Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

132 SHINICHI MOCHIZUKI
[cf. the discussion of Definition 6.1, (v)], which we shall use to identify the group
Aut ± (D ⊚± )/Aut csp (D ⊚± ) with the group F ⋊±
l
. If v ∈ V good ⋂ V non (respectively,
v ∈ V bad ), then the natural restriction functor on finite étale coverings arising from
the natural composite morphism −→v X → X v → X K (respectively, X v
→ X v →
X K ) determines [cf. Examples 3.2, (i); 3.3, (i)] a natural morphism φ Θell
•,v : D v →
D ⊚± [cf. the discussion of Example 4.3, (ii)]. If v ∈ V arc , then [cf. Example 3.4, (i)]
∼
we have a tautological morphism D v = −→v X → X v → X(D ⊚± ,v), hence a morphism
φ Θell
•,v : D v →D ⊚± [cf. the discussion of Example 4.3, (iii)]. For arbitrary v ∈ V,
write
φ Θell
v 0
: D v0 →D ⊚±
for the poly-morphism given by the collection of morphisms D v0
β ◦ φ Θell
•,v
◦ α
→D ⊚± of the form
—whereα ∈ Aut + (D v0 ); β ∈ Aut csp (D ⊚± ); we apply the tautological identification
of D v with D v0 [cf. the discussion of Example 4.3, (ii), (iii), (iv)]. Write
φ Θell
0 : D 0 →D ⊚±
for the poly-morphism determined by the collection {φ Θell
v 0
: D v0 →D ⊚± } v∈V [cf.
the discussion of Example 4.3, (iv)]. Note that the existence of “β” in the definition
of φ Θell
v 0
implies that it makes sense to post-compose φ Θell
0 with an element of
Aut ± (D ⊚± )/Aut csp (D ⊚± ) → ∼ F ⋊±
l
.Thus,foranyt ∈ F l ⊆ F ⋊±
l
,letuswrite
φ Θell
t : D t →D ⊚±
for the result of post-composing φ Θell
0 with the “poly-action” [i.e., action via polyautomorphisms]
of t on D ⊚± [and pre-composing with the tautological identification
of D 0 with D t ]and
φ Θell
± : D ± →D ⊚±
for the collection of arrows {φ Θell
t } t∈Fl .
(ii) Let γ ∈ F ⋊±
l
. Then γ determines a natural poly-automorphism γ ± of D ±
as follows: the automorphism γ ± acts on F l via the usual action of F ⋊±
l
on F l and,
∼
for t ∈ F l , induces the +-full poly-isomorphism D t → Dγ(t) whose sign at every
v ∈ V is equal to the sign of γ [cf. the construction of Example 6.2, (ii)]. Thus, we
obtain a natural poly-action of F ⋊±
l
on D ± . On the other hand, the isomorphism
Aut ± (D ⊚± )/Aut csp (D ⊚± ) → ∼ F ⋊±
l
of (i) determines a natural poly-action of F ⋊±
l
on D ⊚± . Moreover, one verifies immediately that φ Θell
± is equivariant with respect
to these poly-actions of F ⋊±
l
on D ± and D ⊚± ; in particular, we obtain a natural
poly-action
F ⋊±
l
(D ± , D ⊚± ,φ Θell
± )
of F ⋊±
l
on the collection of data (D ± , D ⊚± ,φ Θell
± ) [cf. the discussion of Example
4.3, (iv)].