Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

104 SHINICHI MOCHIZUKI
Remark 4.9.3.
(i) Ultimately, in the theory of the present series of papers [cf. [IUTchIII]],
we would like to apply the mono-anabelian theory of [AbsTopIII] to the various
local and global arithmetic fundamental groups [i.e., isomorphs of Π CK ,Π v for
v ∈ V non ] that appear in a D-ΘNF-Hodge theater [cf. the discussion of Remark
4.3.2]. To do this, it is of essential importance to have available not only the
absolute Galois groups of the various local and global base fields involved, but
also the geometric fundamental groups that lie inside the isomorphs of Π CK ,Π v
involved. Indeed, in the theory of [AbsTopIII], it is precisely the outer Galois action
of the absolute Galois group of the base field on the geometric fundamental group
that allows one to reconstruct the ring structures group-theoretically in a fashion
that is compatible with localization/globalization operations as shown in Fig. 4.3.
Here, we pause to recall that in [AbsTopIII], Remark 5.10.3, (i), one may find a
discussion of the analogy between this phenomenon of “entrusting of arithmetic
moduli” [to the outer Galois action on the geometric fundamental group] and the
Kodaira-Spencer isomorphism of an indigenous bundle — an analogy that
is reminiscent of the discussion of Remark 4.7.2, (iii).
(ii) Next, let us observe that the state of affairs discussed in (i) has important
implications concerning the circumstances that necessitate the use of “X −→v ” [i.e.,
as opposed to “C v ”] in the definition of “D v ” in Examples 3.3, 3.4 [cf. Remark
4.2.1]. Indeed, localization/globalization operations as shown in Fig. 4.3 give rise,
when applied to the various geometric fundamental groups involved, to various
bijections between local and global sets of label classes of cusps. Now suppose that
one uses “C v ” instead of “X −→v ” in the definition of “D v ” in Examples 3.3, 3.4.
Then the existence of v ∈ V of the sort discussed in Remark 4.2.1, together with
the condition of compatiblity with localization/globalization operations as shown in
Fig. 4.3 — where we take, for instance,
(v of Fig. 4.3)
(v ′ of Fig. 4.3)
def
= (v of Remark 4.2.1)
def
= (w of Remark 4.2.1)
— imply that, at a combinatorial level, one is led, in effect, to a situation of the
sort discussed in Remark 4.9.2, (i), i.e., a situation involving nonsynchronized
labels [cf. Fig. 4.5], which, as discussed in Remark 4.9.2, (i), is incompatible with
the construction of the crucial bijection † ζ of Proposition 4.7, (iii), an object which
will play an important role in the theory of the present series of papers.
Definition 4.10. Let C be a category, n a positive integer. Then we shall refer
to as a procession of length n, orn-procession, ofC any diagram of the form
P 1 ↩→ P 2 ↩→ ... ↩→ P n
—whereeachP j [for j =1,... ,n]isaj-capsule [cf. §0] of objects of C; each
arrow P j ↩→ P j+1 [for j =1,... ,n− 1] denotes the collection of all capsule-full