Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

110 SHINICHI MOCHIZUKI
objects of † F ⊛ that lie over Galois objects of † D ⊛ ,weobtainapair [i.e., consisting
of a topological group acting continuously on a discrete abelian group]
π 1 ( † D ⊛ ) Õ⊚×
— which we consider up to the action by the “inner automorphisms of the pair”
arising from conjugation by π 1 ( † D ⊛ ). Write Φ† F for the divisor monoid of the
⊛
Frobenioid † F ⊛ [which is category-theoretic — cf. [FrdI], Corollary 4.11, (iii);
[FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper]. Thus, for each
p ∈ Prime(Φ† F ⊛(A)) [where we use the notation “Prime(−)” as in [FrdI], §0],
the natural homomorphism O × (A birat ) → Φ† F ⊛(A)gp [cf. [FrdI], Proposition 4.4,
(iii)] determines — i.e., by taking the inverse image via this homomorphism of
[the union with {0} of] the subset of Φ† F ⊛(A) constituted by p —asubmonoid
Op
⊲ ⊆O × (A birat ). That is to say, in more intuitive terms, this submonoid is the
submonoid of integral elements of O × (A birat ) with respect to the valuation determined
by p of the number field corresponding to A. Write O p × ⊆Op ⊲ for the
submonoid of invertible elements. Thus, by allowing A to vary among the Frobeniustrivial
objects of † F ⊛ that lie over Galois objects of † D ⊛ and considering the way
in which the natural action of Aut† F ⊛(A) onO× (A birat ) permutes the various submonoids
Op ⊲ , it follows that for each p 0 ∈ Prime(Φ† F ⊛(A 0)), where A 0 ∈ Ob( † F ⊛ )
lies over a terminal object of † D ⊛ ,weobtainaclosed subgroup [well-defined up to
conjugation]
Π p0 ⊆ π 1 ( † D ⊛ )
by considering the elements of Aut† F ⊛(A) thatfix the submonoid O⊲ p ,forsome
system of p’s lying over p 0 . That is to say, in more intuitive terms, the subgroup
Π p0 is simply the decomposition group associated to some v ∈ V mod .Inparticular,
it follows that p 0 is nonarchimedean if and only if the p-cohomological dimension
of Π p0 is equal to 2 + 1 = 3 for infinitely many prime numbers p [cf., e.g., [NSW],
Theorem 7.1.8, (i)].
(v) We continue to use the notation of (iv). Now if “(−)” is a [commutative]
topological monoid, then let us write
μẐ((−)) def
= Hom(Q/Z, (−))
[cf. [AbsTopIII], Definition 3.1, (v); [AbsTopIII], Definition 5.1, (v)]. Also, let
us write μ Θ(π 1( † D ⊛ )) for the cyclotome “μẐ(Π (−) )” of [AbsTopIII], Theorem 1.9,
Ẑ
which we think of as being applied “via the Θ-approach” [cf. Remark 3.1.2] to
π 1 ( † D ⊛ ); write π1 Θ ( † D ⊚ )=π1 Θ ( † D ⊛ ) ⊆ π 1 ( † D ⊚ ) ⊆ π 1 ( † D ⊛ ) for the closed subgroup
— which is characteristic as a subgroup of π 1 ( † D ⊚ )—determinedby“Δ X ”
[cf. [AbsAnab], Lemma 1.1.4, (i); [AbsTopII], Corollary 3.3, (i), (ii); [AbsTopI],
Example 4.8; [EtTh], Proposition 2.2, (i)]. Next, let us observe that there exists a
unique isomorphism of cyclotomes
μ Θ Ẑ (π 1( † D ⊛ )) ∼ → μẐ(Õ⊚× )
such that the image of the natural injection
Õ ⊚× ↩→ lim −→H H 1 (H, μ Θ Ẑ (π 1( † D ⊛ )))