Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 71
by [the positive real number] e v . Next, suppose that v ∈ V arc ; then let us recall
from [AbsTopIII], Proposition 5.8, (vi), that [since, by definition, D ⊢ v ∈ Ob(TM ⊢ )]
one may construct algorithmically from D ⊢ v a topological monoid isomorphic to R ≥0
(R ⊢ ≥0) v
[i.e., the topological monoid determined by the nonnegative elements of the ordered
topological group “R arc (G)” of loc. cit.] equipped with a distinguished “Frobenius
element” ∈ (R ⊢ ≥0 ) v; we shall write log D Φ(p v ) ∈ (R ⊢ ≥0 ) v for the result of dividing this
Frobenius element by [the positive real number] 2π. In particular, for every v ∈ V,
we obtain a uniquely determined isomorphism of topological monoids [which are
isomorphic to R ≥0 ]
ρ D v
:Φ D
⊩
mod ,v
∼
→ (R
⊢
≥0 ) v
by assigning log D mod(p v ) ↦→
1
[K v :(F mod ) v ] logD Φ(p v ). Thus, we obtain data [consisting of
a Frobenioid, a bijection of sets, a collection of data indexed by V, and a collection
of isomorphisms indexed by V]
F ⊩ D
def
= (D ⊩ mod, Prime(D ⊩ mod) ∼ → V, {D ⊢ v } v∈V , {ρ D v } v∈V )
[where we apply the natural bijection V ∼ → V mod ], which, by [AbsTopIII], Proposition
5.8, (iii), (vi), may be reconstructed algorithmically from the data {D ⊢ v } v∈V .
Remark 3.5.1.
(i) The formal symbol “log(Θ)” may be thought of as the result of identifying
the various formal quotients “log(Θ v
)/log Φ (q
v
)”, as v varies over the elements of
V bad .
(ii) The global Frobenioids Cmod ⊩ , C⊩ tht
of Example 3.5 may be thought of as
“devices for currency exchange” between the various “local currencies” constituted
by the divisor monoids at the various v ∈ V.
(iii) One may also formulate the data contained in F ⊩ mod , F⊩ tht
via the language
of poly-Frobenioids as developed in [FrdII], §5, but we shall not pursue this topic in
the present series of papers.
Remark 3.5.2. In Example 3.5, as well as in the following discussion, we shall
often speak of “isomorphisms of collections of data”, relative to the following conventions.
(i) Such isomorphisms are always assumed to satisfy various evident compatibility
conditions, relative to the various relationships stipulated between the various
constituent data, whose explicit mention we shall omit for the sake of simplicity.
(ii) In situations where the collections of data consist partially of various categories,
the portion of the “isomorphism of collections of data” involving corresponding
categories is to be understood as an isomorphism class of equivalences of
categories [cf. §0].