Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

112 SHINICHI MOCHIZUKI
— which [since Π p0 is commensurably terminal in π 1 ( † D ⊛ ) — cf., e.g., [AbsAnab],
Theorem 1.1.1, (i)] we consider up to the action by the “inner automorphisms of
the pair” arising from conjugation by Π p0 . In the language of [AbsTopIII], §3, this
pair is an “MLF-Galois TM-pair of strictly Belyi type” [cf. [AbsTopIII], Definition
3.1, (ii)].
(vi) Before proceeding, we observe that the discussion of (iv), (v) concerning
† F ⊛ , † D ⊛ may also be carried out for † F ⊚ , † D ⊚ . We leave the routine details to
the reader.
(vii) Next, let us consider the index set J of the capsule of D-prime-strips † D J .
def
∼
For j ∈ J, writeV j = {v j } v∈V . Thus, we have a natural bijection V j → V, i.e.,
given by sending v j ↦→ v. These bijections determine a “diagonal subset”
V 〈J〉 ⊆ V J
def
= ∏ j∈J
V j
— which also admits a natural bijection V 〈J〉
∼
→ V. Thus,weobtainnatural bijections
V 〈J〉
∼
→ Vj
∼
→ Prime( † F ⊚ mod ) ∼ → V mod
for j ∈ J. Write
† F ⊚ 〈J〉
† F ⊚ j
def
= { † F ⊚ mod , V 〈J〉
def
= { † F ⊚ mod , V j
∼
→ Prime( † F ⊚ mod )}
∼
→ Prime( † F ⊚ mod )}
for j ∈ J. That is to say, we think of † F ⊚ j
as a copy of † F ⊚ mod
“situated on” the
constituent labeled j of the capsule † D J ; we think of † F ⊚ 〈J〉 as a copy of † F ⊚ mod
“situated in a diagonal fashion on” all the constituents of the capsule † D J . Thus,
we have a natural embedding of categories
† F ⊚ 〈J〉
↩→ † F ⊚ J
def
= ∏ j∈J
† F ⊚ j
— where, by abuse of notation, we write † F ⊚ 〈J〉
for the underlying category of [i.e.,
the first member of the pair] † F ⊚ 〈J〉 . Here, we observe that the category † F ⊚ J
is not
equipped with a Frobenioid structure. Write
† F ⊚R
j ;
† F ⊚R
〈J〉 ;
† F ⊚R
J
def
= ∏ j∈J
† F ⊚R
j
for the respective realifications [or product of the underlying categories of the realifications]
of the corresponding Frobenioids whose notation does not contain a
superscript “R”. [Here, we recall that the theory of realifications of Frobenioids is
discussed in [FrdI], Proposition 5.3.]
Remark 5.1.1. Thus, † F ⊚ 〈J〉
may be thought of as the Frobenioid associated to
divisors on V J [i.e., finite formal sums of elements of this set with coefficients in Z or