Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

60 SHINICHI MOCHIZUKI
the Frobenioid equipped with its pre-Frobenioid structure” [cf. [FrdI], Definition
1.1, (iv)]. Nevertheless, we chose to apply the theory of [FrdI], [FrdII], and [EtTh]
partly because it simplifies the exposition [i.e., reduces the number of auxiliary
structures that one must carry around], but more importantly because it renders
explicit precisely which structures arising from scheme-theory are “categorically
intrinsic” and which merely amount to “arbitrary, non-intrinsic choices” which,
when formulated intrinsically, correspond to various “indeterminacies”. This explicitness
is of particular importance with respect to phenomena related to the unitlinear
Frobenius functor [cf. [FrdI], Proposition 2.5] and the Frobenioid-theoretic
indeterminacies studied in [EtTh], §5.
Remark 3.2.2. Although the submonoid Φ C ⊢
v
is not “absolutely primitive” in the
sense of [FrdII], Example 1.1, (ii), it is “very close to being absolutely primitive”,
in the sense that [as is easily verified] there exists a positive integer N such that
N · Φ C
⊢
v
is absolutely primitive. This proximity to absolute primitiveness may also
be seen in the existence of the characteristic splittings τv ⊢ .
Remark 3.2.3.
(i) Let α ∈ Aut Dv (Ÿ ). Then observe that α determines, in a natural way, an
v
automorphism α D of the functor Dv ⊢ →D v obtained by composing the equivalence
of categories Dv
⊢ ∼
→Dv
Θ [i.e., which maps Ob(Dv ⊢ ) ∋ A ↦→ A Θ ∈ Ob(Dv Θ )] discussed
in Example 3.2, (v), with the natural functor Dv Θ ⊆ (D v )Ÿ →D v . Moreover,
v
α D induces, in a natural way, an isomorphism α O ⊲ of the monoid O ⊲ (−) on
Cv
Θ
D Θ v
associated to Θ v
in Example 3.2, (v), onto the corresponding monoid on Dv
Θ
associated to the α-conjugate Θ α of Θ . Thus, it follows immediately from the
v v
discussion of Example 3.2, (v), that
α O ⊲
Fv
Θ
— hence also α — induces an isomorphism of the split Frobenioid
associated to Θ v
onto the split Frobenioid Fv
Θα associated to Θ α which v
lies over the identity functor on Dv Θ .
In particular, the expression “Fv Θ , regarded up to the l · Z indeterminacy in Θ v
discussed in Example 3.2, (ii)” may be understood as referring to the various split
Frobenioids “Fv
Θα ”, as α ranges over the elements of Aut Dv (Ÿ ), relative to the
v
identifications given by these isomorphisms of split Frobenioids induced by the
various elements of Aut Dv (Ÿ ). v
(ii) Suppose that A ∈ Ob(D v ) lies in the image of the natural functor Dv
Θ ⊆
(D v )Ÿ →D v ,andthatψ : B → T A is a linear morphism in the Frobenioid F
v
v
.
Then ψ induces an injective homomorphism
O × (T ÷ A ) ↩→ O× (B ÷ )