Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

68 SHINICHI MOCHIZUKI
Remark 3.4.2.
One way to think of the Kummer structure
κ v : O ⊲ (C v ) ↩→ A Dv
discussed in Example 3.4, (i), is as follows. In the terminology of [AbsTopIII], Definition
2.1, (i), (iv), the structure of CAF on A Dv determines, via pull-back by κ v ,
an Aut-holomorphic structure on the groupification O ⊲ (C v ) gp of O ⊲ (C v ), together
with a [tautological!] co-holomorphicization O ⊲ (C v ) gp →A Dv . Conversely, if one
starts with this Aut-holomorphic structure on [the groupification of] the topological
monoid O ⊲ (C v ), together with the co-holomorphicization O ⊲ (C v ) gp →A Dv ,then
one verifies immediately that one may recover the inclusion of topological monoids
κ v . [Indeed, this follows immediately from the elementary fact that every holomorphic
automorphism of the complex Lie group C × that preserves the submonoid of
elements of norm ≤ 1 is equal to the identity.] That is to say, in summary,
the Kummer structure κ v is completely equivalent to the collection
of data consisting of the Aut-holomorphic structure [induced by κ v ]on
the groupification O ⊲ (C v ) gp of O ⊲ (C v ), together with the co-holomorphicization
[induced by κ v ] O ⊲ (C v ) gp →A Dv .
The significance of thinking of Kummer structures in this way lies in the observation
that [unlike inclusions of topological monoids!]
the co-holomorphicization induced by κ v is compatible with the logarithm
operation discussed in [AbsTopIII], Corollary 4.5.
Indeed, this observation may be thought of as a rough summary of a substantial
portion of the content of [AbsTopIII], Corollary 4.5. Put another way, thinking of
Kummer structures in terms of co-holomorphicizations allows one to separate out
the portion of the structures involved that is not compatible with this logarithm
operation — i.e., the monoid structures! — from the portion of the structures
involved that is compatible with this logarithm operation — i.e., the tautological
co-holomorphicization.
Example 3.5.
Global Realified Frobenioids.
(i) Write
C ⊩ mod
for the realification [cf. [FrdI], Theorem 6.4, (ii)] of the Frobenioid of [FrdI], Example
6.3 [cf. also Remark 3.1.5 of the present paper], associated to the number field
F mod and the trivial Galois extension of F mod [so the base category of Cmod ⊩ is,
in the terminology of [FrdI], equivalent to a one-morphism category]. Thus, the
divisor monoid Φ C
⊩ of C ⊩
mod
mod
may be thought of a single abstract monoid, whose
set of primes, which we denote Prime(Cmod ⊩ )[cf. [FrdI],§0], is in natural bijective
correspondence with V mod [cf. the discussion of [FrdI], Example 6.3]. Moreover,
the submonoid Φ C
⊩
mod ,v of Φ C
⊩ corresponding to v ∈ V mod is naturally isomorphic
mod
to ord(O ⊲ (F mod ) v
) pf ⊗ R ≥0 ( ∼ = R ≥0 ) [i.e., to ord(O ⊲ (F mod ) v
)( ∼ = R ≥0 )ifv ∈ V arc
mod
]. In