Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

116 SHINICHI MOCHIZUKI
Base( i F ⊛ ) (respectively, Base( i F ⊚ )) for the base category of this Frobenioid. Then
the natural map
Isom( 1 F ⊛ , 2 F ⊛ ) → Isom(Base( 1 F ⊛ ), Base( 2 F ⊛ ))
(respectively, Isom( 1 F ⊚ , 2 F ⊚ ) → Isom(Base( 1 F ⊚ ), Base( 2 F ⊚ )))
[cf. [FrdI], Corollary 4.11; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present
paper] is bijective.
(ii) For i =1, 2, let i F be an F-prime-strip; i D the D-prime-strip associated
to i F [cf. Remark 5.2.1, (i)]. Then the natural map
[cf. Remark 5.2.1, (i)] is bijective.
Isom( 1 F, 2 F) → Isom( 1 D, 2 D)
(iii) For i =1, 2, let i F ⊢ be an F ⊢ -prime-strip; i D ⊢ the D ⊢ -prime-strip
associated to i F ⊢ [cf. Remark 5.2.1, (i)]. Then the natural map
[cf. Remark 5.2.1, (i)] is surjective.
Isom( 1 F ⊢ , 2 F ⊢ ) → Isom( 1 D ⊢ , 2 D ⊢ )
(iv) Let v ∈ V bad . Recall the category F v
of Example 3.2, (i). Thus, F v
is equipped with a natural Frobenioid structure [cf. [FrdI], Corollary 4.11;
[EtTh], Proposition 5.1], with base category D v . Then the natural homomorphism
Aut(F v
) → Aut(D v ) [cf. Example 3.2, (vi), (d)] is bijective.
Proof. Assertion (i) follows immediately from the category-theoreticity of the “natural
injection Õ⊚× ↩→ lim −→H H 1 (H, μ Θ(π 1( † D ⊛ )))” of Example 5.1, (v) [cf. also
Ẑ
the surrounding discussion; Example 5.1, (vi)]. [Here, we note in passing that this
argument is entirely similar to the technique applied to the proof of the equivalence
“Th ⊚ ∼
T
→ EA ⊚ ” of [AbsTopIII], Corollary 5.2, (iv).] Assertion (ii) (respectively, (iii))
follows immediately from [AbsTopIII], Proposition 3.2, (iv); [AbsTopIII], Proposition
4.2, (i) [cf. also [AbsTopIII], Remarks 3.1.1, 4.1.1] (respectively, [AbsTopIII],
Proposition 5.8, (ii), (v)).
Finally, we consider assertion (iv). First, we recall that since automorphisms
of D v = B temp (X v
) 0 necessarily arise from automorphisms of the scheme X v
[cf.,
[AbsTopIII], Theorem 1.9; [AbsTopIII], Remark 1.9.1], surjectivity follows immediately
from the construction of F v
. Thus, it remains to verify injectivity. To this
end, let α ∈ Ker(Aut(F v
) → Aut(D v )). For simplicity, we suppose [without loss of
generality] that α lies over the identity self-equivalence of D v . Then I claim that
to show that α is [isomorphic to — cf. §0] the identity self-equivalence of F v
,it
suffices to verify that
α induces [cf. [FrdI], Corollary 4.11; [EtTh], Proposition 5.1] the identity
on the rational function and divisor monoids of F v
.