Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 111
obtained by applying the inverse of the above displayed isomorphism to the [tautological]
Kummer map Õ⊚× ↩→ lim −→H H 1 (H, μẐ(Õ⊚× )), where H ranges over the
open subgroups of π 1 ( † D ⊛ ) containing π1 Θ ( † D ⊛ ), coincides with the subset “k × NF”
constructed in [AbsTopIII], Theorem 1.9, (d), in a fashion that is compatible with
the integral submonoids “Op ⊲ ” [cf. the discussion of (iv)], relative to the ring structure
on [the union with {0} of] “k × NF” constructed in [AbsTopIII], Theorem 1.9,
(e). [Here, we note in passing that the natural injection of the above display was
constructed in a purely category-theoretic fashion from † F ⊛ [cf. the slightly different
construction discussed in [AbsTopIII], Corollary 5.2, (iii)], while the codomain
of this natural injection was constructed in a purely category-theoretic fashion from
† D ⊛ .] Indeed, the existence portion of the above observation follows immediately
by observing that the isomorphism of cyclotomes under consideration is simply the
“usual common sense identification of cyclotomes” that is typically applied without
mention in discussions of arithmetic geometry over number fields; on the other
hand, the uniqueness portion of the above observation follows immediately, in light
of the condition on the image of the natural injection of the above display [cf. also
the discussion of (iv)], from the elementary observation that
Q >0
⋂ Ẑ× = {1}
[relative to the natural inclusion Q ↩→ Ẑ ⊗ Q]. Thus, by applying the anabelian
result of [AbsTopIII], Theorem 1.9, “via the Θ-approach” [cf. Remark 3.1.2] to
π 1 ( † D ⊛ ), we conclude that
one may reconstruct category-theoretically from † F ⊛ the pair
π 1 ( † D ⊛ ) Õ⊚×
[up to “inner automorphism”], as well as the additive structure on
{0} ⋃ Õ ⊚× , and the topologies on Õ⊚× determined by the set of valuations
of the resulting field {0} ⋃ Õ ⊚× .
In particular, we obtain a purely category-theoretic construction, from † F ⊛ ,ofthe
natural bijection
Prime( † F ⊚ mod ) ∼
→ Vmod
— where we write Prime( † F ⊚ mod
) for the set of primes [cf. [FrdI], §0] of the divisor
monoid of † F ⊚ mod ; we think of V mod as the set of π 1 ( † D ⊛ )-orbits of V( † D ⊚ ). Now,
in the notation of the discussion of (iv), suppose that p is nonarchimedean [i.e.,
lies over a nonarchimedean p 0 ]. Thus, p determines a valuation, hence, in particular,
a topology on the ring {0} ⋃ O × (A birat ). Write O , for the respective
×̂p
O⊲̂p
completions, with respect to this topology, of the monoids O p × , Op ⊲ .
Then O ⊲̂p
may be identified with the multiplicative monoid of nonzero integral elements of
the completion of the number field corresponding to A at the prime of this number
field determined by p. Thus, again by allowing A to vary and considering the
resulting system of topological monoids “O ”, we obtain a construction, for nonar-
⊲̂p
chimedean p 0 ,ofthepair [i.e., consisting of a topological group acting continuously
on a topological monoid]
Π p0 Õ⊲̂p 0