Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

138 SHINICHI MOCHIZUKI
Remark 6.6.1. The underlying combinatorial structure of a D-Θ ±ell -Hodge
theater — or, essentially equivalently [cf. Definition 6.11, Corollary 6.12 below], of
aΘ ±ell -Hodge theater — is illustrated in Fig. 6.1 above. Thus, Fig. 6.1 may be
thought of as a sort of additive analogue of the multiplicative situation illustrated
in Fig. 4.4. In Fig. 6.1, the “⇑” corresponds to the associated [D-]Θ ± -bridge, while
the “⇓” corresponds to the associated [D-]Θ ell -bridge; the“/ ± ’s” denote D-primestrips.
Proposition 6.7. (Base-Θ-Bridges Associated to Base-Θ ± -Bridges) Relative
to a fixed collection of initial Θ-data, let
†
† φ Θ±
±
D T −→
† D ≻
be a D-Θ ± -bridge, as in Definition 6.4, (i). Then by replacing † D T by † D T [cf.
Definition 6.4, (i)], identifying the D-prime-strip † D ≻ with the D-prime-strip † D 0
via † φ Θ±
0 [cf. the discussion of Definition 6.4, (i)] to form a D-prime-strip † D > ,
replacing the various +-full poly-morphisms that occur in † φ Θ±
± at the v ∈ V good
by the corresponding full poly-morphisms, and replacing the various +-full polymorphisms
that occur in † φ Θ±
± at the v ∈ V bad by the poly-morphisms described [via
group-theoretic algorithms!] in Example 4.4, (i), (ii), we obtain a functorial
algorithm for constructing a [well-defined, up to a unique isomorphism!] D-Θbridge
† † φ Θ
D T −→ † D >
as in Definition 4.6, (ii). Thus, the newly constructed D-Θ-bridge is related to the
given D-Θ ± -bridge via the following correspondences:
† D T | (T \{0}) ↦→ † D T ;
† D 0 , † D ≻ ↦→ † D >
— each of which maps precisely two D-prime-strips to a single D-prime-strip.
Proof. The various assertions of Proposition 6.7 follow immediately from the
various definitions involved. ○
Next, we consider additive analogues of Propositions 4.9, 4.11; Corollary 4.12.
Proposition 6.8. (Symmetries arising from Forgetful Functors) Relative
to a fixed collection of initial Θ-data:
(i) (Base-Θ ell -Bridges) The operation of associating to a D-Θ ±ell -Hodge theater
the underlying D-Θ ell -bridge of the D-Θ ±ell -Hodge theater determines a natural
functor
category of
D-Θ ±ell -Hodge theaters
and isomorphisms of
D-Θ ±ell -Hodge theaters
→
category of
D-Θ ell -bridges
and isomorphisms of
D-Θ ell -bridges
† HT D-Θ±ell ↦→ ( † D T † φ Θell
±
−→ † D ⊚± )