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'inpartiiuFui~~nt
o:f the Requirements
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ri~. ' .. .'." . ,,,This' the.sis : is:conb~riied: wit hthe; geii;r~ : form ,of those one,.' , .
aimen~iOI.'"aifo~~'gr~~ a~';~~~e~~rfiei~ ·!;.:~!llCh'je
.. ' . given as .'
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• . r",t'l. :9• n:'.' ~'•• f'.. un• cti"o'.' ns,.\\ ' o, v0e'" r' ,.!';.. ~ .. M• .:t. er h• avin.':g ''. d"~ te"r ni• ine.d'. im '.~e ~ct::'. ex.p'r e~io_l.l","
. • .f.o..r . any • :r:'\ 't•i.•0.I.•l ~ • ~c~ \ .t... i..o.n.. .w'. h,'i.c:h . 'i.s" \ a, . .. ,.f o, rmal. gr'o up la.' .c"w, ,:" over.·!;, .•' 8.Il'" .. :.,
....
investigatio~,is madeintqt~ecategorYvilci~e objects are ,the r~tiona..l
. .' . , : ~ . ~ '.
formill. .g roup .1a\1s,'b;v:er A •.. ;.Jd, whOsemo!]?bisms, .are just their. ,formal ..... ;:
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'./ /' .. ", .' (~"o;,p'm6rphi~m~ Wh~ch ,ai-"e'g~V~ir bir'7tiQna;r fun,.~tions. The iSOlDOrphiSJll, . j .,"
and,.,
" '.' cl-as.ses df. this cat~gorya.I-e. then:c6~pl~tely,deterniiri~d~ e::te show
.,.'.;:. . ' '., .
tobe·.. . 'essentiillyeqlrlv iIl-ent :' .1. :0. the classes'· ofcongruentquaaraticforins .. :,.'"
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'over !!:. .... tIjr6Yid~d •. of.~our.se.tllat th~ c;haracteri·stico:f. . ·!;. is alffei-;mt
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to the case .of senrl.-pi-fue Hngs. • making.a. . ·suita'!:le de,fil}ition of,a '
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ration8.1". fo~ groUp.. la';:6':;er a i?~-:-prime rl.ng. parlicule.::· we :show'·
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. ..... An, attempt haSbeen.ma'de vhile wrt.ing ,tliis ,\:ile~is:to .lIliike ... the .. : ..
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,:.. .. materiu' co,!,efed. • and •.. tne.resUlts pied ce'd.,a¢cessible to,those.r-eiiders:\
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With olily a.slight·lqlovlEidge ofth~th;i, rings. and.thetheor'y-:·oi
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'. cate~o;i~il.F~r: th~ir benefit ;'~d.to.m th~thes:is as self~o.~.t8 i.~.e d\
0,:" ,",_ _ ' ••; .' _'-':':._' • - . •-••• "':.,:": \ . . "
mostof;:th~ ~
•a :spossible; specializedbac gr.s' ..0 •.l.. u.n••g n di·f·.·.·,i:frCa··n· omc.·e:roinfg.... fano·lltgr.e·g ooup..'• '
• theciry';- vh:!,chis needecl'to tri,derstand .the
: '.' .~...' -.' . : .
~:, .·i'ii,vS;:h~S' bee~incl~d~d;' and., cari be foun i~~~et~o' appendices.
.; '": t,' .: ... . ,: .." ." ,. ... '; . . ,..... ','. ", ,"' :<: ' .
.
ThE!: f'irs~. chapter ix:t;oduces the .• c'oiJ.. ept! 6f. a~.fOrm~']~~O~"'Ii.1;
·f ,..... -' ..._ '. \. '. . . - .: . . . .
• and •. thl! ~ev results'vhichshal.J:'be Ileeded.\:!: ,la.ter d1sc)lssions,. Chapter
t~ci,sihe'core~f .th~' ·tMs:i.S:iti,~he;et t~e, n:~inresti1tis':pro\<e;
.,;- • at •
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vhitlh::isl!Lter·generaJ.ized itlcilapter fow.<chapter- t~ee~ this,esult
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. :,' .,... is' th.en' used eJCtensively in the invest:l.gat·ion· of a certain .. naturally·
" •••' • " ," '., I '" ' •••• -- ",,',.'., ". •• ".' '," ' " •
. '~isin~sUbcategciz'y' of the ·:sategory.' of formal. group 'la~s over an _arpiti-~
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: I '~sh' tothiuik every~ne' Wi\;bvhom I c~e ~iito :contact:!,n:the ',,;
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: . ,:prel>8r~tionaz:~'yr~1;.·~n~ d1" thi~tbe~is;the'~;~~l~b;e ~~~nts '~d~
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'oyerwhel.nrl.~g p.atie~ce· h~v:e, b'~~n .more than ·~elconie. lrip~i.ciular! the,' "
f'iLCplt.y arid se~retafie~ o1'.'theIle:Paz:tme~;tofMB.th~atics d.eserve ~sp'ec:i.'Ii:l, ','.
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'
"thanks, as ,Vit hout ,'them; I ~ou" ldD"eyer have been ,!'oble to :t'UJiycOlIlpie~e
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lIlY. ~ork; 'N~tur'::l.l:Y-, thE; greatest part of JJIY thaIlkS is );'served·.for JJIY ,. .
s;k,~i"~ii.~~i,·n:~',r ,riavisori.BO~h energeticenth\l~ia.~~\ci~~dS
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m~tl1ematics " and;, ~i~'9.eepinsight ,not .. :\;'Omention ,bis, ,kind patie.ncE7' ,•
.' \.~erVedto stimul.ate ;.and ,to,maintaiiliny ,mathematical curiosity ~hile
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" I,' itiv~stigati~g ,the,various.:p:f.ob~ems.discussed in this· 'thesis ....
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. ) .. " .' SymboL~a~le.~;'.'" ; ~ ••••.••. ~.;: .;~:. "':'. ;: •• :.\. •••• ~; ': <;V:~;Z vii \
"
. 6hapteI;.J.::.GeneraJ.Theory- o:CFormaJ;: Groups., • '" i. ....', •.. , .: •••. '; .'1
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Grouptaw~~:: ~~;~::;~~ ,";~~; .',,'. ' .
I :\' .:Chapt " . Ra,.,t' 1ona'.J, .F.. o,'n,'n a.J'.. ,., . ••. : ..• ':;'."
Cll!,pter 'TheR.ationiiL ci~e~oJ7 •••••• ~.:~:: •• ~:• • ;: ..•• ,:~ ~:-~" ;'.: ,43 .....
.': , : ,:, ..' "., .~ ~. ,
,"', RatiohaJ. FormaJ. Grmlp La..ts>over '.' . . ..' ..
.C oinniutaiiveRirigs;; : .. ,"' .,; .••••• ,~ ,.; .'; .••' •• ; .• '.' • .'70
,:
some9~~~gory. 'rhe~~ :'~.::.': '.~ .~ ,.~~~~'~
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,APpen~~,
. ~ppendix~:: i, 'S, o!.D...E i.R; .i.n ',g' .." ,T. "h e. or,y. .:.. ., ;: • • ,', .:.-'. •:..• .'• •.•'•,' , ••:• .".,•"" ., .'\'.. ' ..,. :'; . ...... '.": .,..: 8..5. '
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.BibJ.i~graphy .••..••.•••••••. '.;'::~' :102
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.o"'~r1;he :fiel.d,'E .. ,'. ,', "",
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-,the 'algebraic" cl.osUreo:f the fiel~, ,. E ';'.
... :'"
\ . " "'. 'thecateg~ry o:f.a ' ,' J~ ;l':forin~,"g:r 6Uplaws6ver,the,ri., .n... ,~ R. ':,," ". , >:.
, ' . ". " , .. ,' .'; . .... t~;~l~l, re:.'c a'tRe gor•y. ,.ot: .,i.".i .J. .,l., ., .o. '.'n ' e 'diinen!3ione.1:f.orma1:·group laws .!-" •
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. , , ,-the categ.oryci:f, all.rati.onei :rormli.J;, ,gr.oup laws over ,R ~'
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i ',F-F(R) • "';' a c;'i;egory.o:f:f.o, rmli.J;' 'pi:rwer' seriesrfugs over'H~ ,
I " , i'1ltered ,with ,respect to ,the .order filtrati.on, defined
"in §1.1', "", , " , "
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the i'.ield of real number's, , .
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:":,th~, set o:f. pc;>sitive integer:, ;,:'
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" . Property L 1,.1: Given:,tw,,9' R-~igebr.a. s'i~'Cllg(..~ .), Sand, S I, wit hfllt ratipns '
. :' ", . "'~' .:.; '. ' ... :' " "'.'. ','.: .\ :' " '..",' ,;,"., '.
, " ,v, "and.' 'vlr~sp'eCtiveiy, ,then 'a,contibuous:R-,:,algebrah"m6inorpJ:d,sln" '
. " '. . ".~:""i'.. " J', ~ ... ',': .'. :"': .....: ,:'. '.' " ••,'". ', ... ':. . ' ,
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,Now let ,. FP( ,R.): denote'the"full stibcat,egory of,: (jllg(Rr~.',':Whose. ,objects are
/' .• ,.' , ' " . 1 . • ':'. • ....
, the Formhl.PowerSe~ies 'rii:;g~"oy~r, R ,',each considered' filtered ,with resPeC!t, to _
.....
the order ,t:iltraticm" . :•. '"
. ..' ': -t. " ' ; .' \ .
R~arks;:i.. itself.,i,.sObiousl~anob.lect<ofFP(R); tQ~, 'fi~tr,'at,.i, ~n, g e',:',
R b, ein., A,t,h,
.discret!,,~il;~~,at:i,on. ....', , "
11:
,The Objectsof'FP-{R) \ are just, th'e :free ," complete R-algebras· on ' ,
'. " ' .. .. "
,
finitely many gene~ators.,~ince ,all the ~o:phisms, heresa:\isc:." " \
. 'Proper.ty' r.Ll; PropositionB,,3.'l g)larantees the universal,proper;!;! \'
.\ ,
'\
~.
<the, ge~erators unaer its' 'act'ion:' ,
,These reinark~directly U;ply:",
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,ii., FP(R) , bas ;Ul f'in:l.tec9P'rodu'ct,s, ,in particular ,
~icn)i URC(Y~"': ':::~m)) "~R jCn·'Y~,;:·:
R( (Xl,'" ;, ((Xi ,.:, ," Y)) "
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' "':-.
.. ....
.".; ." " "1.'" .".,
..... 'I'"
.,! ..
I .... •....: .
'. .
.,. ,: - '~:" '','''. • . ,' :.:. I'I ' t.·. ". ...~..
. '. ~',~'.' ,
, .:. .
.....: .. >,<.:. ..... ' ".
.. . .... , '" .' ~ .i ..' •. ,' ::'.
..•. ; ......... ..' ..'" , " .: .".' ., ' . ,~:. ".': .;", e': .. :'. '", .:' Rc ("(""x, ": l·; .,•' .,. '; .;"x '.), ")n .-,'- '--.,.' - '>. :: r : ."' ::R',' ._>' · ' .., ' ,". ' ;.' '. .I" , "~ . .. , '>\'. \,;" ···..1. . ..... ..~ ......". --
the :'a~gm~·nt~ti~ifmap. " ' , . ' •
... . . .. -.. ' -. ''' .. -.:~ .' " ' IJ"
ciear~y.~s-;~: ~e:~e~in;:witb.;R~aige~r·a,;;o~;;in~~PhisDis.the.·l!'ap
. . . i
.... '.
, "';-." .... '-'. '''.',"> ..... ~,,~'., .' ..... '. '." '·:, ..a ,' .~>./
.. ..." ." • .. unique, Ori the. tltherhap.a:,th~aUgm'mtation·. e: .• is ;mique in .vi.ew.o!"::·
',.'
. :~. '. -, '.
., . Ptop.erty 1.i.l,·Hemce,·R·.,is a. zero' object,"iri '.' '-, .. _,.::'
',.'- ".'" ..... '. .. ' ." .
.. ' .' '. ':' ·r,·:. '
.. ii""~:AB we aie..d. e-al.ing with free .o. bjec. 1fs, .. tllepro.o . fqf ·ii'. . is :immediate;
.' . ' '. ,; '.' . ".,' ',.' .' ".~ .
" .
';,
". " " 0.'., • ',. '
.1\: .•..
andlett1W.~~b~i ~epresent ther{ngqf .formal. power s.eries. 1D.k.
;':' .
·'indet~minates.( k >:0: ) over· .. R. ;. The:preciseindetermin~t",sof R,: sllall. .,
, . ,. ' '. ., " k ' :.'. .
~_'. ,'.'
b~ indicated~nlYifth~y 61'la~ly'~tidiJaie~
: are not bicontext.J· ,,'.
...... , " ....',.' - ~ '" 0" -... ",'
" .. 1 . '/ ,> .
. ".:
'. '. R ' ': .. .'
...0 ·
. "' s:i.nc~ weha'Ve,.shown thl'tFP(R). ~s ah:~I\itiai ·obj'ect, and. all finite: , ',.
.. '. . .' ".:"'
'coproaU:ctl? , we may consider arb~1<r8i-yo ciogroup~. in , :1
\'. ., '. ',. ,.., , .
general' background material):.
. " . \ '. . '..S uppo,s"e 't.h at .. '<R~;C. ;Tl,...a ,)" '.. 1s a cogtoup' :i.n.FP (R.). ~ .• ,"" '. I ,.: \ . '. .., ~
, The
map, .., . " ,':",.'"
-. , . . c,: .. R .,..- :-,--,-+-, R ·· , .
. ~ " .. (,. n '", '. •.. ' 2n '.' ..' ,: .' ..
~s .compl~t~y·determined by.the follo.wirig ve.ct91' of n fo:r:naJ;. po:wer. serie.s in '2n
indeternunates ::
F(X,y): (FI(X,Y) ••. :~,F (X,y)). , .'
.. . .". ' .. n . , ,"
'.~
. , where: ',X,y a..' re the KiOUS. -a,bb', reviations 'for.' t,h.e. i"nde. t~,rm'i nat'¥s: "Xl,, ', .' •• '~n.'
:. :' ; 'j;
and,. Y1',.: " .~' res~ectiyelY, ·and, for ·ru'.l i, ~ . .$ i ... n , ..
.
"'" ,. 2 '. .., . " .
-1 - /
'\ ....
.,' .' . '
I.
I'
.'
