Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

120 SHINICHI MOCHIZUKI
isomorphisms † D ⊚ →D ∼ ⊚ that maps δ ↦→ [ɛ] ∈ LabCusp(D ⊚ ). We shall refer to
as a δ-valuation ∈ V( † D ⊚ ) [cf. Definition 4.1, (v)] any element that maps to an
element of V ±un [cf. Example 4.3, (i)] via this Aut ɛ ( † D ⊚ )-orbit of isomorphisms.
Note that the notion of a δ-valuation may also be defined intrinsically by means
of the structure of D-NF-bridge † φ NF
. Indeed, [one verifies immediately that] a
δ-valuation maybedefinedasavaluation∈ V( † D ⊚ ) that lies in the “image” [in
the evident sense] via † φ NF
of a D-prime-strip † D j of the capsule † D J such that
the map LabCusp( † D ⊚ ) → LabCusp( † D j ) induced by † φ NF
maps δ to the element
of LabCusp( † D j )thatis“labeled 1”, relative to the bijection of the second display
of Proposition 4.2.
(iv) We continue to use the notation of (iii). Then let us observe that by
localizing at the various δ-valuations ∈ V( † D ⊚ ), one may construct, in a natural
way, an F-prime-strip
† F ⊚ | δ
— which is well-defined up to isomorphism —from † F ⊚ [i.e., in the notation of
Example 5.1, (iv), from Õ⊚× , equipped with its natural π 1 ( † D ⊚ )-action]. Indeed, at
a nonarchimedean δ-valuation v, this follows by considering the p v -adic Frobenioids
[cf. Remark 3.3.2] associated to the restrictions to Π p0
⋂
π1 ( † D ⊚ )(⊆ π 1 ( † D ⊚ ) ⊆
π 1 ( † D ⊛ )) of the pairs
“Π p0 Õ⊲̂p 0
”
of Example 5.1, (v) [cf. also Example 5.1, (vi)]. On the other hand, at an
archimedean δ-valuation v, this follows by applying the functorial algorithm for
reconstructing the Aut-holomorphic orbispace at v given in [AbsTopIII], Corollaries
2.8, 2.9, together with the discussion concerning the “natural injection Õ⊚× ↩→
lim −→H
H1 (H, μ Θ Ẑ (π 1( † D ⊛ )))” in Example 5.1, (v) [cf. also Example 5.1, (vi)]. Here,
we observe that since the natural projection map V ±un → V mod fails to be injective,
in order to relate the restrictions obtained at different elements in a fiber of this
map in a well-defined fashion, it is necessary to regard † F ⊚ | δ as being well-defined
only up to isomorphism. Nevertheless, despite this indeterminacy inherent in the
definition of † F ⊚ | δ , it still makes sense to define, for an F-prime-strip ‡ F with
underlying D-prime-strip ‡ D [cf. Remark 5.2.1, (i)], a poly-morphism
‡ F → † F ⊚
to be a full poly-isomorphism ‡ F ∼ → † F ⊚ | δ for some δ ∈ LabCusp( † D ⊚ ) [cf. Definition
4.1, (vi)]. Moreover, it makes sense to pre-compose such poly-morphisms with
isomorphisms of F-prime-strips and to post-compose such poly-morphisms with isomorphisms
between isomorphs of † F ⊚ . Here, we note that such a poly-morphism
‡ F → † F ⊚ may be thought of as “lying over” an induced poly-morphism ‡ D → † D ⊚
[cf. Definition 4.1, (vi)]. Also, we observe that any poly-morphism ‡ F → † F ⊚ is
fixed by pre-composition with automorphisms of ‡ F, as well as by post-compostion
with automorphisms ∈ Aut ɛ ( † F ⊚ ). In a similar vein, if { e F} e∈E is a capsule of
F-prime-strips whose associated capsule of D-prime-strips [cf. Remark 5.2.1, (i)]
we denote by { e D} e∈E , then we define a poly-morphism
{ e F} e∈E → † F ⊚ (respectively, { e F} e∈E → † F)