Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 57
of “K” [cf. Definition 3.1, (c)], that q v admits a 2l-th root in O ⊲ (T Xv )( ∼ = OK ⊲ v
).
Then one computes immediately from the final formula of [EtTh], Proposition 1.4,
(ii), that the value of Θ v
at √ −q v is equal to
q
v
def
= q 1/2l
v ∈O ⊲ (T Xv )
— where the notation “qv
1/2l ” [hence also q ]iscompletely determined up to a
v
μ 2l (T Xv )-multiple. Write Φ Cv for the divisor monoid [cf. [FrdI], Definition 1.1,
(iv)] of the p v -adic Frobenioid C v . Then the image of q determines a constant
v
section [i.e., a sub-monoid on D v isomorphic to N] “log Φ (q )” of Φ Cv . Moreover,
v
the resulting submonoid [cf. Remark 3.2.2 below]
Φ C ⊢ v
def
= N · log Φ (q
v
)| D ⊢ v
⊆ Φ Cv | D ⊢ v
determines a p v -adic Frobenioid with base category given by D ⊢ v [cf. [FrdII], Example
1.1, (ii)]
C ⊢ v (⊆ C v ⊆ F v
→ F ÷ v )
— which may be thought of as a subcategory of C v . Also, we observe that [since the
q-parameter q ∈ K v , it follows that] q determines a μ 2l (−)-orbit of characteristic
v v
splittings [cf. [FrdI], Definition 2.3]
on C ⊢ v .
τ ⊢ v
(v) Next, let us recall that the base field of Ÿ is equal to K v
v [cf. the discussion
of Definition 3.1, (e)]. Write
D Θ v
⊆ (D v )Ÿ
v
for the full subcategory of the category (D v )Ÿ
v
[cf. the notational conventions
concerning categories discussed in §0] determined by the products in D v of Ÿ v
with objects of Dv ⊢ . Thus, one verifies immediately that “forming the product
with Ÿ ” determines a natural equivalence of categories v D⊢ ∼
v →Dv Θ . Moreover, for
A Θ ∈ Ob(Dv Θ ), the assignment
A Θ ↦→O × (T A Θ) · (Θ N v | T A Θ
) ⊆O × (T ÷ A Θ )
determines a monoid O ⊲ (−) on D Θ Cv
Θ v [in the sense of [FrdI], Definition 1.1, (ii)];
write O × (−) ⊆O ⊲ (−) for the submonoid determined by the invertible elements.
Cv
Θ Cv
Θ
Next, let us observe that, relative to the natural equivalence of categories D ⊢ v
∼
→D Θ v