Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

64 SHINICHI MOCHIZUKI
determines a monoid Φ Cv on [Dv ⊢ , hence, by pull-back via the natural functor D v →
Dv ⊢ ,on]D v ; the assignment
Φ C ⊢ v
: Spec(L) ↦→ ord(Z ⊲ p v
)(⊆ ord(OL ⊲ )pf )
determines an absolutely primitive [cf. [FrdII], Example 1.1, (ii)] submonoid Φ C ⊢
v
Φ Cv | D ⊢
v
on Dv ⊢ ; these monoids Φ C ⊢
v
,Φ Cv determine p v -adic Frobenioids
⊆
C ⊢ v ⊆C v
[cf. [FrdII], Example 1.1, (ii), where we take “Λ” to be Z], whose base categories
are given by D ⊢ v , D v [in a fashion compatible with the natural inclusion D ⊢ v ⊆D v ],
respectively. Also, we shall write
F v
def
= C v
[cf. the notation of Example 3.2, (i)]. Finally, let us observe that the element
p v ∈ Z pv ⊆O ⊲ K v
determines a characteristic splitting
τ ⊢ v
on Cv
⊢ [cf. [FrdII], Theorem 1.2, (v)]. Write Fv
⊢
Frobenioid.
def
=(C ⊢ v ,τ ⊢ v ) for the resulting split
(ii) Next, let us write log(p v )fortheelementp v of (i) considered additively and
consider the monoid on D ⊢ v
O ⊲ C ⊢ v
(−) =O × (−) × (N · log(p
Cv
⊢ v ))
associated to Cv
⊢ [cf. [FrdI], Proposition 2.2]. By replacing “log(p v )” by the formal
symbol “log(p v ) · log(Θ) = log(p log(Θ)
v )”, we obtain a monoid
O ⊲ C Θ v
(−) def
= O × (−) × (N · log(p
Cv
Θ v ) · log(Θ))
[i.e., where O × (−) def
= O × (−)], which is naturally isomorphic to O ⊲ and which
Cv
Θ Cv
⊢ Cv
⊢
arises as the monoid “O ⊲ (−)” of [FrdI], Proposition 2.2, associated to some p v -adic
Frobenioid Cv
Θ with base category Dv
Θ def
= Dv ⊢ equipped with a characteristic splitting
determined by log(p v ) · log(Θ). In particular, we have a natural equivalence
τ Θ v
F ⊢ v
∼
→F Θ v
—whereF Θ v
def
=(C Θ v ,τ Θ v )—ofsplit Frobenioids.
(iii) Here, it is useful to recall that