Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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118 SHINICHI MOCHIZUKI [cf. Examples 3.2, (iii); 3.3, (i); 3.4, (i)] of the various p-adic and archimedean Frobenioids [cf. [FrdII], Example 1.1, (ii); [FrdII], Example 3.3, (ii)] that appear in an F-prime-strip that it makes sense to speak of the “pull-back” — i.e., by forming the “categorical fiber product” [cf. [FrdI], §0; [FrdI], Proposition 1.6] — of the Frobenioids that appear in the F-prime-strip 2 F via the various morphisms at v ∈ V that constitute φ. That is to say, it follows from our assumptions on φ [cf. also [AbsTopIII], Proposition 3.2, (iv)] that φ determines a pulled-back F- prime-strip “φ ∗ ( 2 F)”, whose associated D-prime-strip [cf. Remark 5.2.1, (i)] is tautologically equal to 1 D. On the other hand, by Corollary 5.3, (ii), it follows that this tautological equality of associated D-prime-strips uniquely determines an isomorphism 1 F → ∼ φ ∗ ( 2 F). Then we define the arrow ψ : 1 F → 2 F to be the collection of data consisting of φ and this isomorphism 1 F → ∼ φ ∗ ( 2 F); we refer to ψ as the “morphism uniquely determined by φ” or the “uniquely determined morphism that lies over φ”. Also, we shall apply various terms used to describe a morphism φ of D-prime-strips to the “arrow” of F-prime-strips determined by φ. (ii) The conventions discussed in (i) concerning liftings of morphisms of D- prime-strips may also be applied to poly-morphisms. We leave the routine details to the reader. Remark 5.3.2. Just as in the case of Corollary 5.3, (i), (ii), the rigidity property of Corollary 5.3, (iv), may be regarded as being essentially a consequence of “Kummerreadiness” [cf. Remarks 5.1.3, 5.2.2] of the tempered Frobenioid F v — cf. also the arguments applied in the proofs of [AbsTopIII], Proposition 3.2, (iv); [AbsTopIII], Corollary 5.2, (iv). Remark 5.3.3. We take this opportunity to rectify a minor oversight in [FrdI]. The hypothesis that the Frobenioids under consideration be of “unit-profinite type” in [FrdI], Proposition 5.6 — hence also in [FrdI], Corollary 5.7, (iii) — may be removed. Indeed, if, in the notation of the proof of [FrdI], Proposition 5.6, one writes φ ′ p = c p · φ p ,wherec p ∈O × (A), for p ∈ Primes, then one has c 2 · c 2 p · φ 2 · φ p = c 2 · φ 2 · c p · φ p = φ ′ 2 · φ ′ p = φ ′ p · φ ′ 2 = c p · φ p · c 2 · φ 2 = c p · c p 2 · φ p · φ 2 = c p · c p 2 · φ 2 · φ p —soc 2 · c 2 p = c p · c p 2 , i.e., c p = c p−1 2 ,forp ∈ Primes. Thus,φ ′ p = c −1 2 · φ p · c 2 ,soby taking u def = c −1 2 ,onemayeliminate the final two paragraphs of the proof of [FrdI], Proposition 5.6. Let † HT Θ =({ † F v } v∈V , † F ⊩ mod) be a Θ-Hodge theater [relative to the given initial Θ-data] such that the associated D-prime-strip { † D v } v∈V is [for simplicity] equal to the D-prime-strip † D > of the D-ΘNF-Hodge theater in the discussion preceding Example 5.1. Write † F >
INTER-UNIVERSAL TEICHMÜLLER THEORY I 119 for the F-prime-strip tautologically associated to this Θ-Hodge theater [cf. the data “{ † F v } v∈V ” of Definition 3.6; Definition 5.2, (i)]. Thus, † D > may be identified with the D-prime-strip associated [cf. Remark 5.2.1, (i)] to † F > . Example 5.4. Model Θ- and NF-Bridges. (i) For j ∈ J, let † F j = { † F vj } vj ∈V j be an F-prime-strip whose associated D-prime-strip [cf. Remark 5.2.1, (i)] is equal to † D j , † F 〈J〉 = { † F v〈J〉 } v〈J〉 ∈V 〈J〉 an F-prime-strip whose associated D-prime-strip we denote by † D 〈J〉 .Write † F J def = ∏ j∈J † F j — where the “formal product ∏ ” is to be understood as denoting the capsule with index set J for which the datum indexed by j is given by † F j .Thus, † F 〈J〉 may be related to † F > ,inanatural fashion, via the full poly-isomorphism † F 〈J〉 ∼ → † F > and to † F J via the “diagonal arrow” † F 〈J〉 → † F J = ∏ j∈J † F j — i.e., the arrow defined as the collection of data indexed by J for which the datum indexed by j is given by the full poly-isomorphism † ∼ F 〈J〉 → † F j . Thus, we think of † F j as a copy of † F > “situated on” the constituent labeled j of the capsule † D J ; we think of † F 〈J〉 as a copy of † F > “situated in a diagonal fashion on” all the constituents of the capsule † D J . (ii) Note that in addition to thinking of † F > as being related to † F j [for j ∈ J] via the full poly-isomorphism † ∼ F > → † F j , we may also regard † F > as being related to † F j [for j ∈ J] via the poly-morphism † ψ Θ j : † F j → † F > uniquely determined by † φ Θ j [i.e., as discussed in Remark 5.3.1]. Write † ψ Θ : † F J → † F > for the collection of arrows { † ψj Θ} j∈J collection of arrows † φ Θ = { † φ Θ j } j∈J. —whichwethinkofas“lying over” the (iii) Next, let † F ⊛ , † F ⊚ be as in Example 5.1, (iii); δ ∈ LabCusp( † D ⊚ ). Thus, [cf. the discussion of Example 4.3, (i)] there exists a unique Aut ɛ ( † D ⊚ )-orbit of
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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