^{k})}$
denotes the base change of F by the kth power Frobenius morphism on S for an integer
$k\geq 1$
.
Lemma 3.2. Let G be a log finite group scheme over S. Suppose that there is an integer
$n\geq 1$
such that the homomorphism
$F^{n}\colon G\to G^{(p^{n})}$
is an isomorphism (resp. a zero map). Then G is classical. In particular, G is a finite étale group scheme (resp. a finite radiciel locally free group scheme) over
$\mathring {S}$
.
Proof. We shall prove the assertion just in the case where
$F^{n}$
is an isomorphism because the same argument also works in the other case. By Proposition 2.22, we may assume that
$\mathring {S}=\mathrm {Spec} k$
for a field k. In this case, we have the following diagram:
Since
$G^{\acute{\rm e}\mathrm{t}}$
is a finite étale group scheme over
$\mathring {S}$
, the right vertical map in this diagram is an isomorphism. Hence, the left vertical map is also an isomorphism. On the other hand, the mth power of the Frobenius homomorphism
$G^{0}\to (G^{0})^{(p^{m})}$
is a zero map for some integer
$m\geq 1$
because
$G^{0}$
is a connected finite group scheme over the field k. Therefore,
$G^{0}$
is trivial, and so
$G\cong G^{\acute{\rm e}\mathrm{t}}$
. In particular, G is classical.
Definition 3.3. A log p-divisible group G over S is completely slope divisible if G admits a filtration of log p-divisible subgroups
$$\begin{align*}0=G_0\subset G_1 \subset \dots \subset G_m=G, \end{align*}$$
and if there are integers
$$\begin{align*}s\geq r_1> r_2 >\dots >r_{m}\geq 0 \end{align*}$$
such that the following conditions are satisfied:
-
1. The quasi-isogeny
$p^{-r_{i}}F^{s}$
from
$G_{i}$
to
$G_{i}^{(p^{s})}$
is an isogeny for every
$1\leq i\leq m$
. -
2. The morphism
$G_{i}/G_{i-1}\to (G_{i}/G_{i-1})^{(p^{s})}$
induced from
$p^{-r_{i}}F^{s}\colon G_{i}\to G_{i}^{(p^{s})}$
is an isomorphism for every
$1\leq i\leq m$
.
Proposition 3.4. Let G be a completely slope divisible log p-divisible group over S. Then there exists the unique exact sequence
$$\begin{align*}0\to G^{0}\to G\to G^{\acute{\rm e}\mathrm{t}}\to 0, \end{align*}$$
where
$G^{0}$
is a completely slope divisible formal Lie group over
$\mathring {S}$
and
$G^{\acute{\rm e}\mathrm{t}}$
is an étale p-divisible group over
$\mathring {S}$
.
Proof. We use the notation in Definition 3.3. We may assume that
$r_{m}=0$
. Let
$n\geq 1$
be an integer. By the definition, the Frobenius homomorphism
$F\colon (G/G_{m-1})[p^{n}]\to (G/G_{m-1})[p^{n}]^{(p)}$
is an isomorphism, and the homomorphism
$F^{sn}\colon G_{m-1}[p^{n}]\to G_{m-1}[p^{n}]^{(p^{sn})}$
is a zero map. Hence, by Lemma 3.2,
$G/G_{m-1}$
is an étale p-divisible group over
$\mathring {S}$
, and
$G_{m-1}$
is a formal Lie group over
$\mathring {S}$
. Therefore, the exact sequence
$$\begin{align*}0\to G_{m-1}\to G\to G/G_{m-1}\to 0 \end{align*}$$
is the desired one.
We generalize the notion of Newton polygons to log p-divisible groups.
Definition 3.5. Let S be an fs log scheme such that
$\mathring {S}$
is the spectrum of a field of characteristic
$p>0$
. Let G be a log p-divisible group over S. The Newton polygon of G is the Newton polygon of the classical p-divisible group
$G^{0}\oplus G^{\acute{\rm e}\mathrm{t}}$
.
The notion of Newton polygons for log p-divisible groups enables us to formulate a log version of [Reference Oort and Zink17, Proposition 2.3] as follows.
Corollary 3.6. Let G be a log p-divisible group over S with constant Newton polygon. Suppose that
$\mathring {S}$
is integral, and that the pullback
$G|_{\eta }$
is completely slope divisible, where
$\eta $
is the generic point. Then G is completely slope divisible.
Proof. We take
$G^{0}$
and
$G^{\acute{\rm e}\mathrm{t}}$
as in Lemma 2.30. The formal Lie group
$G^{0}$
has a constant Newton polygon, and its generic fiber
$G^{0}|_{\eta }$
is completely slope divisible. Hence, [Reference Oort and Zink17, Proposition 2.3] implies that
$G^{0}$
is completely slope divisible. Therefore, G is also completely slope divisible.
The following corollary is a log version of [Reference Oort and Zink17, Theorem 2.1].
Corollary 3.7. Let G be a log p-divisible group over S with constant Newton polygon. Suppose that
$\mathring {S}$
is locally Noetherian normal. Then there exist a completely slope divisible log p-divisible group H over S and an isogeny
$G\to H$
.
Proof. We take
$G^{0}$
and
$G^{\acute{\rm e}\mathrm{t}}$
as in Lemma 2.30. Since
$G^{0}$
has a constant Newton polygon, there exist a completely slope divisible classical p-divisible group
$H'$
and an isogeny
$\phi \colon G^{0}\to H'$
by [Reference Oort and Zink17, Theorem 2.1]. Then there is an exact sequence
$$\begin{align*}0\to H'\to G/\mathrm{Ker}(\phi)\to G^{\acute{\rm e}\mathrm{t}}\to 0, \end{align*}$$
and so
$G/\mathrm {Ker}(\phi )$
is a completely slope divisible log p-divisible group. Hence, the canonical surjection
$G\twoheadrightarrow G/\mathrm {Ker}(\phi )$
is the desired isogeny.
The following lemma is a log version of [Reference Oort and Zink17, Lemma 2.5], which we use in the proof of the main theorem.
Lemma 3.8. Suppose that
$\mathring {S}$
is the spectrum of an algebraically closed field k. Let G be a log p-divisible group over S and
$d\geq 1$
be an integer. Then there are only finitely many isomorphism classes of isogenies
$G\to H$
of degree
$p^d$
in which H is a completely slope divisible log p-divisible group.
Proof. Let K be a log finite subgroup scheme of G of order
$p^{d}$
. Since
$G/K$
is completely slope divisible if and only if
$(G/K)^{0}=G^{0}/K^{0} $
and
$(G/K)^{\acute{\rm e}\mathrm{t}}=G^{\acute{\rm e}\mathrm{t}}/K^{\acute{\rm e}\mathrm{t}} $
, there are only finitely many possibilities of
$K^{0}$
and
$K^{\acute{\rm e}\mathrm{t}}$
such that
$G/K$
is completely slope divisible by [Reference Oort and Zink17, Lemma 2.5]. Lemma 2.24 implies that K is determined from
$K^{0}$
and
$K^{\acute{\rm e}\mathrm{t}}$
uniquely. This proves the statement.
4. Some extension properties on log regular schemes
In this section, we study log finite groups and log p-divisible groups over log regular bases.
4.1. Log regular schemes
In this section, we recall the definition of log regularity and some properties of log regular log schemes.
Definition 4.1 [Reference Kato9]
Let S be a locally Noetherian fs log scheme. Let
$s\in S$
and let
$\bar {s}$
be a geometric point on s. Let
$I(\bar {s})$
be the ideal of
$\mathcal {O}_{S, \bar {s}}$
generated by the image of the map
$\mathcal {M}_{S, \bar {s}}\setminus \mathcal {O}_{S, \bar {s}}^{\times }\to \mathcal {O}_{S, \bar {s}}$
. We say that S is log regular at s if the following two conditions are satisfied:
-
1.
$\mathcal {O}_{S, \bar {s}}/I(\bar {s})$
is a regular local ring; -
2.
$\dim (\mathcal {O}_{S, \bar {s}})=\dim (\mathcal {O}_{S, \bar {s}}/I(\bar {s}))+\mathrm {rk}(\mathcal {M}_{S, \bar {s}}^{\mathrm {gp}}/\mathcal {O}_{S, \bar {s}}^{\times })$
.
The log scheme S is called log regular if it is log regular at every point
$s\in S$
.
Proposition 4.2 (Kato)
For a locally Noetherian fs log scheme S, the following statements are true:
(1) The subset of S defined as {
$s\in S | S$
is log regular at s} is stable under generalization.
(2) If S is log regular at
$s\in S$
,
$\mathring {S}$
is normal at s.
Proof. (1) See [Reference Kato9, Proposition 7.1].
(2) See [Reference Kato9, Theorem 4.1].
Lemma 4.3. Let S be a log regular fs log scheme such that
$\mathring {S}=\mathrm {Spec} R$
for a Noetherian strict local ring R. Let s be the unique closed point of S. Fix a chart
$P\to \mathcal {M}_S$
which is neat at s. Then, for a sharp fs monoid Q with a Kummer map
$P\to Q$
, the log scheme 
is also log regular.
Proof. First, we claim that
$R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q]$
is a Noetherian strict local ring. Since
$R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q]$
is finite over R, it suffices to prove that the special fiber of
$T\to S$
consists of a single point. Let k be the residue field of R. Since Q is sharp and
$P\to Q$
is injective, sending
$Q\backslash \{1\}$
to
$0$
defines a ring map
$k\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q] \to k$
. This ring map induces a ring map
$(k\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])_{\mathrm {red}} \to k$
, which is the inverse of the natural map
$k\to (k\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])_{\mathrm {red}}$
. Hence, we get an isomorphism
$k\cong (k\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])_{\mathrm {red}}$
, and so the special fiber of
$T\to S$
consists of a single point.
By Proposition 4.2(1), it suffices to prove that T is log regular at the unique closed point t. Since
$R\to R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q]$
is finite and injective, we have the equation
$\mathrm {dim}(R)=\mathrm {dim}(R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])$
. Since
$P\to Q$
is Kummer, the following equations hold:
$$\begin{align*}\mathrm{rk}(\mathcal{M}_{S,s}^{\mathrm{gp}}/\mathcal{O}_{S,s}^{\times})=\mathrm{rk}(P^{\mathrm{gp}})=\mathrm{rk}(Q^{\mathrm{gp}})=\mathrm{rk}(\mathcal{M}_{T,t}^{\mathrm{gp}}/\mathcal{O}_{T,t}^{\times}). \end{align*}$$
Since S is log regular, it is enough to show the equation
$$ \begin{align} \mathrm{dim}(R/I(s))=\mathrm{dim}((R\otimes_{\mathbb{Z}[P]} \mathbb{Z}[Q])/I(t)). \end{align} $$
The ideal
$I(s)\subset R$
(resp.
$I(t)\subset R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q]$
) is generated by the image of
$P\backslash \{1\}$
(resp.
$Q\backslash \{1\}$
). Sending
$Q\backslash \{1\}$
to
$0$
defines a ring morphism
$(R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])/I(t)\to R/I(s)$
, and it can be checked directly that this map is the inverse of the natural map
$R/I(s)\to (R\otimes _{\mathbb {Z}[P]} \mathbb {Z}[Q])/I(t)$
. In particular, we obtain equation (4.1).
4.2. One-dimensional cases
Lemma 4.4. Let S be a log regular fs log scheme such that
$\mathring {S}=\mathrm {Spec} R$
for a strict local discrete valuation ring R. Then the log structure of S is either the trivial one or the standard one (i.e., the log structure defined by the unique closed point).
Proof. Let s denote the unique closed point of S. The condition
$\mathrm {rk}((\overline {\mathcal {M}_{S, s}})^{\mathrm {gp}})\leq 1$
implies that the monoid
$\overline {\mathcal {M}_{S, s}}$
is isomorphic to either
$0$
or
$\mathbb {N}$
. In the former case, the log structure of S is trivial. In the latter case, we can take a chart
$\mathbb {N}\to \mathcal {M}_{S, s}$
which is neat at s. Since
$R/I(s)$
is a field, the morphism
$\mathbb {N}\to \mathcal {M}_{S, s}\to R$
maps
$1$
to a uniformizer of R. Hence, the log structure of S is the standard one.
Proposition 4.5. Let S be a log regular fs log scheme such that
$\mathring {S}=\mathrm {Spec} R$
for a Henselian discrete valuation ring R. Let K be the fraction field of R. Let G be a log finite group scheme over S and
$H_{K}$
be a log finite subgroup scheme of
$G|_{K}$
. Then
$H_{K}$
uniquely extends to a log finite subgroup scheme of G.
Proof. The limit argument (Proposition 2.14) and Galois descent allow us to assume that R is strictly local. By Lemma 4.4, the log structure of S is either the trivial one or the standard one. In the former case, G is classical, and so the scheme-theoretic closure of
$H_{K}$
in G is the unique extension of
$H_{K}$
.
It is enough to consider the latter case. We fix a uniformizer
$\pi \in R$
. By Lemma 2.13, there exists an integer
$n\geq 1$
such that
$G|_{T}$
is classical, where T is
$\mathrm {Spec} R[\pi ^{1/n}]$
equipped with the standard log structure. Let 
be the fraction field of
$R[\pi ^{1/n}]$
. Let 
. We have the following commutative diagram in which every square is a Cartesian diagram in the category of saturated log schemes:
Here, all log schemes in the right column are endowed with the trivial log structure, and all horizontal arrows are strict open immersion.
By taking the scheme-theoretic closure, we see that
$H_{K}|_{L}\subset G|_{L}$
uniquely extends to a classical finite subgroup scheme
$H_{T}$
of
$G|_{T}$
. As the pullbacks of
$H_{T}$
by two projections
$T'\to T$
coincide on the dense open subscheme
$\mathrm {Spec} (L\otimes _{K} L)$
as subobjects of
$G|_{L\otimes _{K} L}$
, they coincide on the whole of
$T'$
. Hence, by Kummer log flat descent,
$H_{T}$
descends to the subobject
$H\subset G$
in the category
$(\mathrm {fin}/S)_{\mathrm {f}}$
with an isomorphism
$H_{K}\cong H|_{K}$
. By Corollary 2.25, H is a log finite group scheme.
Remark 4.6. Proposition 4.5 does not hold without the assumption that S is log regular. Let R be a Henselian discrete valuation ring over
$\mathbb {F}_{p}$
with a uniformizer
$\pi $
, and K be the fraction field of R. Let S be the fs log scheme
$\mathrm {Spec}R$
equipped with the log structure defined by
$\mathbb {N}\to R$
mapping
$1$
to
$\pi ^{p}$
(which is not log regular). Let
$\eta $
be the generic point of S and s be the closed point of S. By abuse of notation, we simply write s for the log scheme
$(\mathrm {Spec}k(s),\mathcal {M}_{\mathrm {Spec}k(s)})$
. As in Example 2.11, we can associate to 
an exact sequence of log finite group schemes over S
$$ \begin{align} 0\to \mu_{p}\to G_{q}\to \mathbb{Z}/p\mathbb{Z}\to 0. \end{align} $$
The Kummer sequence
$$\begin{align*}0\to \mu_{p}\to \mathbb{G}_{m,\mathrm{log}}\to \mathbb{G}_{m,\mathrm{log}}\to 0 \end{align*}$$
induces the following commutative diagram, in which both rows are exact:
By this diagram, we see that the generic fiber of the exact sequence (4.2) splits. We claim that the unique section
$(\mathbb {Z}/p\mathbb {Z})_{\eta }\rightarrowtail (G_{q})|_{\eta }$
does not extend to a log finite subgroup scheme of
$G_{q}$
. Assume that there exists such an extension
$H\subset G_{q}$
. Since the order of H is a prime number p, H is classical by Lemma 2.23. Since the surjection
$G_{q}\twoheadrightarrow \mathbb {Z}/p\mathbb {Z}$
admits no section, the composite
$H\hookrightarrow G_{q}\twoheadrightarrow \mathbb {Z}/p\mathbb {Z}$
is not an isomorphism. Hence, the composite
$H|_{s}\rightarrowtail G_{q}|_{s}\twoheadrightarrow (\mathbb {Z}/p\mathbb {Z})_{s}$
is not an isomorphism, and so it is trivial. Therefore, we obtain
$H|_{s}\cong \mu _{p,s}$
by (4.2). In particular, the multiplicative rank of H jumps under the specialization. This is a contradiction.
4.3. Extension properties on log regular schemes
Lemma 4.7. Let X be a locally Noetherian normal scheme and
$f\colon Y\to X$
be a flat morphism of schemes. Let U be a dense open subset of X containing all points on X of codimension
$1$
and 
. Then the restriction functors
$$ \begin{align*} \mathcal{LF}(Y)&\to \mathcal{LF}(V) \\ (\mathrm{fin}/Y)&\to (\mathrm{fin}/V) \end{align*} $$
are fully faithful.
Proof. It is enough to prove the fully faithfulness of the former functor. By taking internal homomorphisms, the problem is reduced to showing that, for a vector bundle
$\mathcal {E}$
on Y, the restriction map
$\Gamma (Y,\mathcal {E})\to \Gamma (V,\mathcal {E})$
is an isomorphism. Let
$i\colon V\hookrightarrow Y$
and
$j\colon U\hookrightarrow X$
be natural open immersions. Then we have isomorphisms of
$\mathcal {O}_{Y}$
-modules
$$ \begin{align*} i_{*}i^{*}\mathcal{E}\cong \mathcal{E}\otimes i_{*}\mathcal{O}_{V}\cong \mathcal{E}\otimes f^{*}j_{*}\mathcal{O}_{U}\cong \mathcal{E}\otimes f^{*}\mathcal{O}_{X}\cong \mathcal{E}, \end{align*} $$
where the first isomorphism is the projection formula, the second one is the flat base change, and the third one follows from the assumption that U is an open subset of a locally Noetherian normal scheme X containing all points of codimension
$1$
. Taking global sections on both sides, we obtain the statement.
Proposition 4.8. Let S be a log regular fs log scheme and U be a dense open subset of S.
(1) The restriction functors
$$ \begin{align*} \mathcal{LLF}(S)&\to \mathcal{LLF}(U) \\ (\mathrm{fin}/S)_{\mathrm{f}}&\to (\mathrm{fin}/U)_{\mathrm{f}} \end{align*} $$
are faithful.
(2) Suppose that U contains all points on S of codimension
$1$
. Then the restriction functors
$$ \begin{align*} \mathcal{LLF}(S)&\to \mathcal{LLF}(U) \\ (\mathrm{fin}/S)_{\mathrm{f}}&\to (\mathrm{fin}/U)_{\mathrm{f}} \end{align*} $$
are fully faithful.
Proof. We shall prove only (2) because the same argument also works for (1). By Proposition 2.16, it is enough to prove the fully faithfulness of the former functor. Let
$\mathcal {E}_1, \mathcal {E}_2$
be log vector bundles on S. It suffices to show that the map
$$\begin{align*}\mathrm{Hom}(\mathcal{E}_1, \mathcal{E}_2)\to \mathrm{Hom}(\mathcal{E}_1|_U, \mathcal{E}_2|_U) \end{align*}$$
is bijective. By the limit argument (Proposition 2.5), we may assume that
$\mathring {S}$
is the spectrum of a strict local ring. Let s be the unique closed point of S. Fix a chart
$P\to \mathcal {M}_S$
which is neat at s.
By Lemma 2.4, there exists an integer
$n\geq 1$
such that
$\mathcal {E}_{1}|_{T}$
and
$\mathcal {E}_{2}|_{T}$
are classical, where we put 
. Since T is log regular by Lemma 4.3, it follows from Lemma 4.2 (2) that T is normal. Let V be the preimage of U by
$T\to S$
. Since
$\mathring {T}\to \mathring {S}$
is a finite dominant morphism of normal schemes, the going down property holds for
$\mathring {T}\to \mathring {S}$
by [19, Proposition 00H8]. Hence, V contains all points on T of codimension
$1$
, and so the restriction map
$$\begin{align*}\mathrm{Hom}(\mathcal{E}_1|_T, \mathcal{E}_2|_T)\to \mathrm{Hom}(\mathcal{E}_1|_V, \mathcal{E}_2|_V) \end{align*}$$
is bijective.
Let
$T':=T\times _S T$
and
$V':=V\times _{U} V$
. Note that
$T'$
is not necessarily normal. Since there are isomorphisms
$$ \begin{align*} T'&\cong S\times_{\mathbb{A}_{P}} \mathbb{A}_{P^{1/n}}\times_{\mathbb{A}_{P}} \mathbb{A}_{P^{1/n}} \\ &\cong S\times_{\mathbb{A}_{P}} \mathbb{A}_{P^{1/n}}\times \mathbb{A}_{P^{\mathrm{gp}}/nP^{\mathrm{gp}}} \\ &\cong T\times \mathbb{A}_{P^{\mathrm{gp}}/nP^{\mathrm{gp}}}, \end{align*} $$
natural projection morphisms
$T'\to T$
are strict finite free. Hence, by Lemma 4.7, the restriction map
$$\begin{align*}\mathrm{Hom}(\mathcal{E}_1|_{T'}, \mathcal{E}_2|_{T'})\to \mathrm{Hom}(\mathcal{E}_1|_{V'}, \mathcal{E}_2|_{V'}) \end{align*}$$
is bijective. Then the statement follows from the following diagram, in which both rows are exact:
The following proposition is a log version of the theorem of Tate and de Jong ([Reference Tate18, Theorem 4] and [Reference de Jong3, Corollary 1.2]). This is proved in the case where
$\mathring {S}$
is the spectrum of a discrete valuation ring by Bertapelle–Wang–Zhao in [Reference Bertapelle, Wang and Zhao1, Theorem 5.19]. In general, we use the reduction to one-dimensional cases.
Proposition 4.9. Let S be a log regular fs log scheme and U be a dense open subset of S. Then the restriction functor
$$\begin{align*}(\mathrm{BT}/S)_{\mathrm{d}}\to (\mathrm{BT}/U)_{\mathrm{d}} \end{align*}$$
is fully faithful.
Proof. Let G and H be log p-divisible groups over S. We shall prove the natural map
$$\begin{align*}\mathrm{Hom}(G,H)\to \mathrm{Hom}(G|_{U},H|_{U}) \end{align*}$$
is an isomorphism. The injectivity follows from Proposition 4.8 (1). To show the surjectivity, we take a homomorphism
$f_{U}\colon G|_{U}\to H|_{U}$
. Fix an integer
$n\geq 1$
. By [Reference Bertapelle, Wang and Zhao1, Theorem 5.19] and Lemma 4.4, there exists an open subset V of S containing U and all points of codimension
$1$
such that the homomorphism
$f_{U,n}\colon G[p^{n}]|_{U}\to H[p^{n}]|_{U}$
induced from
$f_{U}$
extends to a homomorphism
$f_{V,n}\colon G[p^{n}]|_{V}\to H[p^{n}]|_{V}$
. Furthermore,
$f_{V,n}$
extends to a homomorphism
$f_{n}\colon G[p^{n}]\to H[p^{n}]$
by Proposition 4.8 (2). By Proposition 4.8 (1),
$f_{n}$
is the unique extension of
$f_{U,n}$
, and so, when n varies, the system
$\{f_{n}\}_{n\geq 1}$
gives a homomorphism
$f\colon G\to H$
restricting to
$f_{U}$
.
Proposition 4.10. Let S be a log regular fs log scheme and U be a dense open subset of S. Let
$f\colon G_{1}\to G_{2}$
be a homomorphism of log p-divisible groups over S. Suppose that
$f|_{U}\colon G_{1}|_{U}\to G_{2}|_{U}$
is an isogeny. Then f itself is an isogeny.
Proof. We may assume that S is quasi-compact. Take an integer
$d\geq 1$
such that
$\mathrm {Ker}(f|_{U})$
is killed by
$p^{d}$
. Since
$f|_{U}$
is surjective, there exists a unique homomorphism
$g_{U}\colon G_{2}|_{U}\to G_{1}|_{U}$
such that
$f|_{U}\circ g_{U}$
and
$g_{U}\circ f|_{U}$
are the multiplication by
$p^{d}$
. By Proposition 4.9,
$g_{U}$
uniquely extends to a homomorphism
$g\colon G_{2}\to G_{1}$
, and both
$fg$
and
$gf$
are the multiplication by
$p^{d}$
. In particular,
$\mathrm {Ker}(f)$
is killed by
$p^{d}$
. Therefore, it follows from Proposition 2.33 that f is an isogeny.
5. Slope filtrations of log p-divisible groups
The goal of this section is to prove Theorem 5.4. We start with some technical lemmas.
Lemma 5.1. Let
$f\colon Y\to X$
be a faithfully flat morphism of locally Noetherian normal schemes. Let U be a dense open subset of X containing all points on X of codimension
$1$
, and 
. Let
$G_{U}$
be a finite locally free group scheme over U. Suppose that 
extends to a finite locally free group scheme
$G_{Y}$
over Y. Then
$G_{U}$
itself uniquely extends to a finite locally free group scheme over X.
Proof. Let
$j\colon U\hookrightarrow X$
and
$j'\colon V\hookrightarrow Y$
denote the natural open immersions. Let
$f_{U}\colon V\to U$
be the restriction of f. For a finite locally free group scheme H over a scheme S, we write
$\mathcal {A}_{H}$
for the Hopf algebra object in
$\mathcal {LF}(S)$
corresponding to H (cf. Proposition 2.16). By the flat base change theorem, there are isomorphisms
$$ \begin{align} f^{*}j_{*}\mathcal{A}_{G_{U}}\cong j^{\prime}_{*}(f_{U})^{*}\mathcal{A}_{G_{U}}\cong j^{\prime}_{*}\mathcal{A}_{G_{V}}. \end{align} $$
Since f has the going down property by the flatness of f, the open subset V contains all points on Y of codimension
$1$
. Hence, there are isomorphisms
$$ \begin{align} j^{\prime}_{*}\mathcal{A}_{G_{V}}\cong j^{\prime}_{*}{j'}^{*}\mathcal{A}_{G_{Y}}\cong \mathcal{A}_{G_{Y}}, \end{align} $$
by the normality of Y. It follows from (5.1) and (5.2) that
$f^{*}j_{*}\mathcal {A}_{G_{U}}$
is a vector bundle on Y, and so
$j_{*}\mathcal {A}_{G_{U}}$
is also a vector bundle on X. Since the restriction functor
$\mathcal {LF}(X)\to \mathcal {LF}(U)$
is fully faithful, the Hopf algebra structure on
$\mathcal {A}_{G_{U}}$
uniquely induces a Hopf algebra structure on
$j_{*}\mathcal {A}_{G_{U}}$
, which corresponds to the desired finite locally free group scheme over X extending
$G_{U}$
.
Lemma 5.2. Let X be a locally Noetherian normal scheme and U be a dense open subset of X containing all points of codimension
$1$
. Let
$G'\to G\to G"$
be a sequence of finite locally free group schemes over X. Suppose that
$G"$
is finite étale over X, and that the sequence
$$\begin{align*}0\to G'|_{U}\to G|_{U}\to G"|_{U}\to 0 \end{align*}$$
is exact. Then the sequence
$$\begin{align*}0\to G'\to G\to G"\to 0 \end{align*}$$
is also exact.
Proof. Since
$G"$
is finite étale over X, the morphism
$G\to G"$
is finite flat. The image of
$G\to G"$
is closed and contains a dense open subset
$G^{\prime \prime }_{U}=G\times _{X} U$
, and so
$G\to G"$
is faithfully flat. What remains to be proved is that the composite
$G'\to G\to G"$
is a zero map and that the induced homomorphism
$G'\to \mathrm {Ker}(G\to G")$
is an isomorphism. Both of them follow from the fully faithfulness of the restriction functor
$(\mathrm {fin}/X)\to (\mathrm {fin}/U)$
(Lemma 4.7).
Proposition 5.3. Let S be a log regular fs log scheme over
$\mathbb {F}_{p}$
with
$\mathring {S}$
being regular, and let U be a dense open subset of S containing all points on S of codimension
$1$
. Let
$G_{U}$
be a completely slope divisible log p-divisible group over U. Then
$G_{U}$
uniquely extends to a log p-divisible group over S.
Proof. Since
$G_{U}$
is completely slope divisible, there exists an exact sequence
$$\begin{align*}0\to G_{U}^{0}\to G_{U}\to G_{U}^{\acute{\rm e}\mathrm{t}}\to 0 \end{align*}$$
as in Proposition 3.4. By [Reference Zink21, Proposition 14],
$G_{U}^{0}$
(resp.
$G_{U}^{\acute{\rm e}\mathrm{t}}$
) extends to a formal Lie group
$H_{1}$
(resp. an étale p-divisible group
$H_{2}$
) over
$\mathring {S}$
.
Note that
$\mathcal {M}_{S,\bar {s}}/\mathcal {O}_{S, \bar {s}}^{\times }$
is a free monoid for all
$s\in S$
by [Reference Ogus14, Chapter III, Theorem 1.11.6]. It is enough to prove the following two claims:
-
1.
$G_{U}[p^{n}]$
uniquely extends to a log finite group scheme
$G_{n}$
over S for each
$n\geq 1$
; -
2.
is a log p-divisible group over S.
is a log p-divisible group over S.Hence, the limit argument (Proposition 2.14) allows us to assume that
$\mathring {S}=\mathrm {Spec}R$
for a strict local ring R. Let
$s\in \mathring {S}$
be the unique closed point. Take a chart
$\mathbb {N}^{r}\to \mathcal {M}_{S,s}$
which is neat at s. Let
$\alpha \colon \mathbb {N}^{r}\to R$
denote the composite map
$\mathbb {N}^{r}\to \mathcal {M}_{S,s}\to R$
. By the log regularity of S, the sequence
$(\alpha (e_{1}),\dots ,\alpha (e_{r}))$
can be extended to a system of parameter of R, where
$e_{i}$
is the ith standard basis of
$\mathbb {N}^{r}$
for
$1\leq i\leq r$
.
We shall prove the claim (1) for a fixed
$n\geq 1$
. By Lemma 2.13, there exists an integer
$m\geq 1$
such that
$G_{U}[p^{n}]|_{V}$
is classical, where we set 
and V to be the preimage of U by
$T\to S$
. Since the sequence
$(\alpha (e_{1}),\dots ,\alpha (e_{r}))$
can be extended to a system of parameter of R, the scheme
$\mathring {T}$
is regular. We have an exact sequence of finite locally free group schemes over
$\mathring {V}$
$$ \begin{align} 0\to H_{1}[p^{n}]|_{V}\to G_{U}[p^{n}]|_{V}\to H_{2}[p^{n}]|_{V}\to 0. \end{align} $$
By the argument in [Reference Zink21, p. 7] (or the fifth paragraph of the proof of [Reference Zink21, Proposition 14]), this sequence splits after taking the pullback by the kth power of the Frobenius morphism
$F_{\mathring {V}}^{k}\colon \mathring {V}\to \mathring {V}$
(which is an fpqc covering by the regularity of
$\mathring {V}$
) for some integer
$k\geq 1$
. In particular, the pullback of
$G_{U}[p^{n}]|_{V}$
by
$F_{\mathring {V}}^{k}$
extends to a finite locally free group scheme over
$\mathring {T}$
. Hence, by Lemma 5.1,
$G_{U}[p^{n}]|_{V}$
itself also extends to a finite locally free group scheme
$G_{n,T}$
over
$\mathring {T}$
. By Lemma 5.2, the exact sequence (5.3) uniquely extends to an exact sequence of finite locally free group schemes over
$\mathring {T}$
$$ \begin{align} 0\to H_{1}[p^{n}]|_{T}\to G_{n,T}\to H_{2}[p^{n}]|_{T}\to 0. \end{align} $$
Since projection morphisms
$T\times _{S} T\to T$
are strict finite free (see the proof of Proposition 4.8), the descent datum on
$G_{U}[p^{n}]|_{V}$
over
$V\times _{U} V$
uniquely extends to the descent datum on
$G_{n,T}$
over
$T\times _{S} T$
by Lemma 4.7. Therefore,
$G_{n,T}$
descends to an object
$G_{n}\in (\mathrm {fin}/S)_{\mathrm {f}}$
by Kummer log flat descent. Since the exact sequence (5.4) descends to an exact sequence
$$ \begin{align} 0\to H_{1}[p^{n}]\to G_{n}\to H_{2}[p^{n}]\to 0, \end{align} $$
$G_{n}$
is a log finite group scheme over S by [Reference Kato11, Proposition 2.3]. This proves the claim (1).
By construction,
$G_{n}$
is killed by
$p^{n}$
for any integer
$n\geq 1$
. Due to Proposition 4.8 (2), the sequence
$G_{U}[p]\hookrightarrow G_{U}[p^{2}]\hookrightarrow \dots $
uniquely extends to a sequence
$$\begin{align*}G_{1}\to G_{2}\to \dots. \end{align*}$$
To prove the claim (2), it is enough to prove that the following claims are true:
-
•
$G_{n}\to G_{n+1}$
is an injection for any integer
$n\geq 1$
; -
• the sequence
is exact.
$$\begin{align*}0\to G_{n}\to G_{n+m}\stackrel{\times p^{n}}{\to} G_{m}\to 0 \end{align*}$$
These claims follow from the exact sequence (5.5) and the fact that similar properties hold for systems
$\{H_{1}[p^{n}]\}$
and
$\{H_{2}[p^{n}]\}$
.
Theorem 5.4. Let S be a locally Noetherian log regular fs log scheme over
$\mathbb {F}_p$
and U be a dense open subscheme of S. Let G be a log p-divisible group over S. Suppose that we are given a completely slope divisible log p-divisible group
$H_{U}$
over U and an isogeny
$f_{U}\colon G|_{U}\to H_{U}$
. Then there exist a completely slope divisible log p-divisible group H over S and an isogeny
$f\colon G\to H$
extending
$f_{U}$
.
Proof. By the limit argument (Proposition 2.14), we may assume that the
$\mathring {S}$
is the spectrum of a strict local ring with the unique closed point s. Fix a chart
$P\to \mathcal {M}_{S}$
which is neat at s and an integer
$d\geq 1$
with
$\mathrm {Ker}(f_{U})\subset G[p^d]|_{U}$
. By Lemma 2.13, there exists an integer
$n\geq 1$
such that
$G[p^d]|_{T}$
is classical, where we set 
. Let V be the preimage of U by
$T\to S$
.
Then, Nizioł’s desingularization theorem [Reference Nizioł12, Theorem 5.10] gives a log regular fs log scheme
$T'$
with
$\mathring {T'}$
being regular and a proper birational morphism
$\pi \colon T'\to T$
. By Proposition 4.5 and the limit argument, there exist an open subset
$V'$
containing
$\pi ^{-1}(V)$
and all points on
$T'$
of codimension
$1$
and an isogeny
$f^{\prime }_{V'}\colon G|_{V'}\to H_{V'}$
extending
$f|_{\pi ^{-1}(V)}$
. The log p-divisible group
$H_{V'}$
uniquely extends to a log p-divisible group
$H_{T'}$
over
$T'$
by Proposition 5.3, and the isogeny
$f^{\prime }_{V'}$
uniquely extends to an isogeny
$f'\colon G|_{T'}\to H_{T'}$
by Propositions 4.9 and 4.10.
We shall prove that
$f'$
descends to T by using the method of Oort–Zink (see [Reference Oort and Zink17, Proposition 2.7]). Let
$\mathcal {M}\to T$
be the moduli (non-log) scheme of isogenies from
$G|_{T}$
whose kernel is contained in
$G[p^d]|_{T}$
. The isogeny
$f'$
induces a morphism
$\varphi \colon T'\to \mathcal {M}$
of schemes. It suffices to prove that
$\varphi $
factors as
$T'\to T\to \mathcal {M}$
. Since
$\mathcal {O}_{T}\cong \pi _{*}\mathcal {O}_{T'}$
by the normality of T, it is enough to show that, for
$x\in T$
, the morphism
$\varphi $
maps
$\pi ^{-1}(x)$
to a single point. This follows from Lemma 3.8 and the connectivity of
$\pi ^{-1}(x)$
.
As a result, we obtain a log p-divisible group
$H_{T}$
over T with
$(H_{T})|_{T'}\cong H_{T'}$
and an isogeny
$f_{T}\colon G|_{T}\to H_{T}$
with
$(f_{T})|_{T'}=f'$
. Since
$\pi $
is birational,
$f_{T}$
is the extension of
$(f_{U})|_{V}$
by Proposition 4.8 (1). Hence, by Lemma 4.7, the two pullbacks of
$\mathrm {Ker}(f_{T})$
by two projection morphisms
$T\times _{S} T\to T$
are equal as subobjects of
$G[p^{d}]|_{T\times _{S} T}$
. Therefore, by Kummer log flat descent,
$\mathrm {Ker}(f_{T})$
descends to a log finite subgroup of
$G[p^{d}]$
. Let 
and
$f\colon G\to H$
be the natural surjection. Then the isogeny f restricts to the given isogeny
$f_{U}$
. By Corollary 3.6, H is completely slope divisible. This proves the statement.
A. Limit arguments for log schemes
In this appendix, we shall prove some fundamental results on limits of log schemes which are well-known to experts. We note that Lemma A.2 for fine log schemes can be deduced easily from the fact that Olsson’s stack
$\mathcal {L}og_{S}$
is of finite presentation [Reference Olsson15, Theorem 1.1]. Here, we shall give a more direct proof.
Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of coherent log schemes. We say that
$\{ S_{i}\}$
satisfies the condition
$(\ast )$
if the following conditions are satisfied:
-
• The log scheme
$S_i$
is quasi-compact and quasi-separated for all
$i\in I$
. -
• An arbitrary transition morphism
$S_i\to S_j$
is affine and strict.
When
$\{ S_{i}\}$
satisfies the condition
$(\ast )$
, there exists a coherent log scheme S which is the limit of
$\{S_{i}\}$
in the category of log schemes, and S is described explicitly as follows. The underlying scheme
$\mathring {S}$
is the limit of the underlying schemes
$\varprojlim _{i\in I}\mathring {S_{i}}$
. The log structure
$\mathcal {M}_{S}$
is the pullback log structure of
$\mathcal {M}_{S_{i}}$
by the natural projection morphism
$S\to S_{i}$
for some
$i\in I$
, which is independent of the choice of
$i\in I$
by the condition
$(\ast )$
. Obviously, if
$S_{i}$
is fine (resp. fs) for every
$i\in I$
, the log scheme S is also fine (resp. fs).
Lemma A.1. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of coherent log schemes satisfying the condition
$(\ast )$
. Set 
. Then the canonical morphism
$$\begin{align*}\varinjlim_{i\in I} \Gamma(S_i, \mathcal{M}_{S_i})\to \Gamma(S, \mathcal{M}_{S}) \end{align*}$$
is an isomorphism.
Proof. We have the following diagram:
where both rows are exact. Here, note that
$p_{i}^{-1}\mathcal {M}_{S_{i}}$
is not the pullback log structure but the inverse image of
$\mathcal {M}_{S_{i}}$
by
$p_{i}$
as a sheaf. Hence, the canonical morphism
$\varinjlim p_{i}^{-1}\mathcal {M}_{S_{i}}\to \mathcal {M}_{S}$
is an isomorphism.
For a quasi-compact and quasi-separated étale morphism
$U\to S$
, we define a monoid
$\mathcal {M}^{\prime }_{S}(U)$
as follows. We take
$j\in I$
, a quasi-compact and quasi-separated étale morphism
$U_{j}\to S_{j}$
, and an isomorphism
$U\cong U_{j}\times _{S_{j}} S$
over S. Here, we put
$\mathcal {M}^{\prime }_{S}(U):=\displaystyle \varinjlim _{i\geq j} \mathcal {M}_{S_{i}}(U_{j}\times _{S_{j}} S_{i})$
. This definition is independent of the choice of j. Then
$U\mapsto \mathcal {M}^{\prime }_{S}(U)$
defines a presheaf
$\mathcal {M}^{\prime }_{S}$
on the category of quasi-compact and quasi-separated étale morphisms to S. Since
$\mathcal {M}^{\prime }_{S}$
satisfies the sheaf condition for quasi-compact and quasi-separated étale coverings,
$\mathcal {M}^{\prime }_{S}$
uniquely extends to an étale sheaf on S which is also denoted by
$\mathcal {M}^{\prime }_{S}$
. It follows from the definition that
$\mathcal {M}^{\prime }_{S}$
satisfies the same universal mapping property as
$\varinjlim p_{i}^{-1}\mathcal {M}_{S_{i}}$
. Hence, the natural morphism
$\varinjlim p_{i}^{-1}\mathcal {M}_{S_{i}}\to \mathcal {M}^{\prime }_{S}$
is an isomorphism. The isomorphisms
$$\begin{align*}\mathcal{M}_{S}\cong \varinjlim p_{i}^{-1}\mathcal{M}_{S_{i}}\cong \mathcal{M}^{\prime}_{S} \end{align*}$$
give the desired isomorphism
$\varinjlim \Gamma (S_i, \mathcal {M}_{S_i})\cong \Gamma (S, \mathcal {M}_{S}).$
Lemma A.2. Let
$\{S_{i}\}_{i\in I}$
be a cofiltered system of quasi-compact and quasi-separated schemes in which every transition morphism is affine. We put 
.
(1) Suppose that the cofiltered category I has a final object
$0$
. Let
$\mathcal {M}_{0}, \mathcal {N}_{0}$
be log structures on
$S_{0}$
. Let
$\mathcal {M}_{i}$
(resp.
$\mathcal {N}_{i}$
) denote the pullback log structure of
$\mathcal {M}_{0}$
(resp.
$\mathcal {N}_{0}$
) by
$S_{i}\to S_{0}$
, and let
$\mathcal {M}$
(resp.
$\mathcal {N}$
) denote the pullback log structure of
$\mathcal {M}_{0}$
(resp.
$\mathcal {N}_{0}$
) by
$S\to S_{0}$
. Suppose that
$\mathcal {M}_{0}$
is coherent. Then the natural map
$$\begin{align*}\varinjlim_{i\in I} \mathrm{Hom}_{S_i}(\mathcal{M}_{i},\mathcal{N}_{i}) \to \mathrm{Hom}_{S}(\mathcal{M},\mathcal{N}) \end{align*}$$
is bijective, where
$\mathrm {Hom}_{S_i}(\mathcal {M}_{i},\mathcal {N}_{i})$
(resp.
$\mathrm {Hom}_{S}(\mathcal {M},\mathcal {N})$
) is the set of morphisms of log structures
$\mathcal {M}_{i}\to \mathcal {N}_{i}$
(resp.
$\mathcal {M}\to \mathcal {N}$
).
(2) Let
$\mathcal {M}$
be a coherent (resp. fine) (resp. fs) log structure on S. Then there exist
$i\in I$
and a coherent (resp. fine) (resp. fs) log structure
$\mathcal {M}_{i}$
on
$S_{i}$
such that
$\mathcal {M}$
is isomorphic to the pullback log structure of
$\mathcal {M}_{i}$
.
Proof. (1) By working étale locally on
$S_{0}$
, we may assume that there are a finitely generated monoid P and a chart
$P\to \mathcal {M}_{0}$
. Then the map in the statement factors as follows:
$$ \begin{align*} \displaystyle \varinjlim \mathrm{Hom}_{S_{i}}(\mathcal{M}_{i}, \mathcal{N}_{i}) &\cong \varinjlim \mathrm{Hom}(P,\Gamma(S_{i},\mathcal{N}_{i})) \to \mathrm{Hom}(P,\varinjlim \Gamma(S_{i},\mathcal{N}_{i})) \to \mathrm{Hom}(P,\Gamma(S,\mathcal{N})) \\ &\cong \mathrm{Hom}_{S}(\mathcal{M},\mathcal{N}). \end{align*} $$
Here, the second morphism is bijective by the fact that P is finitely presented, and the third morphism is bijective by Lemma A.1. This proves (1).
(2) We prove only the assertion for coherent log structures because other assertions follow from the same argument. First, we consider the case, where
$\mathcal {M}$
admits a chart
$P\to \mathcal {M}$
for a finitely generated monoid P. Since P is finitely presented and
$\varinjlim \Gamma (S_{i},\mathcal {O}_{S_{i}})$
is isomorphic to
$\Gamma (S,\mathcal {O}_{S})$
, the morphism
$P\to \mathcal {M}\to \mathcal {O}_{T}$
descends to a morphism
$P\to \mathcal {O}_{T_{i}}$
for some
$i\in I$
. Then the associated log structure to the morphism
$P\to \mathcal {O}_{T_{i}}$
is a desired one.
We consider the general case. Take a quasi-compact and quasi-separated étale covering
$\pi \colon U\to T$
such that 
admits a chart. For some
$i\in I$
, the morphism
$\pi $
descends to an étale covering
$\pi _{i}\colon U_{i}\to T_{i}$
and
$\mathcal {M}_{U}$
descends to a coherent log structure
$\mathcal {M}_{U_{i}}$
on
$U_{i}$
. Let
$p_{k}\colon U\times _{S} U\to U$
and
$p_{i,k}\colon U_{i}\times _{S_{i}} U_{i}\to U_{i}$
be natural projection morphisms for
$k=1,2$
. Due to the assertion (1), we may assume that the isomorphism
$p_{1}^{*}\mathcal {M}_{U}\cong p_{2}^{*}\mathcal {M}_{U}$
descends to an isomorphism
$p_{i,1}^{*}\mathcal {M}_{U_{i}}\cong p_{i,2}^{*}\mathcal {M}_{U_{i}}$
satisfying the cocycle condition over
$U_{i}\times _{S_{i}} U_{i}\times _{S_{i}} U_{i}$
by replacing i with a bigger one. This descent datum gives a coherent log structure
$\mathcal {M}_{i}$
on
$S_{i}$
such that the pullback log structure of
$\mathcal {M}_{i}$
to S is isomorphic to
$\mathcal {M}$
.
Proposition A.3. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of coherent log schemes satisfying the condition
$(\ast )$
. We put 
.
(1) Suppose that the cofiltered category I has a final object
$0$
. Let
$X_{0},Y_{0}$
be coherent log schemes over
$S_{0}$
. Suppose that
$Y_{0}$
is of finite presentation over
$S_{0}$
. Then the natural map
$$\begin{align*}\varinjlim_{i\in I} \mathrm{Mor}_{S_{i}}(X_{0}\times_{S_{0}}S_{i}, Y_{0}\times_{S_{0}} S_i)\to \mathrm{Mor}_{S}(X_{0}\times_{S_{0}}S, Y_{0}\times_{S_{0}} S) \end{align*}$$
is bijective.
(2) Let X be a coherent (resp. fine) (resp. fs) log scheme of finite presentation over S. Then there exist
$i\in I$
and a coherent (resp. fine) (resp. fs) log scheme
$X_i$
of finite presentation over
$S_i$
with an isomorphism
$X\cong X_{i}\times _{S_{i}} S$
.
Proof. These statements are formal consequences of the limit argument for non-log schemes and Lemma A.2.
Lemma A.4. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of coherent log schemes satisfying the condition
$(\ast )$
. Set 
. Suppose that S admits a chart
$\alpha \colon P\to \mathcal {M}_{S}$
for a finitely generated monoid P. Then
$\alpha $
arises from a chart
$P\to \mathcal {M}_{S_{i}}$
for some
$i\in I$
.
Proof. Since P is finitely presented, the chart
$\alpha $
descends to a monoid map
$P\to \mathcal {M}_{S_{i}}$
for some
$i\in I$
by Lemma A.1. Let
$\mathcal {N}_{i}$
be the log structure associated with the composite
$P\to \mathcal {M}_{S_{i}}\to \mathcal {O}_{S_{i}}$
and let
$\phi _{i}\colon \mathcal {N}_{i}\to \mathcal {M}_{S_{i}}$
be the morphism of coherent log structures induced from
$P\to \mathcal {M}_{S_{i}}$
. Since the pullback of
$\phi _{i}$
to S is an isomorphism, the pullback of
$\phi _{i}$
to
$S_{j}$
is an isomorphism for some
$j\geq i$
by Lemma A.2 (1). Then the monoid map
$P\to \mathcal {M}_{S_{j}}$
induced from
$P\to \mathcal {M}_{S_{i}}$
is the desired chart.
Proposition A.5. Let
$\{ S_{i}\}_{i\in I}$
be a cofiltered system of fs log schemes satisfying the condition (
$\ast $
). Set 
. Suppose that the cofiltered category I has a final object
$0$
. Let
$X_{0}$
be an fs log scheme of finite presentation over
$S_{0}$
. We put
$X_{i}:=X_{0}\times _{S_{0}} S_{i}$
and
$X:=X_{0}\times _{S_{0}} S$
. Let
$f_{i}\colon X_{i}\to S_{i}$
and
$f\colon X\to S$
denote the morphisms induced from
$X_{0}\to S_{0}$
.
Let
$\mathcal {P}$
be one of the following properties of morphisms of fs log schemes. Then, if f satisfies the property
$\mathcal {P}$
, the morphism
$f_{i}$
also satisfies the property
$\mathcal {P}$
for some
$i\in I$
:
-
1. log flat;
-
2. log smooth;
-
3. log étale;
-
4. Kummer log flat.
Proof. These properties have a criterion using a chart of a morphism. For (1), this is just the definition. For (2) and (3), see [Reference Kato8, Theorem 3.5]. For (4), see [Reference Illusie, Nakayama and Tsuji4, Proposition 1.3]. Then the statement can be deduced from these criteria and Lemma A.4.
Acknowledgements
The author would like to thank his advisor, Tetsushi Ito, for useful discussions and warm encouragement. The author is also grateful to Kazuya Kato for useful discussions and to the referee for carefully reading the manuscript and pointing out errors. This work was supported by JSPS KAKENHI Grant Number 23KJ1325 and the Graduate School of Science, Kyoto University under the Ginpu Fund.
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