Inter-universal Teichmuller Theory I
------------------------------------
・In \S I1, rewrote misleading wording concerning the gluing operation that occurs in the construction
of Hodge theaters in the paragraph that discusses of "bookkeeping of labels", as well as in the
statement of Theorem A, (i)
・In the paragraph of \S I1 that begins "Ultimately, ...", replaced the word "height" by the
expression "square root of the height" and added a reference to [IUTchIV], Corollary 2.2, (ii),
(C1)
・In the discussion immediately preceding the statement of Theorem A in \S I1, slightly modified
the reference to [FrdI]
・Throughout the statement of Theorem A, added various references to the main text of the paper
・Slightly modified the final sentence of the paragraph immediately following the statement of
Theorem B
・Corrected a misprint in the second to last paragraph of \S I3 ("types of mathematical object"
---> "types of mathematical objects")
・In the first paragraph of \S I4 concerning p-adic Teichmuller theory, corrected misprints
("horizontal arrow" ---> "arrow", 2 instances) in the explanation immediately following
the first display
・In the discussion towards the end of \S I5,, replaced the phrase "Congruence Subgroup Problem"
by the phrase "Congruence Subgroup Property"
・Slightly modified the phrasing ("passing" ---> "operation of passing", 2 instances) in the
final paragraph of \S I5
・In \S 0, added a reference to [Cusp], \S 0, following the term "generically scheme-like"
・Deleted the explanation of the abbreviations "RAF" and "AF" from the \S 0, since these
abbreviations are never used in the present series of papers
・Added some explanatory notation involving the k-core "C" to the first and second sentences
of the second paragraph of \S 1
・Corrected a misprint in the final sentence of the statement of Corollary 1.2 (i.e., a
missing "\Pi_{\underrightarrow{C}}")
・Corrected a misprint in the first sentence of Remark 1.2.2, (ii) (deleted the phrase
"of the second paragraph")
・Corrected a misprint in the first sentence of the second paragraph of \S 2 ("underlying
graph" ---> "underlying semi-graph")
・Corrected a misprint in the second sentence of the second paragraph of \S 2 ("Remark 2.5.3, (ii)"
---> "Remark 2.5.3, (i), (T6)")
・Replaced "Remark 2.5.3, (ii)" by "Remark 2.5.3, (ii), (E7)" in the third sentence of the second
paragraph of \S 2; the fourth sentence of the proof of Proposition 2.1; the final sentence
of the paragraph immediately following Remark 2.2.2
・Slightly modified the phrasing of the second sentence of the third paragraph of \S 2
・Added a more detailed explanation to the explanation following the second display of the
third paragraph of \S 2
・In the first sentence of the proof of Proposition 2.2, replaced "it follows from" by
"we conclude from"
・Added a supplementary explanatory phrase to the second to last sentence of the proof of
Proposition 2.2
・Corrected slight inaccuracies in the final portion of the paragraph immediately following
Remark 2.2.2
・Replaced "profinite" by "pro-{\widehat \Sigma}" in the first and final paragraphs of the proof
of Corollary 2.3
・Slightly modified the phrasing of the final sentence of the second to last paragraph of
the proof of Corollary 2.3
・Rephrased various portions of the proof of Corollary 2.3, (iii)
・In the proof of Proposition 2.4, rephrased the second sentence of the first paragraph, the
final sentence of the second paragraph, and the second sentence of the third paragraph
・Added an explanatory phrase ("on pro-... completions") to the second to last sentence of
the first paragraph of the proof of Proposition 2.4
・Replaced "Remark 2.5.3, (ii)" by "Remark 2.5.3, (ii), (E5), (E7)" in the second sentence
of the second paragraph of the proof of Proposition 2.4; Remark 2.5.1, (i)
・Corrected a misprint in Remark 2.5.3, (i), (T2) ("each vertex" ---> "each component")
・Slightly modified the text of Remark 2.5.3, (iii)
・Corrected a misprint in Remark 2.5.3, (v), ("restriction of" ---> "{\it splits} the
restriction of", 2 instances)
・Corrected a misprint in the paragraph preceding Theorem 2.6 ("2.17" ---> "2.17, (i)")
・Slightly modified the phrasing of the fifth, sixth, and ninth sentences of the final paragraph
of the proof of Lemma 2.7
・Slightly modified the phrasing of Definition 3.1, (b) ("{\it bad} reduction at the elements of"
---> "{\it bad} reduction at some of the elements of")
・Slightly modified the phrasing of the explanations immediately following the unique displays of
Definition 3.1, (d), (f), and the second display of Definition 3.1, (e)
・In Remark 3.1.2, (iii), added a reference to [AbsTopIII], Remark 1.10.1, (ii), as well as
supplementary explanatory sentence concerning "C_{F_{mod}}"
・Rewrote various portions of Remark 3.1.7, (i), and the beginning of Remark 3.1.7, (ii)
・Corrected a misprint in the upper left-hand corner of the commutative diagram of Remark 3.1.7, (iv)
("Gal({\overline L}/L_C)" ---> "Gal({\overline L}_C/L_C)")
・Corrected a misprint in Remark 3.1.7, (iv) ("Lemma 2.6" ---> "Lemma 2.7")
・Added a supplementary explanation at the beginning of Example 3.2
・Added a supplementary explanation concerning Frobenius-trivial objects in Example 3.2, (i); Example 5.1, (iv)
・Corrected a misprint in Example 3.2, (ii) ("A_\circledcirc" ---> "A^{bs}_\circledcirc")
・Corrected a misprint in the explanation immediately preceding the final display of Example 3.3, (i)
(added a missing superscript "\vartriangleright")
・In the explanation immediately preceding Example 3.3, (iii), (d), added a supplementary reference to
[AbsAnab], Proposition 1.2.1, (v)
・In Remark 3.4.2, added a supplementary reference to [AbsTopIII], Corollary 2.3
・Corrected a misprint in Example 3.5, (i) ("thought of a single" ---> "thought of as a single")
・Corrected misprints (deleted unnecessary underlines from the certain of the "v's" that occurs as
subscripts to "\Phi's") in Definition 3.6, (c) (1 instance) and Definition 5.2, (iv), (e) (2 instances)
・In Corollary 3.7, (ii), added a supplmentary reference to Examples 3.3, (ii); 3.4, (iii)
・In Corollary 3.8, slightly modified the reference to [FrdI] ("Introduction" ---> "\S I4")
・Corrected misprints in Remark 3.9.3, (ii) ("moves along the" ---> "moves along a" (first sentence);
"Examples 3.2, 3.3" ---> "Examples 3.2, 3.3, 3.4")
・Corrected misprints (missing underlines on calligraphic "D"'s: 2 instances) in the final sentence of
Remark 4.2.1
・In Example 4.3, (iii), replaced "2.3, (i)" by "2.3"
・Corrected a misprint in the third display of Remark 4.3.3, (ii) (a missing exponent "2")
・Corrected a misprint in the statement of Proposition 4.7, (iii) ("4.3, (iii)" ---> "4.3, (iv) ")
・Slightly modified the third sentence of Remark 4.7.2, (ii) ("In particular," ---> "That is to say,")
・Added a supplementary reference to Example 4.5, (i), (ii) in the statement of Proposition 4.8, (iv)
・Slightly modified the wording of the first sentences of Propositions 4.11, (i); 6.9, (i), and
Corollaries 4.12, (i); 6.10, (i)
・In the statements of Propositions 4.11, (i); 6.9, (i), added an explanation of "n" immediately
following the first display
・Corrected misprints (the "NF-" preceding the word "link" should be replaced by "$\Theta$NF-") in
Corollary 4.12, (i), (ii)
・Corrected a misprint in the text between the third and fourth displays of Example 5.1, (i) (added
the notation "\cup \{ 0 \}")
・Replaced (erroneous) "$\subseteq$'s" by arrows in the second display of Example 5.1, (iii), and the
first and second displays of Example 5.4, (vi)
・In Example 5.1, (v), replaced 6 instances (all of which are related to the argument of "\mu^\Th(-)")
of "\circledast" by "\circedcirc")
・Added supplementary references to Remark 3.1.7, (i), in Example 5.1, (v) (2 instances); Definition 5.2, (vi)
(2 instances)
・In Remark 5.1.3, replaced references to Example 5.1, (v), by references to Example 5.1, (v), (vi)
・Added a reference to [AbsTopIII], Theorem 1.9, to the discussion preceding the first display of
Definition 5.2, (v)
・In Definition 5.2, (v), (vi), replaced the "(d)" or "(d')" in references to [AbsTopIII], Corollary 1.10,
by "(d), (d')"
・In the discussion immediately following the third to last display of Definition 5.2, (vi), replaced
the word "subset" by the word " "pseudo-monoid" "
・In Definition 5.2, (vii), replaced "[AbsTopIII], Proposition 2.2, (i)" by "[AbsTopIII], Corollary 2.3"
・Corrected a misprint ("topological space" ---> "topological orbispace") in Definition 5.2, (vii)
・In Remark 5.2.1, (ii), added some supplementary explanatory notation following the phrase "indexed by"
・In the final portion of the second paragraph of the proof of Corollary 5.3, moved the reference to
Remark 5.3.3 to a slightly different location
・In the third paragraph of the proof of Corollary 5.3, added a reference to Remark 3.2.4, (vi)
・Corrected a misprint in the first sentence of Remark 5.3.2 ("consequence of" ---> "consequence of the")
・In the discussion immediately preceding Example 5.4, added supplementary references to Definition 3.6;
Example 3.2, (iii); Example 3.3, (i)
・In Example 5.4, (i), added a supplementary reference to Example 5.1, (vii)
・Slightly rephrased the final sentence of Example 5.4, (iii)
・Added a supplementary explanatory phrase ("where ...") to the discussion immediately preceding the
second display of Example 5.4, (iv)
・Corrected a misprint ("follwing sense" ---> "following sense") in Example 5.4, (iv)
・Deleted an unnecessary reference to Corollary 5.3, (i), in Example 5.4, (v); Definition 5.5, (i), (d)
・Slightly rephrased portions of the discussion surrounding the first display of Example 5.4, (vi), as
well as the discussion immediately following the final display of Example 5.4, (vi)
・Added "(vii)" to the resp'd portion of Remark 5.4.1
・Corrected a misprint ("a full" ---> "the full") in the discussion immediately following the final display
of Definition 5.5, (ii)
・Slightly reworded the portion following the (unique) display of the proof of Corollary 5.6
・Slightly modified the reference to Definition 4.1, (v), at the beginning of Example 6.1, (v)
・In Example 6.1, (vi), slightly modified the notation concerning "${\underline v}$"
・Slightly modified the reference to Remark 3.1.2 at the beginning of Example 6.1, (vii)
・In Remark 6.1.1, added a supplementary reference to Remark 3.1.2, (i)
・In Example 6.3, (i), replaced the word "existence" by the word "presence"
・Corrected a misprint ("...-group LabCusp..." ---> "...-group T") in the explanation immediately
following the second display of Proposition 6.5, (ii), and added a similar explanation immediately
following the first display of Proposition 6.5, (iii)
・In Proposition 6.5, (iv), added a reference to Definition 6.1, (v)
・In Proposition 6.8, (ii), (iii); 6.9, (i), (ii), deleted the expression "[well-defined ...]"
・Corrected misprints in the portion of Definition 6.11, (i) (an unnecessary calligraphic "F": 2 instances),
(ii) (an unnecessary calligraphic "D": 1 instance), concerning morphisms
・Slightly modified the wording of the explanation immediately following the display of Definition 6.11, (iii)
・Switched the order of the final two sentences of Remark 6.12.2, (i)
・Corrected a misprint concerning types of Hodge theaters in the first sentence of Remark 6.12.3, (iii)
・In Remark 6.12.3, (iii), corrected misprints (omitted "below"'s: 2 instances) in the references to
Remark 6.12.6
・Corrected a misprint ("Proposition 4.9.1" ---> "Proposition 4.9") in Remark 6.12.4, (ii)
・In Remark 6.12.6, (iii), added references to Definition 6.1, (v); Example 6.3; Definition 6.4, (ii)
・In Remark 6.12.6, (v), (a), replaced ", and" by "and"
・In Fig. 6.6, added a reference to "I, 3.6, (a)"
・Updated the Bibiographical entries for [IUTchII], [IUTchIII], [IUTchIV] and added a relevant
webpage address at the end of the Bibliography
Inter-universal Teichmuller Theory II
-------------------------------------
・Corrected a misprint in the first sentence of Remark 3.6.2 ("thought of a" ---> "thought of as a")
・Updated the Bibiographical entries for [IUTchI], [IUTchIII], [IUTchIV] and added a relevant
webpage address at the end of the Bibliography
Inter-universal Teichmuller Theory III
--------------------------------------
・Updated the Ackowledgements
・Added supplementary explanations concerning the various poly-isomorphisms that appear in the
discussions immediately following the fifth and sixth displays of Theorem 1.5, (iii); in the
first sentence of Theorem 1.5, (iv), deleted the word "full"
・Corrected a misprint in Remark 1.5.3, (iii) ("thought of a" ---> "thought of as a")
・Updated the Bibiographical entries for [IUTchI], [IUTchII], [IUTchIV] and added a relevant
webpage address at the end of the Bibliography
Inter-universal Teichmuller Theory IV
-------------------------------------
・Updated the Bibiographical entries for [IUTchI], [IUTchII], [IUTchIII] and added a relevant
webpage address at the end of the Bibliography