%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Infinito.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips Infinito.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.05.10:1239 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (Infinito.dvi) @start /Fa 19 123 df45 D<127012F8A312700505788416>I< 13F8EA03FC487EEA0F07381C3B80EA38FF12793873C7C01383EAE701A73873838013C738 79FF00EA38FEEA1C38380F03C0EA07FF6C1300EA00FC12197E9816>64 D97 D99 D<133FA31307A4EA03C7EA0F F748B4FCEA3C1F487EEA700712E0A6EA700F12786C5A381FFFE0EA0FF7EA07C713197F98 16>II<3803E3C03807F7E0EA0FFF381C1CC038 380E00A56C5AEA0FF8485AEA1BE00038C7FC1218EA1FFC13FF481380387003C038E000E0 A4387001C0EA7C07383FFF80380FFE00EA03F8131C7F9116>103 D<12FCA3121CA41378EA1DFCEA1FFE130FEA1E07121CAA38FF8FE0139F138F13197F9816 >I<1203EA0780A2EA0300C7FCA4EAFF80A31203ACEAFFFC13FE13FC0F1A7C9916>I<127E 12FE127E120EA4EB7FE0A3EB0F00131E5B5B5B120F7F13BC131EEA0E0E7F1480387F87F0 EAFFCFEA7F871419809816>107 D<38F9C38038FFEFC0EBFFE0EA3C78A2EA3870AA38FE 7CF8A31512809116>109 DI112 D<387F0FC038FF3FE0EA7F7F3807F040EBC0005BA290C7FCA8EA7FFC12FF127F13127F91 16>114 DI<12035AA4EA7FFFB5FCA20007 C7FCA75BEB0380A3EB8700EA03FE6C5A6C5A11177F9616>II<383FFFC05AA238700780EB0F00131EC65A5B485A 485AEA078048C7FC381E01C0123C1278B5FCA312127F9116>122 D E /Fb 43 128 df<12E0A212F012381218120C1206120207087A9C17>18 D<391F1003E23930F0061E3960300C0600C0131801101302A239E0001C00A2007EEB0FC0 393FC007F8391FE003FC3903F0007ED80078130F013813070118130300801310A2011013 0239C030180639F0601E0C398F8011F020157E9426>25 D<126012F012F812681208A312 10A2122012401280050C7B9C0D>39 D<13401380EA0100120212065AA25AA25AA2127012 60A312E0AC1260A312701230A27EA27EA27E12027EEA008013400A2A7B9E12>I<7E1240 7E7E12187EA27EA27EA213801201A313C0AC1380A312031300A21206A25AA25A12105A5A 5A0A2A7D9E12>I<126012F0A212701210A41220A212401280040C7B830D>44 DI48 D<12035A123F12C71207 B3A4EA0F80EAFFF80D1C7B9B17>I51 D<13F8EA0306EA0602EA0C07485A1238EA30060070C7FCA21260 EAE3E0EAE418EAE80CEAF00613077F00E01380A41260A212701400EA3007EA18065B6C5A EA03E0111D7D9B17>54 D56 DI<1303A3497EA2497E130BA2EB11E0A2EB31F01320A2EB4078 A3497EA23801003EEBFFFEEB001E00027FA348EB0780A2000C14C0121E39FF803FFC1E1D 7E9C22>65 D<90380FE02090387018603801C00439030003E000061301000E1300481460 5A15201278127000F01400A80070142012781238A26C14407E000614806CEB01003801C0 0638007018EB0FE01B1E7D9C21>67 DII<90381FC04090387030C03801C00C38030003000E13 01120C001C13005A15401278127000F01400A6EC7FF8EC07C00070130312781238A27E12 0C120E000313053801C008390070304090381FC0001D1E7D9C23>71 D73 D77 D80 D<3803F040380C0CC0EA1002EA30 01EA600012E01440A36C13007E127EEA7FE0EA3FFC6CB4FC00071380EA007FEB07C0EB03 E0130113007EA36C13C0A238E0018038D00300EACE06EA81F8131E7D9C19>83 D<007FB512C038700F010060130000401440A200C014201280A300001400B1497E3803FF FC1B1C7D9B21>I<39FFF01FF0390F000380EC0100B3A26C1302138000035BEA01C03800 E018EB7060EB0F801C1D7D9B22>I<13201370A313B8A3EA011CA2EA031EEA020EA2487E EA07FFEA040738080380A2001813C01301123838FC07F815157F9419>97 D99 DII103 D<38FF8FF8381C01C0A9EA1FFFEA1C01A938FF8FF815157F9419>II108 D<00FEEB0FE0001E140000171317A338138027A23811C047A33810E087A2EB7107A3133A A2131CA2123839FE083FE01B157F941F>I<38FC03F8381E00E014401217EA138013C012 11EA10E01370A21338131CA2130E130714C0130313011300123800FE134015157F9419> III114 DI<387FFFF038607030004013 10A200801308A300001300ADEA07FF15157F9419>I<38FF83F8381C00E01440AE000C13 C0000E138038060100EA0386EA00FC15157F9419>I<38FF01F8383C0070001C13601440 A26C1380A238070100A3EA0382A2EA01C4A3EA00E8A21370A3132015157F9419>I<38FF 80FE381E0038000E1320000F13606C13403803808013C03801C10013E212001374137C13 38A848B4FC1715809419>121 D127 D E /Fc 3 113 df<12C012F0123C120FEA03C0EA00F01338130E6D7EEB01E0EB0078141EEC 0780A2EC1E001478EB01E0EB0780010EC7FC133813F0EA03C0000FC8FC123C12F012C019 1A7D9620>62 D110 D<380787803809C8603808 D03013E0EA11C014381201A238038070A31460380700E014C0EB0180EB8300EA0E861378 90C7FCA25AA4123CB4FC151A819115>112 D E /Fd 8 58 df<1360EA01E0120F12FF12 F31203B3A2387FFF80A2111B7D9A18>49 DIII<38380180383FFF005B5B5B13C00030C7FCA4EA31F8EA361E38380F80EA300700 0013C014E0A3127812F8A214C012F038600F8038381F00EA1FFEEA07F0131B7E9A18>I< 1260387FFFE0A214C01480A238E00300EAC0065B5BC65AA25B13E0A212015B1203A41207 A66C5A131C7D9B18>55 DII E /Fe 1 1 df0 D E /Ff 1 66 df<1320A21350A21348138813C8EA014413 241322120213121311EA0409EB0880EA07F8EA0C04000813401218EB0220A200281310A2 38FE0FFC16187F9710>65 D E /Fg 2 79 df<8013018013031460A2EB0630A2EB0E18A2 130FEB1B0CEB198C1486EB30C614C31370EB6063EC6180EBE0319038C030C01418000114 60EBFFF84813FC9038800C30A248486C7EA28148130381381980063AFFC01FFF80A22123 7FA216>65 D<3AFFE001FF8013F03A3818003800D80C0C1318A2EA0E06EA0F03380D8180 A2380CC0C0EB6060EB303013181418EB0C0CEB0606130314039038018198903800C0D814 601578EC3038EC1818140CA214061403EC0198EC00D8A2000E1478D87FE013381518C812 0821237EA123>78 D E /Fh 2 110 df103 D109 D E /Fi 1 110 df109 D E /Fj 49 123 df12 D<91380FF3809138383F00 EC607F14C001011337EC800E1303A35DEB0700A35D90B512F890380E0038A25DA35B5DA4 49485AA315C24A5A1370A215881401EC00F04990C7FCA35BA21271EAF18090C9FC126212 3C212D82A21D>I<1480EB010013025B5B5B13305B5BA2485A48C7FCA21206A2120E120C 121C1218A212381230A21270A21260A212E0A35AAD12401260A21220123012107E113278 A414>40 D<13087F130613021303A27F1480AD1303A31400A25BA21306A2130E130CA213 1C131813381330A25BA25B485AA248C7FC120612045A5A5A5A5A113280A414>I<120E12 1EA41202A21204A21208A21210122012401280070F7D840F>44 DI<127012F8A212F012E005057A840F>I<13011303A21306131E132EEA03CEEA001C A41338A41370A413E0A4EA01C0A4EA0380A41207EAFFFC10217AA019>49 DI<14181438A21470A3 14E0A314C01301148013031400A21306A25BA25B1310EB3180EB61C0EB438013831201EA 03033802070012041208EA3FC7EA403E38800FF038000E00A25BA45BA31330152B7EA019 >52 D<1207EA0F80A21300120EC7FCAB127012F8A25A5A09157A940F>58 D<1403A25CA25CA25C142FA2EC4F80A21487A2EB01071302A21304A21308131813101320 A290387FFFC0EB40031380EA0100A21202A25AA2120C003CEB07E0B4EB3FFC1E237DA224 >65 D<90B512F090380F003C150E81011EEB0380A2ED01C0A25B16E0A35BA449EB03C0A4 4848EB0780A216005D4848130E5D153C153848485B5DEC03804AC7FC000F131CB512F023 227DA125>68 D<027F138090390380810090380E00630138132749131F49130E485A485A 48C7FC481404120E121E5A5D4891C7FCA35AA4EC3FFC48EB01E0A34A5AA27E12704A5A7E 0018130F001C131300060123C7FC380381C1D800FEC8FC212479A226>71 D<9039FFF87FFC90390F000780A3011EEB0F00A449131EA4495BA490B512F89038F00078 A348485BA44848485AA44848485AA4000F130739FFF07FF826227DA124>II<9039FFF8 07FC90390F0001E016C01600011E13025D15105D495B5D4AC7FC1404495A14181438147C EBF0BCEBF23CEBF41E13F83801F01F497EA2813803C007A26E7EA2EA07806E7EA3000F80 39FFF00FFE26227DA125>75 DI<90B512 E090380F0038151E150E011E1307A449130FA3151E5B153C157815E09038F003C09038FF FE0001F0C7FCA2485AA4485AA4485AA4120FEAFFF020227DA121>80 D<903801F02090380E0C4090381802C0EB3001136001E0138013C01201A200031400A291 C7FCA27FEA01F813FF6C13E06D7EEB1FF8EB03FCEB007C143C80A30020131CA314180060 1338143000705B5C38C80180D8C607C7FCEA81FC1B247DA21B>83 D<001FB512F8391E03C03800181418123038200780A200401410A2EB0F001280A2000014 00131EA45BA45BA45BA4485AA41203B5FC1D2277A123>I91 D93 D97 DI<137EEA01C13803 0180EA0703EA0E07121C003CC7FC12381278A35AA45B12701302EA300CEA1830EA0FC011 157B9416>I<143CEB03F8EB0038A31470A414E0A4EB01C013F9EA0185EA0705380E0380 A2121C123C383807001278A3EAF00EA31410EB1C201270133C38305C40138C380F078016 237BA219>I<13F8EA0384EA0E02121C123C1238EA7804EAF018EAFFE0EAF000A25AA413 02A2EA6004EA7018EA3060EA0F800F157A9416>I<143E144714CFEB018F1486EB0380A3 EB0700A5130EEBFFF0EB0E00A35BA55BA55BA55BA45B1201A2EA718012F100F3C7FC1262 123C182D82A20F>II<13F0EA0FE01200A3485AA4485AA448C7FC131FEB2180EBC0C0380F00 E0A2120EA2381C01C0A438380380A3EB070400701308130E1410130600E01320386003C0 16237DA219>I<13C0EA01E013C0A2C7FCA8121E12231243A25AA3120EA25AA35AA21340 EA7080A3EA71001232121C0B217BA00F>I<14E01301A2EB00C01400A8131E1323EB4380 1383A2EA0103A238000700A4130EA45BA45BA45BA3EA70E0EAF0C0EAF1800063C7FC123C 132B82A00F>I<13F0EA07E01200A3485AA4485AA448C7FCEB01E0EB0210EB0470380E08 F01310EB2060EB4000EA1D80001EC7FCEA1FC0EA1C70487EA27F142038703840A3EB1880 12E038600F0014237DA216>II<391C0F80F8392610C10C39476066063987 807807A2EB0070A2000EEBE00EA44848485AA3ED38202638038013401570168015303A70 07003100D83003131E23157B9428>I<38380F80384C30C0384E4060388E8070EA8F0012 8EA24813E0A4383801C0A3EB03840070138814081307EB031012E0386001E016157B941B >I<137EEA01C338038180380701C0120E001C13E0123C12381278A338F003C0A2148013 0700701300130E130CEA3018EA1870EA07C013157B9419>I<3801C1F038026218380474 1C3808780CEB700EA2141EEA00E0A43801C03CA3147838038070A2EBC0E0EBC1C0380723 80EB1E0090C7FCA2120EA45AA3EAFFC0171F7F9419>III<13FCEA018338020080EA0401EA0C03140090C7FC120F13 F0EA07FC6C7EEA003E130F7F1270EAF006A2EAE004EA4008EA2030EA1FC011157D9414> I<13C01201A4EA0380A4EA0700EAFFF8EA0700A2120EA45AA45AA31310EA7020A2134013 80EA3100121E0D1F7C9E10>I<001E1360002313E0EA4380EB81C01283EA8701A2380703 80120EA3381C0700A31408EB0E101218121CEB1E20EA0C263807C3C015157B941A>I<38 1C0180382603C0EA47071303EA8701EA8E00A2000E13805AA338380100A31302A25B5B12 18EA0C30EA07C012157B9416>I<001EEB60E00023EBE0F0384380E1EB81C000831470D8 87011330A23907038020120EA3391C070040A31580A2EC0100130F380C0B023806138438 03E0F81C157B9420>I<3803C1E0380462103808347038103CF0EA203814601400C65AA4 5BA314203861C04012F1148038E2C100EA4462EA383C14157D9416>I<001E1330002313 70EA438014E01283EA8700A2380701C0120EA3381C0380A4EB0700A35BEA0C3EEA03CEEA 000EA25B1260EAF0381330485AEA80C0EA4380003EC7FC141F7B9418>I<3801E0203803 F0603807F8C038041F80380801001302C65A5B5B5B5B5B48C7FC12024813803808010048 5AEA3F06EA61FEEA40FCEA807013157D9414>I E /Fk 30 128 df45 D48 D<12035AB4FC1207B3A2EA7FF80D187D9713> III<1318A21338137813F813B8EA0138 1202A212041208121812101220124012C0B5FCEA0038A6EA03FF10187F9713>III<1240EA7FFF13FEA2EA4004EA80081310A2EA00201340A21380 120113005AA25A1206A2120EA5120410197E9813>III<130CA3131EA2132F1327A2EB4380A3EB81C0A200017F1300A2 48B47E38020070A2487FA3487FA2003C131EB4EBFFC01A1A7F991D>65 D67 DIIII<39FFE1FFC0390E001C00 AB380FFFFC380E001CAC39FFE1FFC01A1A7F991D>III76 DI<00FEEB7FC0000FEB0E001404EA0B80EA09C0A2EA08 E01370A21338131CA2130E1307EB0384A2EB01C4EB00E4A21474143CA2141C140C121C38 FF80041A1A7F991D>I<137F3801C1C038070070000E7F487F003C131E0038130E007813 0F00707F00F01480A80078EB0F00A20038130E003C131E001C131C6C5B6C5B3801C1C0D8 007FC7FC191A7E991E>II82 DI< 007FB5FC38701C0700401301A200C0148000801300A300001400B13803FFE0191A7F991C >I<39FFE07FC0390E000E001404B200065B12076C5B6C6C5A3800E0C0013FC7FC1A1A7F 991D>I127 D E /Fl 34 122 df<14FE90380301801306EB0C03 EB1C0191C7FC13181338A43803FFFE3800700EA35CA213E0A25CA3EA01C01472A4380380 34141891C7FC90C8FCA25A12C612E65A12781925819C17>12 D 45 D<1230127812F0126005047C830D>I<1418A21438A21478A214B8EB0138A2EB023C14 1C1304130C13081310A21320A2EB7FFCEBC01C1380EA0100141E0002130EA25A120C001C 131EB4EBFFC01A1D7E9C1F>65 D<903803F02090381E0C6090383002E09038E003C03801 C001EA038048C7FC000E1480121E121C123C15005AA35AA41404A35C12705C6C5B00185B 6C485AD80706C7FCEA01F81B1E7A9C1E>67 D<48B512F038003C00013813301520A35BA2 14081500495AA21430EBFFF03801C020A439038040801400A2EC0100EA07005C14021406 000E133CB512FC1C1C7E9B1C>69 D73 DI<3A01FFC07F 803A003C001E000138131815205D5DD97002C7FC5C5C5CEBE04014C013E1EBE2E0EA01C4 EBD07013E013C048487EA21418141CEA070080A348130F39FFE07FC0211C7E9B20>I77 D<3801FFFE39003C038090383801C0EC00E0 A3EB7001A315C0EBE0031580EC0700141C3801FFF001C0C7FCA3485AA448C8FCA45AEAFF E01B1C7E9B1C>80 D<3801FFFE39003C078090383801C015E01400A2EB7001A3EC03C001 E01380EC0700141CEBFFE03801C03080141CA2EA0380A43807003C1520A348144039FFE0 1E80C7EA0F001B1D7E9B1E>82 DI97 D<123F1207A2120EA45AA4 EA39E0EA3A18EA3C0C12381270130EA3EAE01CA31318133813301360EA60C0EA3180EA1E 000F1D7C9C13>I<13F8EA0304120EEA1C0EEA181CEA30001270A25AA51304EA60081310 EA3060EA0F800F127C9113>II<13F8EA 0704120CEA1802EA38041230EA7008EA7FF0EAE000A5EA60041308EA30101360EA0F800F 127C9113>I103 DII<130313071303 1300A71378138CEA010C1202131C12041200A21338A41370A413E0A4EA01C0A2EAC180EA E30012C612781024819B0D>I108 D<391C1E078039266318C0394683A0E0384703C0008E13 80A2120EA2391C0701C0A3EC0380D8380E1388A2EC0708151039701C032039300C01C01D 127C9122>II<13F8EA030CEA0E06487E12181230007013 80A238E00700A3130EA25BEA60185BEA30E0EA0F8011127C9115>I<380387803804C860 EBD03013E0EA09C014381201A238038070A31460380700E014C0EB0180EB8300EA0E8613 7890C7FCA25AA45AB4FC151A809115>I114 DI<1203 5AA3120EA4EAFFE0EA1C00A35AA45AA4EAE080A2EAE100A2126612380B1A7C990E>I<38 1C0180EA2E03124EA2388E0700A2121CA2EA380EA438301C80A3EA383C38184D00EA0F86 11127C9116>II<381E0183382703871247148338870701A2120EA238 1C0E02A31404EA180C131C1408EA1C1E380C26303807C3C018127C911C>I<381C0180EA 2E03124EA2388E0700A2121CA2EA380EA4EA301CA3EA383CEA1878EA0FB8EA0038133013 70EAE0605BEA81800043C7FC123C111A7C9114>121 D E /Fm 9 107 df0 D<1202A3EAC218EAF278EA3AE0EA0F80A2EA3AE0EAF2 78EAC218EA0200A30D0E7E8E12>3 D<143014F0EB01C0EB0700131E1378EA01E0EA0380 000EC7FC123C12F0A21238120E6C7EEA01E0EA0078131C1307EB03C0EB00F014301400A6 387FFFE0B512F0141E7D951B>20 D<12C012F01238120E6C7EEA01E0EA0078131C1307EB 03C0EB00F0A2EB01C0EB0700131E1378EA01E0EA0380000EC7FC123C127012C0C8FCA638 7FFFE0B512F0141E7D951B>I<001F1304EA3FC0EA7FE038E0F00C38C03C1C38801FF8EB 0FF0EB03E0C8FCA7387FFFFCB5FC16117E8F1B>39 D<3801FF801207000EC7FC12185A5A A35AA2B51280A200C0C7FCA21260A37E7E120E3807FF80120111167D9218>50 D<137813C0EA0180EA0300AB12065A12F0120C7E7EABEA0180EA00C013780D217E9812> 102 D<12F0120C7E7EABEA0180EA00C0137813C0EA0180EA0300AB12065A12F00D217E98 12>I<12C0B3AF02217C980A>106 D E /Fn 7 102 df<146014C0EB0180EB0300130613 0E5B131813385B136013E0485AA2485AA2120790C7FCA25A120E121EA2121CA2123CA212 38A21278A5127012F0B312701278A51238A2123CA2121CA2121EA2120E120F7EA27F1203 A26C7EA26C7E136013707F1318131C7F13067FEB0180EB00C01460135977811E>16 D<12C012607E7E7E120E7E7E7F6C7E12007F1370A27FA2133C131CA2131E130E130FA27F A21480A21303A214C0A5130114E0B314C01303A51480A21307A21400A25BA2130E131E13 1CA2133C1338A25BA25B5B1201485A90C7FC5A120E120C5A5A5A5A13597F811E>I80 D88 DI<1304131FEB7FC0EBF1E03803C078380F001E001C13070070EB01C000C0 EB00601B0980A41C>98 D101 D E /Fo 18 107 df0 D<127012F8A3127005057C8E0E>I<6C13026C13060060130C6C13186C13306C13606C13 C03803018038018300EA00C6136C1338A2136C13C6EA018338030180380600C048136048 133048131848130C4813064813021718789727>I10 D15 D17 D<150C153C15F0EC03C0EC0F00143C14F0EB03C0010FC7FC133C13F0EA03C0000FC8FC12 3C12F0A2123C120FEA03C0EA00F0133C130FEB03C0EB00F0143C140FEC03C0EC00F0153C 150C1500A8007FB512F8B612FC1E287C9F27>20 D<12C012F0123C120FEA03C0EA00F013 3C130FEB03C0EB00F0143C140FEC03C0EC00F0153CA215F0EC03C0EC0F00143C14F0EB03 C0010FC7FC133C13F0EA03C0000FC8FC123C127012C0C9FCA8007FB512F8B612FC1E287C 9F27>I<90380FFFFC137FD801F0C7FCEA03800006C8FC5A5A5AA25AA25AA81260A27EA2 7E7E7E6C7EEA01E039007FFFFC131F1E1E7C9A27>26 D<4B7EA46F7EA2166082A2161C82 82B812E0A2C9EA0700160E5E1630A25E5EA24B5AA42B1A7D9832>33 D39 D<156081A281A2818181 B77E16E0C91270161CEE0F80EE03E0EE0780EE1E0016381660B75A5EC80003C7FC15065D 5DA25DA25D2B1C7D9932>41 D50 D<1403A21406A2140CA21418A21430A21460A214C0A3EB0180A2EB0300A21306A25BA25B A25BA25BA25BA2485AA248C7FCA31206A25AA25AA25AA25AA25A1240183079A300>54 D<12C0A812E0A212C0A803127D9400>I102 D<12F8120FEA03806C7E6C7EB113707F131EEB03C0EB1E0013385B5BB1485A485A000FC7 FC12F812317DA419>I<12C0B3B3AD02317AA40E>106 D E /Fp 4 51 df<120412081210123012201260A2124012C0AA12401260A212201230121012081204 061A7D920C>40 D<128012401220123012101218A21208120CAA12081218A21210123012 2012401280061A7F920C>I<1218127812981218AC12FF08107D8F0F>49 D<121FEA6180EA40C0EA806012C01200A213C0EA0180EA030012065AEA10201220EA7FC0 12FF0B107F8F0F>I E /Fq 9 116 df11 D27 DI97 D<1204120C1200A5123012581298A21230A212601264A21268123006127E910B>105 D<12781218A21230A41260A412C0A212C8A212D0126005117E900A>108 D<3871F1F0389A1A18EA9C1CEA9818121838303030A214321462386060641438170B7E8A 1B>II<120F EA10801221EA2000123E7EEA01801280EAC0001283127C090B7D8A0F>115 D E /Fr 26 121 df11 D<130FEB3080EBC0C0380100401460 000213C05A1480130138083F00133E130114801210A438300300A21306EA280CEA4418EA 43E00040C7FCA25AA4131D7F9614>I<38078040EA1FC03838E080EA6030EAC010388011 00EA0009130AA3130CA31308A35BA41214808D12>II28 D<126012F0A2126004047D830A>58 D<126012F0A212701210A21220A21240A204 0A7D830A>I64 D<14C0A21301A21303130514E01308131813101320A213401380A23801FFF0EB00701202 5AA25A121838FE03FE17177F961A>I<3807FFF83800E00E140315801401D801C013C0A2 1400A238038001A43907000380A215005C000E130E140C5C1470381C01C0B5C7FC1A177F 961D>68 D<3907FE1FF83900E00380A43901C00700A43803800EA2EBFFFEEB800E48485A A4000E5BA4485B38FF83FE1D177F961D>72 D<3907F007F80000EB00C001B81380A2139C 39011C0100131E130EA238020702A2EB0382A2380401C4A2EB00E4A2481378A21438A200 18131012FE1D177F961C>78 D<3807FFF03800E01C14061407A2EA01C0A3140E48485A14 70EBFF80EB80E03807007080A3000E5BA21580A248EB310038FF801E19177F961B>82 D97 D99 D<133E130CA41318A4EA0730EA18F0EA30701260136012C0A3EA80C013C4A212C1 EA46C8EA38700F177E9612>I<120313801300C7FCA6121C12241246A25A120C5AA31231 A21232A2121C09177F960C>105 D<1318133813101300A6EA01C0EA0220EA0430A2EA08 601200A313C0A4EA0180A4EA630012E312C612780D1D80960E>I<123E120CA41218A412 30A41260A412C012C8A312D0127007177E960B>108 D<38383C1E3844C6633847028138 460301388E0703EA0C06A338180C061520140C154039301804C0EC07001B0E7F8D1F>I< EA383CEA44C6EA47021246EA8E06120CA3485A144013181480EA3009EB0E00120E7F8D15 >III115 D<1203A21206A4EAFFC0EA0C00 A35AA45A1380A2EA31001232121C0A147F930D>I120 D E /Fs 11 94 df<126012F0A51260A91200A4126012F0A2126004177D960A>33 D<120112021204120C1218A21230A212701260A312E0AA1260A312701230A21218A2120C 12041202120108227D980E>40 D<12801240122012301218A2120CA2120E1206A31207AA 1206A3120E120CA21218A2123012201240128008227E980E>I<1330ABB512FCA2380030 00AB16187E931B>43 D48 D<1206120E12FE120EB1EAFFE00B157D9412> III61 D<12FCA212C0B3AB12FCA2 06217D980A>91 D<12FCA2120CB3AB12FCA2062180980A>93 D E /Ft 53 123 df<133FEBE180390380E0203907006040000E1370481330003CEB3880A248 EB3900A2143A5A143C1438A212701478003013B83938031840D81C1C13803907E007001B 157E941F>11 D<143EECC18090380300C013044913E05BA25B5BEC01C013801580140339 01000700EB01FEEB021CEB03EE38020007A21580A25AA448EB0F00A2140E141E0014131C 5C00125B00135B382081C0017EC7FC90C8FCA25AA45AA41B2D7FA21C>I<3801E001EA07 F8380FFC02121F38300E04EA600638400308EA8001A200001310EB009014A0A314C0A314 80A31301A3EB0300A41306A31304A218207F9419>II22 D<000FB5FC5A5A3870418000401300EA8081A21200EA01831303 A21203A21206A2120EA2000C1380121CA2EA180118157E941C>25 D<90387FFF8048B5FC5A390783C000EA0E01486C7E5AA25AA348485AA3495A91C7FCEA60 07130EEA3018EA1870EA07C019157E941C>27 D<380FFFF84813FC4813F8387020001240 128013601200A25BA412015BA21203A348C7FC7E16157E9415>I<133F3801FFC0380381 E0380400404813005AA31360EA0B98EA0FF80010C7FC5A5AA35A6C1380124038600100EA 300EEA1FFCEA07E013177F9517>34 D<127012F8A3127005057C840E>58 D<127012F812FCA212741204A41208A21210A212201240060F7C840E>I<15181578EC01 E0EC0780EC1E001478EB03E0EB0F80013CC7FC13F0EA03C0000FC8FC123C12F0A2123C12 0FEA03C0EA00F0133CEB0F80EB03E0EB0078141EEC0780EC01E0EC007815181D1C7C9926 >I<14801301A2EB0300A31306A35BA35BA35BA35BA35BA3485AA448C7FCA31206A35AA3 5AA35AA35AA35AA311317DA418>I<12C012F0123C120FEA03C0EA00F0133EEB0F80EB01 E0EB0078141EEC0780EC01E0EC0078A2EC01E0EC0780EC1E001478EB01E0EB0F80013EC7 FC13F0EA03C0000FC8FC123C12F012C01D1C7C9926>I64 D<8114018114031407A2140BA2141314331423EC43E0A2 1481EB0101A21302A213041308A201107FEB1FFFEB20005BA25BA248C7FC120281481478 120C001E14F83AFF800FFF8021237EA225>I<90B512F090380F001E81ED0780011E1303 A216C0A24914801507A2ED0F0049131E5D5DEC03E090B55A9038F000F0157881485A151C 151EA248485BA35D485A5D4A5AEC0380000F010FC7FCB512F822227DA125>I<90B512F0 90380F003C150E81011EEB0380A2ED01C0A25B16E0A35BA449EB03C0A44848EB0780A216 005D4848130E5D153C153848485B5D4A5A0207C7FC000F131CB512F023227DA128>68 D<90B6128090380F00071501A2131EA21600A25BA2140192C7FCEB7802A21406140EEBFF FCEBF00CA33801E008A391C8FC485AA4485AA4120FEAFFFC21227DA120>70 D<027F1340903903C0C08090380E00214913130170130F49EB0700485A485A48C7FC4814 02120E121E5A5D4891C7FCA35AA4EC3FFF48EB00F0A34A5A7EA212704A5A7E001813076C 13096CEB1180380380E0D8007FC8FC22247DA226>I<9039FFF83FFE90390F0003C0A301 1EEB0780A449EB0F00A449131EA490B512FC9038F0003CA348485BA448485BA44848485A A4000F130339FFF83FFE27227DA128>I<9039FFF801FF010FC71278166016C0011EEB01 0015025D5D4913205D5D0202C7FC495A5C141C147CEBF0BEEBF11E13F2EBF80FEA01F001 E07F1407A248486C7EA36E7EEA0780811400A2000F497E39FFF80FFF28227DA129>75 D<9039FF8007FE010FEB00F016C0D90BC0134001131480A2EB11E0A2903921F001001320 A2147801401302147C143CA2496C5AA3140FD801005B15881407A20002EB03D0A215F014 01485C1400A2120C001E1440EAFFC027227DA127>78 D<90B512E090380F0038151E150E 011E1307A449130FA3151E5B153C157815E09038F003C09038FFFE0001F0C7FCA2485AA4 485AA4485AA4120FEAFFF820227DA11F>80 D<90B512C090380F0078151C81011E130F81 1680A249EB0F00A3151E495B5D15E0EC0380D9FFFCC7FCEBF0076E7E8148486C7EA44848 485AA44848485A1680A29138038100120F39FFF801C6C8127821237DA125>82 D<903803F81090380E0420903818026090382001E0EB400001C013C05B1201A200031480 A21500A27FEA01F013FE3800FFE06D7EEB1FF8EB01FCEB003C141C80A30020130CA31408 00601318141000705B5C00C85BD8C603C7FCEA81FC1C247DA21E>I<001FB512FE391E01 E00E001814061230382003C0A200401404A2EB07801280A20000140049C7FCA4131EA45B A45BA45BA41201387FFFC01F227EA11D>I86 D<90397FF803FF903907E000F84A13E001 0314C09138E001800101EB03001506ECF00401005B6E5AEC7820EC7C405D023DC7FC143E 141E141FA2142FEC6F8014C790380187C0140301027F1304EB080101107FEB2000497F49 137848C7FC48147CD80F8013FC3AFFE003FFC028227FA128>88 DI<90387FFF FE90387E001E0170133C4913784913F090388001E000011303010013C0EC07800002EB0F 00141EC75A147C14785C495A495A495A130F91C7FC131E4913205B49134012015B484813 80485A380F0001001EEB03005C485B48137EB512FE1F227DA121>I97 DI<133FEBE080380380C0EA0701EA0E03121C003CC7FCA25AA35A A400701340A23830018038380200EA1C1CEA07E012157E9415>I<141E14FC141CA31438 A41470A414E01378EA01C4EA0302380601C0120E121C123C383803801278A338F00700A3 1408EB0E101270131E38302620EA18C6380F03C017237EA219>I<137EEA038138070080 120E5A5A38780100EA7006EAFFF800F0C7FCA25AA41480A238700300EA3004EA1838EA0F C011157D9417>I<141EEC638014C71301ECC30014801303A449C7FCA4EBFFF8010EC7FC A65BA55BA55BA4136013E0A25BA21271EAF18090C8FC1262123C192D7EA218>I I<13F0EA07E01200A3485AA4485AA448C7FCEB0F80EB30C0EB4060380E8070EA0F00120E A24813E0A4383801C0A2EB0380148200701384EB07041408130300E01310386001E01723 7EA21C>I<13E0A21201EA00C01300A9121E1223EA4380A21283EA8700A21207120EA35A A3EA38201340127013801230EA3100121E0B227EA111>I<147014F0A214601400A9130F EB3180EB41C01381A2EA0101A238000380A4EB0700A4130EA45BA45BA3EA7070EAF0605B EA6380003EC7FC142C81A114>I<13F0EA0FE01200A3485AA4485AA448C7FC1478EB0184 EB021C380E0C3C1310EB2018EB4000485A001FC7FC13E0EA1C38487EA27F140838701C10 A3EB0C20EAE006386003C016237EA219>II<393C07E01F3A46183061 803A47201880C03A87401D00E0EB801E141C1300000E90383801C0A4489038700380A2ED 070016044801E01308150EA2ED0610267001C01320D83000EB03C026157E942B>I<383C 07C038461860384720303887403813801300A2000E1370A44813E0A2EB01C014C1003813 C2EB03821484130100701388383000F018157E941D>I<3803C0F03804631CEB740EEA08 78EB7007A2140FEA00E0A43801C01EA3143C38038038A2EBC07014E038072180EB1E0090 C7FCA2120EA45AA3B47E181F819418>112 D114 D<137E138138030080EA0201EA0603 140090C7FC120713F0EA03FC6CB4FCEA003FEB07801303127000F01300A2EAE002EA4004 EA3018EA0FE011157E9417>I<136013E0A4EA01C0A4EA0380EAFFFCEA0380A2EA0700A4 120EA45AA31308EA3810A21320EA184013C0EA0F000E1F7F9E12>I<001E13E0EA230138 4381F01380008313701430EA870000071320120EA3481340A21480A2EB0100A21302EA0C 04EA0618EA03E014157E9418>118 D<3801E0F03806310C38081A1C0010133CEA201C14 181400C65AA45BA314083860E01012F0142038E1704038423080383C1F0016157E941C> 120 D<001E131800231338EA438014701283EA8700A2000713E0120EA3381C01C0A4EB03 80A21307EA0C0B380E1700EA03E7EA0007A2130E1260EAF01C1318485AEA8060EA41C000 3FC7FC151F7E9418>II E /Fu 29 121 df45 D<144014E0A3497EA2497EEB0278 A2497EA3497EA2497EA3496C7EA201407F1403A290B57EA239018001F090C7FCA2000214 78A34880A2001E143E3AFFC003FFE0A223237DA229>65 D<903803F80290381FFF069038 7E03863901F000CE4848133ED80780131E48C7FC48140E001E1406123E123C007C1402A2 127800F891C7FCA7913807FFE01278007C9038001E00A2123C123E121E121F6C7E6C7E6C 6C132ED801F8136E39007E01C690381FFF02D903FCC7FC23247CA22A>71 D<3AFFFC1FFF80A23A078000F000AD90B5FCA2EB8000AF3AFFFC1FFF80A221227CA129> II<3801FFF8A238000780B3A512301278 12FCA21400485AEA400E6C5AEA1838EA07E015237DA11C>I80 D82 D<007FB6FCA2397801E00F0060140381124000C01580A2 00801400A400001500B3A290B512C0A221227DA127>84 D<1304130EA3131FA2EB2F8013 27A2EB43C0A2EBC3E01381A248C67EA2487F13FF38020078487FA3487F1218003C131F00 FEEB7FE01B1A7F991F>97 DIIIIII<39FFC1FF80391E003C00AB38 1FFFFC381E003CAC39FFC1FF80191A7E991F>I I108 D<00FEEB01FE001E14F0A200171302A238138004A33811C008A23810E010A3EB7020A3EB 3840A2EB1C80A3EB0F00A21306123800FEEB07FE1F1A7E9925>I<00FEEB3F80001FEB0E 00EB80041217EA13C0EA11E013F012101378137C133C131E130F14841307EB03C4EB01E4 A2EB00F4147CA2143C141C140C123800FE1304191A7E991F>III114 DI<007FB5FC38701E0700601301124000C0148000801300A30000 1400B0133F3803FFF0191A7F991D>I<39FFC03F80391E000E001404B2000E5B120F6C5B 6C6C5A3800E0C0013FC7FC191A7E991F>I<39FFC00FE0393F000380001E14007E140213 8000075BA26C6C5AA2EBE01800011310A26C6C5AA2EB7840A2137CEB3C80A2011FC7FCA3 130EA213041B1A7F991F>I<39FFE07F80391F803E00000F1318000713106C6C5A13E06C 6C5A00005B13F9017DC7FC7FA27F7FEB1780EB37C01323EB41E0EB81F0EA0180EB007800 02137C00067F000E131E001E133FB4EB7FE01B1A7F991F>120 D E /Fv 75 123 df<1460A214F0A2497E1478EB027C143CEB043E141EEB081F8001107F14 0701207F140301407F140101807F140048C77E15780002147C153C48143E151E48141F81 48158015074815C01503007FB612E0A2B712F024237EA229>1 D<5B497EA3497EA4EB09 E0A3EB10F0A3EB2078A3497EA3497EA348487EA30002EB0780A348EB03C0A3000C14E012 1E39FF801FFE1F237FA222>3 D12 DI<90380FC07F90 397031C0809039E00B00402601801E13E00003EB3E013807003C91381C00C01600A7B712 E03907001C011500B23A7FF1FFCFFE272380A229>I<127012F8A71270AC1220A61200A5 127012F8A3127005247CA30E>33 D<127012F812FCA212741204A41208A21210A2122012 40060F7CA20E>39 D<132013401380EA01005A12061204120CA25AA25AA312701260A312 E0AE1260A312701230A37EA27EA2120412067E7EEA0080134013200B327CA413>I<7E12 407E7E12187E12041206A27EA2EA0180A313C01200A313E0AE13C0A312011380A3EA0300 A21206A21204120C5A12105A5A5A0B327DA413>I<497EB0B612FEA23900018000B01F22 7D9C26>43 D<127012F812FCA212741204A41208A21210A212201240060F7C840E>II<127012F8A3127005057C840E>I48 D<13801203120F12F31203B3A9EA07C0EAFFFE0F217CA018>I< EA03F0EA0C1CEA100700201380384003C0A2008013E012F0EAF801A3EA2003120014C0A2 EB07801400130E5B13185B5B5B485A90C7FC000213205A5A00181360481340383FFFC05A B5FC13217EA018>II<1303A25BA25B1317A2 1327136713471387120113071202120612041208A212101220A2124012C0B512F8380007 00A7EB0F80EB7FF015217FA018>I<00101380381E0700EA1FFF5B13F8EA17E00010C7FC A6EA11F8EA120CEA1C07381803801210380001C0A214E0A4127012F0A200E013C01280EA 4003148038200700EA1006EA0C1CEA03F013227EA018>I<137EEA01C138030080380601 C0EA0C03121C381801800038C7FCA212781270A2EAF0F8EAF30CEAF4067F00F81380EB01 C012F014E0A51270A3003813C0A238180380001C1300EA0C06EA070CEA01F013227EA018 >I<12401260387FFFE014C0A23840008038C0010012801302A2485A5BA25B5BA2136013 4013C0A21201A25B1203A41207A76CC7FC13237DA118>III<127012F8A312701200AB127012F8A3127005157C 940E>I<127012F8A312701200AB127012F8A312781208A41210A312201240A2051F7C94 0E>I61 D<497EA3497EA3EB05E0A2EB09F013 08A2EB1078A3497EA3497EA2EBC01F497EA248B51280EB0007A20002EB03C0A348EB01E0 A348EB00F0121C003EEB01F839FF800FFF20237EA225>65 DI<9038 07E0109038381830EBE0063901C0017039038000F048C7FC000E1470121E001C1430123C A2007C14101278A200F81400A812781510127C123CA2001C1420121E000E14407E6C6C13 803901C001003800E002EB381CEB07E01C247DA223>IIII<903807F0 0890383C0C18EBE0023901C001B839038000F848C71278481438121E15185AA2007C1408 1278A200F81400A7EC1FFF0078EB00F81578127C123CA27EA27E7E6C6C13B86C7E3900E0 031890383C0C08903807F00020247DA226>I<39FFFC3FFF390FC003F039078001E0AE90 B5FCEB8001AF390FC003F039FFFC3FFF20227EA125>II<3803FFE038001F007FB3A6127012F8A2130EEAF01EEA401C6C 5AEA1870EA07C013237EA119>IIII< 39FF8007FF3907C000F81570D805E01320EA04F0A21378137C133C7F131F7FEB0780A2EB 03C0EB01E0A2EB00F014F81478143C143E141E140FA2EC07A0EC03E0A21401A21400000E 1460121FD8FFE0132020227EA125>II82 D<3803F020380C0C60EA1802383001E0EA700000601360 12E0A21420A36C1300A21278127FEA3FF0EA1FFE6C7E0003138038003FC0EB07E01301EB 00F0A214707EA46C1360A26C13C07E38C8018038C60700EA81FC14247DA21B>I<007FB5 12F839780780780060141800401408A300C0140C00801404A400001400B3A3497E3801FF FE1E227EA123>I<39FFFC07FF390FC000F86C4813701520B3A5000314407FA200011480 6C7E9038600100EB3006EB1C08EB03F020237EA125>I<3BFFF03FFC03FE3B1F8007E000 F86C486C48137017206E7ED807801540A24A7E2603C0021480A39039E004780100011600 A2EC083CD800F01402A2EC101E01785CA2EC200F013C5CA20260138890391E400790A216 D090391F8003F0010F5CA2EC00016D5CA20106130001025C2F237FA132>87 D<387FFFFE387E003E0078133C007013781260004013F012C0EB01E0388003C0A2EB0780 1200EB0F005B131E5BA25BA25B1201EBE001EA03C0A2EA07801403EA0F00001E1302A248 1306140E48131E00F8137EB512FE18227DA11E>90 D<12FEA212C0B3B3A912FEA207317B A40E>I<12FEA21206B3B3A912FEA207317FA40E>93 D97 D<120E12FE121E120EAB131FEB61C0EB8060380F0030000E1338143C14 1C141EA7141C143C1438000F1370380C8060EB41C038083F0017237FA21B>II<14E0130F13011300ABEA01F8EA0704EA0C02EA1C01EA38 001278127012F0A7127012781238EA1801EA0C0238070CF03801F0FE17237EA21B>II<133E13E33801C780EA03871307 48C7FCA9EAFFF80007C7FCB27FEA7FF0112380A20F>I<14703803F198380E1E18EA1C0E 38380700A200781380A400381300A2EA1C0EEA1E1CEA33F00020C7FCA212301238EA3FFE 381FFFC06C13E0383000F0481330481318A400601330A2003813E0380E03803803FE0015 217F9518>I<120E12FE121E120EABEB1F80EB60C0EB80E0380F0070A2120EAF38FFE7FF 18237FA21B>I<121C123EA3121CC7FCA8120E127E121E120EB1EAFFC00A227FA10E>I<13 E0EA01F0A3EA00E01300A81370EA07F012001370B3A51260EAF0E013C0EA6180EA3F000C 2C83A10F>I<120E12FE121E120EABEB03FCEB01F014C01480EB02005B5B5B133813F8EA 0F1CEA0E1E130E7F1480EB03C0130114E0EB00F014F838FFE3FE17237FA21A>I<120E12 FE121E120EB3ADEAFFE00B237FA20E>I<390E1FC07F3AFE60E183803A1E807201C03A0F 003C00E0A2000E1338AF3AFFE3FF8FFE27157F942A>I<380E1F8038FE60C0381E80E038 0F0070A2120EAF38FFE7FF18157F941B>III<3801F820380704 60EA0E02EA1C01003813E0EA7800A25AA71278A2EA3801121CEA0C02EA070CEA01F0C7FC A9EB0FFE171F7E941A>III<1202A41206A312 0E121E123EEAFFFCEA0E00AB1304A6EA07081203EA01F00E1F7F9E13>I<000E137038FE 07F0EA1E00000E1370AD14F0A238060170380382783800FC7F18157F941B>I<38FF80FE 381E00781430000E1320A26C1340A2EB80C000031380A23801C100A2EA00E2A31374A213 38A3131017157F941A>I<39FF8FF87F393E01E03C001CEBC01814E0000E1410EB026014 7000071420EB04301438D803841340EB8818141CD801C81380EBD00C140E3900F00F0049 7EA2EB6006EB400220157F9423>I<38FF83FE381F00F0000E13C06C1380EB8100EA0383 EA01C2EA00E41378A21338133C134E138FEA0187EB0380380201C0000413E0EA0C00383E 01F038FF03FE17157F941A>I<38FF80FE381E00781430000E1320A26C1340A2EB80C000 031380A23801C100A2EA00E2A31374A21338A31310A25BA35B12F05B12F10043C7FC123C 171F7F941A>I<383FFFC038380380EA300700201300EA600EEA401C133C1338C65A5B12 015B38038040EA07005A000E13C04813805AEA7801EA7007B5FC12157F9416>I E /Fw 70 128 df<137E3801C180EA0301380703C0120EEB018090C7FCA5B512C0EA0E01 B0387F87F8151D809C17>12 D<90383F07E03901C09C18380380F0D80701133C000E13E0 0100131892C7FCA5B612FC390E00E01CB03A7FC7FCFF80211D809C23>14 D<120EA2121E1238127012E012800707779C15>19 D<1380EA0100120212065AA25AA25A A35AA412E0AC1260A47EA37EA27EA27E12027EEA0080092A7C9E10>40 D<7E12407E12307EA27EA27EA37EA41380AC1300A41206A35AA25AA25A12205A5A092A7E 9E10>I<126012F0A212701210A41220A212401280040C7C830C>44 DI<126012F0A2126004047C830C>I48 D<5A1207123F12C71207B3A5EAFFF80D1C7C9B15>III<130CA2131C133CA2135C13DC139C EA011C120312021204120C1208121012301220124012C0B512C038001C00A73801FFC012 1C7F9B15>II<13F0EA03 0CEA0404EA0C0EEA181E1230130CEA7000A21260EAE3E0EAE430EAE818EAF00C130EEAE0 061307A51260A2EA7006EA300E130CEA1818EA0C30EA03E0101D7E9B15>I<1240387FFF 801400A2EA4002485AA25B485AA25B1360134013C0A212015BA21203A41207A66CC7FC11 1D7E9B15>III<126012F0A212601200AA126012F0A2126004127C910C>I<126012F0A2 12601200AA126012F0A212701210A41220A212401280041A7C910C>I<1306A3130FA3EB 1780A2EB37C01323A2EB43E01341A2EB80F0A338010078A2EBFFF83802003CA3487FA200 0C131F80001E5BB4EBFFF01C1D7F9C1F>65 DI<9038 1F8080EBE0613801801938070007000E13035A14015A00781300A2127000F01400A80070 14801278A212386CEB0100A26C13026C5B380180083800E030EB1FC0191E7E9C1E>III I<90381F8080EBE0613801801938070007000E13035A14015A00781300A2127000F01400 A6ECFFF0EC0F80007013071278A212387EA27E6C130B380180113800E06090381F80001C 1E7E9C21>I<39FFF0FFF0390F000F00AC90B5FCEB000FAD39FFF0FFF01C1C7F9B1F>II<3807FF8038007C00133CB3127012F8A21338EA 7078EA4070EA30E0EA0F80111D7F9B15>I76 DII80 D82 D<3807E080EA1C19EA30051303EA600112E01300A36C13007E127CEA7FC0EA3FF8EA 1FFEEA07FFC61380130FEB07C0130313011280A300C01380A238E00300EAD002EACC0CEA 83F8121E7E9C17>I<007FB512C038700F010060130000401440A200C014201280A30000 1400B1497E3803FFFC1B1C7F9B1E>I<39FFF01FF0390F000380EC0100B3A26C13021380 00035BEA01C03800E018EB7060EB0F801C1D7F9B1F>I<39FFE00FF0391F0003C0EC0180 6C1400A238078002A213C000035BA2EBE00C00011308A26C6C5AA213F8EB7820A26D5AA3 6D5AA2131F6DC7FCA21306A31C1D7F9B1F>I<3AFFE1FFC0FF3A1F003E003C001E013C13 186C6D1310A32607801F1320A33A03C0278040A33A01E043C080A33A00F081E100A39038 F900F3017913F2A2017E137E013E137CA2013C133C011C1338A20118131801081310281D 7F9B2B>I<39FFF07FC0390FC01E003807800CEBC00800035B6C6C5A13F000005BEB7880 137C013DC7FC133E7F7F80A2EB13C0EB23E01321EB40F0497E14783801007C00027F141E 0006131F001F148039FF807FF01C1C7F9B1F>I<12FEA212C0B3B312FEA207297C9E0C> 91 D<12FEA21206B3B312FEA20729809E0C>93 D97 D<12FC121CAA137CEA1D87381E0180381C00C014E014601470A6146014E014C0381E0180 38190700EA10FC141D7F9C17>IIII<13F8EA018CEA071E1206EA0E0C1300A6EAFFE0EA0E00B0EA7F E00F1D809C0D>II<12FC121CAA137C1387EA1D03001E1380121CAD38 FF9FF0141D7F9C17>I<1218123CA21218C7FCA712FC121CB0EAFF80091D7F9C0C>I<12FC 121CAAEB0FE0EB0780EB06005B13105B5B13E0121DEA1E70EA1C781338133C131C7F130F 148038FF9FE0131D7F9C16>107 D<12FC121CB3A9EAFF80091D7F9C0C>I<39FC7E07E039 1C838838391D019018001EEBE01C001C13C0AD3AFF8FF8FF8021127F9124>II II<3803E080EA0E19EA1805EA3807EA7003A212E0A6 1270A2EA38071218EA0E1BEA03E3EA0003A7EB1FF0141A7F9116>III< 1204A4120CA2121C123CEAFFE0EA1C00A91310A5120CEA0E20EA03C00C1A7F9910>I<38 FC1F80EA1C03AD1307120CEA0E1B3803E3F014127F9117>I<38FF07E0383C0380381C01 00A2EA0E02A2EA0F06EA0704A2EA0388A213C8EA01D0A2EA00E0A3134013127F9116>I< 39FF3FC7E0393C0703C0001CEB01801500130B000E1382A21311000713C4A213203803A0 E8A2EBC06800011370A2EB8030000013201B127F911E>I<38FF0FE0381E0700EA1C06EA 0E046C5AEA039013B0EA01E012007F12011338EA021C1204EA0C0E487E003C138038FE1F F014127F9116>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2EA0388A2 13C8EA01D0A2EA00E0A31340A25BA212F000F1C7FC12F312661238131A7F9116>III127 D E /Fx 47 122 df12 D45 D<1238127C12FEA3127C123807077C8610>I<13181378EA 01F812FFA21201B3A7387FFFE0A213207C9F1C>49 DI<13 FE3807FFC0380F07E0381E03F0123FEB81F8A3EA1F0314F0120014E0EB07C0EB1F803801 FE007F380007C0EB01F014F8EB00FCA2003C13FE127EB4FCA314FCEA7E01007813F8381E 07F0380FFFC03801FE0017207E9F1C>I<14E013011303A21307130F131FA21337137713 E7EA01C71387EA03071207120E120C12181238127012E0B6FCA2380007E0A790B5FCA218 207E9F1C>I<00301320383E01E0383FFFC0148014005B13F8EA33C00030C7FCA4EA31FC EA37FF383E0FC0383807E0EA3003000013F0A214F8A21238127C12FEA200FC13F0A23870 07E0003013C0383C1F80380FFF00EA03F815207D9F1C>II<1470A214F8A3497EA2497EA3EB067FA2010C7F143FA2496C7EA201307F 140F01707FEB6007A201C07F90B5FC4880EB8001A2D803007F14004880000680A23AFFE0 07FFF8A225227EA12A>65 D67 DIIIIII76 DIIII82 D<3801FE023807FF86381F01FE383C007E007C131E0078130EA200F8 1306A27E1400B4FC13E06CB4FC14C06C13F06C13F86C13FC000313FEEA003F1303EB007F 143FA200C0131FA36C131EA26C133C12FCB413F838C7FFE00080138018227DA11F>I<00 7FB61280A2397E03F80F00781407007014030060140100E015C0A200C01400A400001500 B3A248B512F0A222227EA127>II97 DIII<13FE3807FF80380F87C0381E01E0003E13F0EA 7C0014F812FCA2B5FCA200FCC7FCA3127CA2127E003E13186C1330380FC0703803FFC0C6 130015167E951A>II<3801FE0F3907FFBF80380F87C7381F03E7391E01 E000003E7FA5001E5BEA1F03380F87C0EBFF80D809FEC7FC0018C8FCA2121C381FFFE06C 13F86C13FE001F7F383C003F48EB0F80481307A40078EB0F006C131E001F137C6CB45A00 0113C019217F951C>II<121C123E127FA3123E121CC7FCA7B4FCA2121FB2EAFFE0A2 0B247EA310>I107 DI<3AFF07F007F090391FFC1FFC3A1F303E30 3E01401340496C487EA201001300AE3BFFE0FFE0FFE0A22B167E9530>I<38FF07E0EB1F F8381F307CEB403CEB803EA21300AE39FFE1FFC0A21A167E951F>I<13FE3807FFC0380F 83E0381E00F0003E13F848137CA300FC137EA7007C137CA26C13F8381F01F0380F83E038 07FFC03800FE0017167E951C>I<38FF0FE0EB3FF8381FE07CEB803E497E1580A2EC0FC0 A8EC1F80A29038803F00EBC03EEBE0FCEB3FF8EB0FC090C8FCA8EAFFE0A21A207E951F> I114 DI<487EA412 03A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB8300EA07C3EA03FEEA00F811207F9F 16>I<38FF01FEA2381F003EAF147E14FE380F81BE3907FF3FC0EA01FC1A167E951F>I<39 FFE07FC0A2390F801C006C6C5A6C6C5AEBF0606C6C5A3800F980137F6DC7FC7F80497E13 37EB63E0EBC1F03801C0F848487E3807007E000E133E39FF80FFE0A21B167F951E>120 D<39FFE01FE0A2391F800700000F1306EBC00E0007130C13E000035BA26C6C5AA26C6C5A A2EB7CC0A2137F6D5AA26DC7FCA2130EA2130CA25B1278EAFC3813305BEA69C0EA7F8000 1FC8FC1B207F951E>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 207 350 a Fx(ON)24 b(A)n(CTIONS)i(OF)e(INFINITESIMAL)g(GR)n (OUP)h(SCHEMES)350 474 y Fw(GAET)m(ANA)18 b(RESTUCCIA)g(AND)g(HANS-J) 1115 464 y(\177)1110 474 y(UR)o(GEN)g(SCHNEIDER)738 675 y Fv(1.)28 b Fu(Intr)o(oduction)243 762 y Fv(Let)19 b Ft(k)i Fv(b)q(e)f(a)f(\014eld)g(of)g(c)o(haracteristic)f Ft(p)h(>)g Fv(0)h(and)f Ft(H)24 b Fv(a)19 b(\014nite-dimensional)193 820 y(comm)o(utativ)n(e)13 b(Hopf)j(algebra)h(o)o(v)o(er)f Ft(k)i Fv(with)e(underlying)f(algebra)332 903 y Ft(H)j Fv(=)13 b Ft(k)r Fv([)p Ft(X)522 910 y Fs(1)542 903 y Ft(;)8 b(:)g(:)g(:)g(;)g(X)692 910 y Fr(n)715 903 y Fv(])p Ft(=)p Fv(\()p Ft(X)816 879 y Fr(p)834 868 y Fq(s)849 875 y Fp(1)812 915 y Fs(1)871 903 y Ft(;)g(:)g(:)g(:)f(;)h(X)1024 882 y Fr(p)1042 871 y Fq(s)1057 875 y(n)1020 915 y Fr(n)1082 903 y Fv(\))p Ft(;)g(n)14 b Fo(\025)f Fv(1)p Ft(;)8 b(s)1287 910 y Fs(1)1321 903 y Fo(\025)14 b(\001)8 b(\001)g(\001)14 b(\025)f Ft(s)1521 910 y Fr(n)1558 903 y Fo(\025)h Fv(1)p Ft(:)-1456 b Fv(\(1.1\))243 984 y(By)11 b(the)g(structure)g(theorem)f (of)i(in\014nitesimal,)e(that)i(is)f(\014nite)g(and)h(connected)193 1042 y(group)g(sc)o(hemes)d(\(see)h([W,)i(14.4]\),)f(an)o(y)g (\014nite-dimensional,)f(comm)o(utativ)n(e)e(and)193 1100 y(lo)q(cal)16 b(Hopf)g(algebra)h(o)o(v)o(er)e(a)i(p)q(erfect)f (\014eld)f(has)i(this)f(form.)243 1158 y(Let)j Ft(A)g Fv(b)q(e)h(a)f(comm)o(utativ)o(e)d(algebra,)k(and)g Ft(\016)g Fv(:)f Ft(A)g Fo(!)g Ft(A)12 b Fo(\012)h Ft(H)t Fv(,)20 b(a)g(righ)o(t)f Ft(H)t Fv(-)193 1216 y(como)q(dule)c(algebra)i (structure)f(on)g Ft(A)p Fv(.)243 1297 y(In)g(other)g(w)o(ords,)g(w)o (e)g(are)h(considering)f(an)g(action)807 1378 y Ft(X)g Fo(\002)11 b Ft(G)j Fo(!)g Ft(X)193 1459 y Fv(of)e(the)g (in\014nitesimal)d(group)k(sc)o(heme)d(represen)o(ted)h(b)o(y)g Ft(H)17 b Fv(on)12 b(the)g(a\016ne)g(sc)o(heme)193 1518 y(represen)o(ted)j(b)o(y)h Ft(A)p Fv(.)243 1599 y(F)l(or)g(an)o(y)g Ft(a)e Fo(2)g Ft(A)p Fv(,)h(w)o(e)h(can)g(write)241 1688 y Ft(\016)r Fv(\()p Ft(a)p Fv(\))d(=)g Ft(a)e Fo(\012)g Fv(1)g(+)581 1640 y Fn(X)564 1745 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)679 1688 y Ft(D)719 1695 y Fr(i)733 1688 y Fv(\()p Ft(a)p Fv(\))g Fo(\012)g Ft(x)886 1695 y Fr(i)910 1688 y Fv(+)28 b(terms)14 b(of)j(higher)f(order)g(in)g(the)g Ft(x)1612 1695 y Fr(i)1626 1688 y Ft(;)193 1818 y Fv(where)d(the)g Ft(x)440 1825 y Fr(i)467 1818 y Fv(are)h(the)f(residue)g(classes)g(of)h (the)f Ft(X)1116 1825 y Fr(i)1144 1818 y Fv(in)g(H,)f(and)i(where)f Ft(D)1531 1825 y Fr(i)1560 1818 y Fv(:)g Ft(A)h Fo(!)193 1876 y Ft(A;)8 b Fv(1)14 b Fo(\024)f Ft(i)h Fo(\024)f Ft(n)p Fv(,)j(are)g Ft(k)r Fv(-linear)g(deriv)m(ations)g(of)h Ft(A)p Fv(.)243 1934 y(Let)g Ft(R)d Fv(=)h Ft(A)472 1916 y Fr(coH)554 1934 y Fv(b)q(e)i(the)f(subalgebra)i(of)f Ft(A)f Fv(of)h(coin)o(v)m(arian)o(t)f(elemen)o(ts)d(under)193 1992 y(the)k(coaction)h Ft(\016)r Fv(.)25 b(The)18 b(extension)f Ft(R)g Fo(\032)f Ft(A)h Fv(is)h(called)e(an)i Ft(H)t Fv(-Galois)h(extension)193 2050 y(\(see)k([M)o(]\))f(\(and)i(the)f(pro) s(jection)f Ft(X)30 b Fo(!)c Ft(Y)34 b Fv(represen)o(ted)22 b(b)o(y)g(the)h(inclusion)193 2108 y Ft(R)15 b Fo(\032)f Ft(A)i Fv(is)g(called)f(a)i(principal)f(homogeneous)g(space)h(for)f Ft(G)p Fv(\))h(if)f(the)h(canonical)193 2166 y(map)587 2227 y Ft(A)10 b Fo(\012)673 2234 y Fr(R)713 2227 y Ft(A)j Fo(!)h Ft(A)d Fo(\012)f Ft(H)q(;)e(a)j Fo(\012)g Ft(b)j Fo(7!)f Ft(a\016)r Fv(\()p Ft(b)p Fv(\))p Ft(;)193 2298 y Fv(is)j(bijectiv)o(e.)243 2356 y(In)f(Theorem)f(4.1)h(w)o(e)g(pro)o (v)o(e)g(a)h(Jacobi)f(criterion)f(for)i Ft(H)t Fv(-Galois)g (extensions.)193 2414 y(F)l(or)j(lo)q(cal)h(algebras)g Ft(A)e Fv(this)h(criterion)g(sa)o(ys)g(that)h(the)f(follo)o(wing)g (statemen)o(ts)193 2472 y(are)d(equiv)m(alen)o(t:)p 193 2517 250 2 v 243 2563 a Fw(1991)e Fl(Mathematics)h(Subje)n(ct)g (Classi\014c)n(ation.)20 b Fw(Primary:)d(17B37;)c(Secondary:)18 b(16W30.)243 2613 y Fl(Key)d(wor)n(ds)f(and)i(phr)n(ases.)k Fw(Hopf)14 b(Galois)e(extensions,)j(in\014nitesimal)c(group)j(sc)o (hemes.)257 2662 y(V)m(ersion)g(of)f(Ma)o(y)g(10,)g(2001.)193 2712 y(This)h(w)o(ork)g(w)o(as)g(partially)f(supp)q(orted)j(b)o(y)e (the)h(Graduiertenk)o(olleg)e(of)h(the)h(Math.)k(Institut)193 2762 y(\(Univ)o(ersit\177)-21 b(at)28 b(M)q(\177)-22 b(unc)o(hen\))193 2811 y(.)931 2861 y Fk(1)p eop %%Page: 2 2 2 1 bop 193 131 a Fk(2)182 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h(HANS-J) 1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)256 217 y Fo(\017)k Fv(R)d Fo(\032)c Ft(A)j Fv(is)g(an)h Ft(H)t Fv(-Galois)g(extension.)256 275 y Fo(\017)j Fv(There)e(are)g(elemen)o (ts)d Ft(y)752 282 y Fs(1)772 275 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)905 282 y Fr(n)945 275 y Fo(2)17 b Ft(A)h Fv(suc)o(h)f(that)i(the)f(matrix) e(\()p Ft(D)1574 282 y Fr(i)1588 275 y Fv(\()p Ft(y)1631 282 y Fr(j)1649 275 y Fv(\)\))301 333 y(is)h(in)o(v)o(ertible)c(o)o(v)o (er)i Ft(A)p Fv(.)193 403 y(By)h(ren)o(um)o(b)q(ering)f(the)h Ft(y)662 410 y Fr(i)693 403 y Fv(w)o(e)g(can)h(assume)f(that)h(for)g (all)f(1)f Fo(\024)f Ft(m)g Fo(\024)h Ft(n)i Fv(the)f(ma-)193 461 y(trices)g(\()p Ft(D)383 468 y Fr(i)398 461 y Fv(\()p Ft(y)441 468 y Fr(j)459 461 y Fv(\)\))497 468 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)658 461 y Fv(are)h(in)o(v)o(ertible.) k(Assuming)c(this)g(stronger)h(condition)193 520 y(w)o(e)c(sho)o(w)g (as)h(an)f(application)g(of)h(our)f(criterion)f(for)h Ft(H)t Fv(-Galois)h(extensions)f(that)193 578 y(the)i(elemen)o(ts)528 660 y Ft(y)554 638 y Fr(\013)577 643 y Fp(1)552 672 y Fs(1)603 660 y Fo(\001)8 b(\001)g(\001)h Ft(y)696 639 y Fr(\013)719 643 y Fq(n)694 672 y Fr(n)741 660 y Ft(;)f Fv(0)14 b Fo(\024)g Ft(\013)885 667 y Fr(i)913 660 y Ft(<)g(p)989 639 y Fr(s)1005 644 y Fq(i)1032 660 y Fo(\000)d Fv(1)p Ft(;)d Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)-1152 b Fv(\(1.2\))193 745 y(form)15 b(an)i Ft(R)p Fv(-basis)g(of)g Ft(A)f Fv(\(Theorem)f(4.3\).)243 807 y(Under)23 b(the)g(additional)g (assumption)h(that)g(all)f(the)g(p)q(o)o(w)o(ers)h Ft(y)1466 784 y Fr(p)1484 772 y Fq(s)1499 780 y(i)1464 820 y Fr(i)1538 807 y Fv(are)g Ft(H)t Fv(-)193 866 y(coin)o(v)m(arian)o(t)16 b(w)o(e)g(obtain)h(rather)g(precise)e(results)h(ab)q(out)i(the)f (structure)f(of)h(the)193 924 y(ring)22 b(extension)f Ft(R)i Fo(\032)g Ft(A)e Fv(for)h(regular)f(lo)q(cal)h(rings)f Ft(A)h Fv(\(Theorem)e(5.1\).)37 b(F)l(or)193 982 y(example)20 b(if)i Ft(A)f Fv(is)h(lo)q(cal)h(regular)f(and)h(complete,)e(then)h (the)g(ring)g(extension)193 1040 y Ft(R)14 b Fo(\032)g Ft(A)i Fv(is)g(isomorphic)f(to)h(the)g(extension)282 1124 y Ft(F)7 b Fv([[)p Ft(Y)387 1100 y Fr(p)405 1089 y Fq(s)420 1096 y Fp(1)377 1136 y Fs(1)440 1124 y Ft(;)h(:)g(:)g(:)g (:Y)580 1103 y Fr(p)598 1092 y Fq(s)613 1096 y(n)570 1136 y Fr(n)638 1124 y Ft(;)g(Z)693 1131 y Fr(n)p Fs(+1)762 1124 y Ft(;)g(:)g(:)g(:)f(;)h(Z)904 1131 y Fr(d)925 1124 y Fv(]])13 b Fo(\032)g Ft(F)7 b Fv([[)p Ft(Y)1113 1131 y Fs(1)1132 1124 y Ft(;)h(:)g(:)g(:)f(:Y)1261 1131 y Fr(n)1285 1124 y Ft(;)h(Z)1340 1131 y Fr(n)p Fs(+1)1408 1124 y Ft(;)g(:)g(:)g(:)g(;)g(Z)1551 1131 y Fr(d)1571 1124 y Fv(]])193 1206 y(of)14 b(formal)f(p)q(o)o(w)o(er)i(series)e (rings)i(o)o(v)o(er)e(a)h(co)q(e\016cien)o(t)f(\014eld)h Ft(k)h Fo(\032)f Ft(F)20 b Fo(\032)14 b Ft(R)h Fv(of)f Ft(R)h Fv(and)193 1264 y Ft(A)p Fv(.)243 1346 y(Theorem)g(5.1)i(w)o(as) g(kno)o(wn)g(b)q(efore)f(in)g(some)g(sp)q(ecial)g(cases.)22 b(The)17 b(Lie)f(alge-)193 1404 y(bra)e(case,)f(that)g(is)g(when)h Ft(s)687 1411 y Fs(1)720 1404 y Fv(=)g Fo(\001)8 b(\001)g(\001)14 b Fv(=)g Ft(s)919 1411 y Fr(n)956 1404 y Fv(=)g(1,)f(w)o(as)h(sho)o(wn) g(in)f([RM)o(])g(in)f(analogy)193 1463 y(to)17 b(a)f(theorem)f(of)h (Lipman)g([L,)g(Theorem)e(2])j(on)f(in)o(v)m(arian)o(ts)g(of)h(deriv)m (ations)f(in)193 1521 y(c)o(haracteristic)k(zero.)36 b(The)21 b(case)g Ft(n)i Fv(=)f(1)f(of)k(\(1.1\))d(but)f(for)h(co)q (comm)o(utativ)o(e)193 1579 y(Hopf)d(algebras)h Ft(H)k Fv(o)o(v)o(er)18 b(separably)i(closed)f(\014elds)g(w)o(as)h(treated)f (in)g([R)l(T].)30 b(In)193 1637 y(this)18 b(pap)q(er)h Ft(H)k Fv(w)o(as)18 b(co)q(comm)o(utativ)o(e)d(in)j(order)g(to)h(apply) f(the)g(theory)g(of)h(for-)193 1695 y(mal)13 b(groups)j(in)f(an)g (essen)o(tial)f(w)o(a)o(y)l(.)20 b(Then)15 b(the)f(results)h(in)f([R)l (T])g(w)o(ere)g(partially)193 1753 y(generalized)f(in)h([R)o(U)o(])f (to)i(the)f(case)g(when)g Ft(n)h Fv(is)f(arbitrary)g(and)h Ft(H)k Fv(is)14 b(co)q(comm)o(u-)193 1811 y(tativ)o(e.)243 1893 y(The)21 b(condition)g(ab)q(out)h(the)f(in)o(v)o(ertibili)o(t)o(y) d(of)j(the)g(matrix)e(\()p Ft(D)1440 1900 y Fr(i)1454 1893 y Fv(\()p Ft(y)1497 1900 y Fr(j)1515 1893 y Fv(\)\))1553 1900 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)193 1952 y Fv(app)q(ears)c(in)f(all)f(these)g(pap)q(ers.)22 b(W)l(e)13 b(explain)g(this)h(condition)g(as)g(a)g(criterion)f(for)193 2010 y Ft(H)t Fv(-Galois)19 b(extensions,)f(and)h(then)g(w)o(e)e(can)i (use)f(the)g(prop)q(erties)h(of)f Ft(H)t Fv(-Galois)193 2068 y(extensions)f(to)g(ac)o(hiev)o(e)f(inductiv)o(e)f(argumen)o(ts)h (for)h(arbitrary)h(Hopf)f(algebras)193 2126 y(satisfying)g(\(1.1\).)243 2208 y(In)h(Section)g(3)h(w)o(e)f(consider)g(coactions)h(of)g(the)g (additiv)o(e)e(group)i(where)g(the)193 2266 y Ft(x)221 2273 y Fr(i)254 2266 y Fv(are)h(primitiv)n(e)c(elemen)o(ts)h(of)i(the)h (Hopf)f(algebra)h Ft(H)t Fv(.)31 b(In)19 b(this)g(Section)g(the)193 2324 y(algebra)g Ft(A)f Fv(is)g(an)g(arbitrary)h(not)f(necessarily)g (comm)o(utati)o(v)o(e)d(algebra.)28 b(F)l(or)18 b(a)193 2382 y(general)h(Hopf)h(algebra)g(H)g(satisfying)g(\(1.1\))g(w)o(e)f (do)h(not)h(kno)o(w)e(whether)h(our)193 2440 y(results)c(can)g(b)q(e)h (generalized)e(to)h(coactions)h(on)g(non-comm)o(utativ)o(e)c(algebras.) 264 2550 y(2.)27 b Fu(Preliminaries)17 b(on)h(Hopf)g(Galois)g (extensions)e(and)i(free)879 2608 y(a)o(ctions)243 2695 y Fv(Let)g Ft(H)k Fv(b)q(e)c(a)h(Hopf)f(algebra)g(o)o(v)o(er)f(the)h (\014eld)g Ft(k)i Fv(with)e(com)o(ultipli)o(cation)d(\001)i(:)193 2753 y Ft(H)23 b Fo(!)c Ft(H)e Fo(\012)c Ft(H)t Fv(,)20 b(counit)f Ft(")g Fv(:)f Ft(H)24 b Fo(!)18 b Ft(k)k Fv(and)e(an)o(tip)q (o)q(de)g Ft(S)h Fv(:)e Ft(H)k Fo(!)c Ft(H)t Fv(.)30 b(W)l(e)19 b(will)193 2811 y(use)j(the)f(Sw)o(eedler)f(notation)j (\001\()p Ft(x)p Fv(\))f(=)i Ft(x)999 2819 y Fs(\(1\))1060 2811 y Fo(\012)15 b Ft(x)1142 2819 y Fs(\(2\))1210 2811 y Fv(for)22 b Ft(x)h Fo(2)g Ft(H)t Fv(.)38 b(If)21 b Ft(V)33 b Fv(is)22 b(a)p eop %%Page: 3 3 3 2 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)227 b(3)193 217 y Fv(righ)o(t)16 b Ft(H)t Fv(-como)q(dule)f(with)h(structure)g(map)f Ft(\016)g Fv(:)f Ft(V)25 b Fo(!)14 b Ft(V)22 b Fo(\012)10 b Ft(H)t Fv(,)16 b(w)o(e)g(will)f(use)h(the)193 275 y(notation)h Ft(\016)r Fv(\()p Ft(v)r Fv(\))c(=)h Ft(v)565 283 y Fs(\(0\))623 275 y Fo(\012)c Ft(v)696 283 y Fs(\(1\))759 275 y Fv(for)17 b(all)f Ft(v)f Fo(2)f Ft(V)e Fv(.)21 b(The)16 b(augmen)o(tation)g (ideal)f(of)i Ft(H)193 333 y Fv(will)e(b)q(e)h(denoted)h(b)o(y)e Ft(H)647 315 y Fs(+)691 333 y Fv(=)f(k)o(er)o(\()p Ft(")p Fv(\))p Ft(:)243 391 y Fv(Let)g(us)h(\014rst)g(recall)e(some)g(notions) i(and)g(results)f(from)f(Hopf)i(algebra)g(theory)193 450 y([M)o(].)193 539 y Fx(De\014nition)j(2.1.)23 b Fv(Let)16 b Ft(A)g Fv(b)q(e)h(an)f(algebra,)h(and)780 622 y Ft(\016)e Fv(:)f Ft(A)f Fo(!)h Ft(A)c Fo(\012)h Ft(H)193 704 y Fv(an)18 b(algebra)g(map)f(and)i(a)f(righ)o(t)f Ft(H)t Fv(-como)q(dule)h(structure)f(on)h Ft(A)p Fv(.)26 b(Then)18 b(\()p Ft(A;)8 b(\016)r Fv(\))193 763 y(is)16 b(called)f(a)i(righ)o(t)f Ft(H)t Fx(-como)r(dule)g(algebra)p Fv(,)f(and)617 845 y Ft(A)654 825 y Fr(coH)734 845 y Fv(=)e Fo(f)p Ft(a)h Fo(2)g Ft(A)f Fo(j)h Ft(\016)r Fv(\()p Ft(a)p Fv(\))f(=)g Ft(a)e Fo(\012)g Fv(1)p Fo(g)193 928 y Fv(is)16 b(the)g(subalgebra)h (of)g Ft(H)t Fv(-coin)o(v)m(arian)o(t)f(elemen)o(ts.)243 986 y(Let)23 b(\()p Ft(A;)8 b(\016)r Fv(\))22 b(b)q(e)h(a)h(righ)o(t)e Ft(H)t Fv(-como)q(dule)h(algebra,)i(and)e Ft(R)j Fv(:=)f Ft(A)1482 968 y Fr(coH)1548 986 y Fv(.)41 b(The)193 1044 y(extension)22 b Ft(R)i Fo(\032)g Ft(A)e Fv(is)g(called)f(an)h Ft(H)t Fx(-Galois)j(extension)c Fv(and)i Ft(\016)g Fv(is)f(called)193 1102 y Ft(H)t Fv(-Galois,)17 b(if)e(the)h(canonical)g(map)374 1185 y(can)e(:)g Ft(A)c Fo(\012)575 1192 y Fr(R)615 1185 y Ft(A)j Fo(!)h Ft(A)d Fo(\012)g Ft(H)q(;)d(x)j Fo(\012)g Ft(y)k Fo(7!)f Ft(x\016)r Fv(\()p Ft(y)r Fv(\))e(=)i Ft(xy)1314 1193 y Fs(\(0\))1372 1185 y Fo(\012)c Ft(y)1445 1193 y Fs(\(1\))1492 1185 y Ft(;)193 1268 y Fv(is)16 b(bijectiv)o(e.)243 1357 y(Let)g(no)o(w)h Ft(H)j Fv(b)q(e)d(a)f (\014nite-dimensional)e(Hopf)j(algebra.)243 1415 y(By)k(a)h(fundamen)o (tal)e(theorem)h(of)h(Larson)h(and)g(Sw)o(eedler)d([M)o(,)j(2.1.3])f (the)193 1473 y(space)c(of)f(left)g(in)o(tegrals)g(of)h Ft(H)k Fv(is)17 b(one-dimensional.)24 b(A)17 b(left)f(in)o(tegral)h (\003)g(of)h Ft(H)193 1532 y Fv(is)e(an)h(elemen)o(t)c(of)j Ft(H)21 b Fv(suc)o(h)16 b(that)h Ft(x)p Fv(\003)c(=)h Ft(")p Fv(\()p Ft(x)p Fv(\)\003)h(for)i(all)f Ft(x)d Fo(2)h Ft(H)t Fv(.)193 1621 y Fx(Theorem)i(2.2.)24 b Fj([Kr)n(eimer)d(and)g(T)l(akeuchi])h(L)n(et)e Fv(\003)h Fj(b)n(e)g(a)g(non-zer)n(o)g(inte)n(gr)n(al)193 1679 y(of)d Ft(H)t Fj(,)g(and)g Ft(\016)d Fv(:)f Ft(A)g Fo(!)h Ft(A)c Fo(\012)g Ft(H)22 b Fj(a)17 b(right)h Ft(H)t Fj(-c)n(omo)n(dule) g(algebr)n(a)h(with)f Ft(R)d Fv(=)f Ft(A)1608 1661 y Fr(coH)1674 1679 y Ft(:)193 1737 y Fj(Then)k(the)g(fol)r(lowing)i(ar)n (e)c(e)n(quivalent:)218 1808 y Fv(\(1\))21 b Ft(R)15 b Fo(\032)e Ft(A)k Fj(is)h(an)f Ft(H)t Fj(-Galois)h(extension.)218 1866 y Fv(\(2\))j Fj(The)d(c)n(anonic)n(al)g(map)f(c)n(an)d Fv(:)g Ft(A)c Fo(\012)922 1873 y Fr(R)962 1866 y Ft(A)k Fo(!)f Ft(A)e Fo(\012)g Ft(H)21 b Fj(is)d(surje)n(ctive.)218 1924 y Fv(\(3\))j(1)12 b Fo(\012)f Fv(\003)i Fo(2)h Ft(A)d Fo(\012)g Ft(H)22 b Fj(is)17 b(in)h(the)g(image)g(of)f(the)h(c)n (anonic)n(al)g(map.)193 1994 y(In)f(this)f(c)n(ase,)h Ft(A)g Fj(is)f(a)h(\014nitely)h(gener)n(ate)n(d)f(and)g(pr)n(oje)n (ctive)g Ft(R)p Fj(-left)h(and)f Ft(R)p Fj(-right)193 2052 y(mo)n(dule.)23 b(In)17 b(p)n(articular,)g Ft(A)g Fj(is)g(faithful)r(ly)i(\015at)e(over)h Ft(R)g Fj(if)f Ft(R)h Fj(is)f(c)n(ommutative.)193 2142 y(Pr)n(o)n(of.)i Fv(The)g(equiv)m(alence)f(of)h(\(1\))h(and)f(\(2\),)h(and)g(the)f(fact) g(that)h Ft(A)f Fv(is)g(\014nitely)193 2200 y(generated)g(and)h(pro)s (jectiv)o(e)e(o)o(v)o(er)g Ft(R)i Fv(on)g(b)q(oth)h(sides)e(is)g(a)h (result)f(of)h(Kreimer)193 2258 y(and)h(T)l(ak)o(euc)o(hi)e(\(see)h([M) o(,)h(8.3.1]\).)33 b(Since)20 b(can)h(:)f Ft(A)14 b Fo(\012)1227 2265 y Fr(R)1269 2258 y Ft(A)21 b Fo(!)f Ft(A)14 b Fo(\012)f Ft(H)25 b Fv(is)20 b(left)193 2316 y Ft(A)p Fv(-linear)15 b(and)i(righ)o(t)f Ft(H)t Fv(-colinear)g(with)h(the)f(ob)o(vious)g (structure)g(maps,)g(and)h(\003)193 2374 y(generates)d Ft(H)19 b Fv(as)c(a)g(left)e Ft(H)688 2356 y Fm(\003)708 2374 y Fv(-mo)q(dule)g(or)i(as)g(a)g(righ)o(t)f Ft(H)t Fv(-como)q(dule,)f(\(2\))i(and)g(\(3\))193 2433 y(are)k(equiv)m(alen)o (t.)29 b(Finally)l(,)18 b(if)h Ft(R)h Fv(is)f(comm)o(utativ)n(e,)e (then)i Ft(A)g Fv(is)g(faithfully)f(\015at)193 2491 y(o)o(v)o(er)c Ft(R)p Fv(,)h(since)f Ft(A)517 2498 y Fi(m)560 2491 y Fv(is)h(non-zero)g(and)g(free)f(o)o(v)o(er)g Ft(R)1129 2498 y Fi(m)1173 2491 y Fv(for)h(an)o(y)f(maximal)e(ideal)i Fh(m)193 2549 y Fv(of)j Ft(R)p Fv(.)p 1652 2607 2 33 v 1654 2576 30 2 v 1654 2607 V 1683 2607 2 33 v 243 2695 a(The)e(next)g(result)g(is)g(useful)g(for)g(induction)g(argumen)o(ts)g (on)g(the)h(order)f(of)h Ft(H)t Fv(.)193 2753 y(A)d(Hopf)h(subalgebra)h Ft(K)j Fo(\032)13 b Ft(A)h Fv(is)g(called)e(normal)h(if)h(for)g(all)f Ft(x)h Fo(2)g Ft(H)t Fv(,)g(and)g Ft(y)i Fo(2)e Ft(K)t Fv(,)193 2811 y Ft(x)221 2819 y Fs(\(1\))268 2811 y Ft(y)r(S)s Fv(\()p Ft(x)374 2819 y Fs(\(2\))420 2811 y Fv(\))22 b Fo(2)g Ft(K)j Fv(\(or)d(equiv)m(alen)o(tly)c Ft(S)s Fv(\()p Ft(x)1021 2819 y Fs(\(1\))1068 2811 y Fv(\))p Ft(y)r(x)1141 2819 y Fs(\(2\))1210 2811 y Fo(2)k Ft(K)t Fv(\).)35 b(Note)21 b(that)g(an)o(y)p eop %%Page: 4 4 4 3 bop 193 131 a Fk(4)182 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h(HANS-J) 1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)193 217 y Fv(Hopf)h(subalgebra)h(is)f(trivially)e(normal)h(if)h Ft(H)k Fv(is)c(comm)o(utativ)o(e)o(.)k(If)c Ft(K)i Fo(\032)c Ft(H)22 b Fv(is)193 275 y(normal,)15 b(then)p 483 235 45 2 v 16 w Ft(H)j Fv(=)c Ft(H)q(=H)t(K)747 257 y Fs(+)794 275 y Fv(is)i(a)h(quotien)o(t)e(Hopf)h(algebra)h(of)f Ft(H)t Fv(.)243 333 y(W)l(e)g(will)f(sa)o(y)h(that)h(an)g Ft(H)t Fv(-Galois)g(extension)f Ft(R)e Fo(\032)g Ft(A)i Fv(is)g(faithfully)f(\015at)i(if)f Ft(A)193 391 y Fv(is)f(faithfully)g (\015at)h(o)o(v)o(er)e Ft(R)i Fv(as)g(a)g(left)f(\(or)h(equiv)m(alen)o (tly)d(righ)o(t\))i(mo)q(dule)g(o)o(v)o(er)f Ft(R)p Fv(.)193 485 y Fx(Theorem)i(2.3.)24 b Fv([MS)o(,)13 b(6.1,)h(6.2])g Fj(L)n(et)f Ft(K)18 b Fo(\032)c Ft(H)k Fj(b)n(e)d(a)f(normal)g(Hopf)g (sub)n(algebr)n(a,)193 543 y(and)22 b Ft(R)g Fo(\032)g Ft(A)f Fj(a)h(faithful)r(ly)h(\015at)e Ft(H)t Fj(-)h(Galois)g (extension)i(with)e(structur)n(e)g(map)193 601 y Ft(\016)17 b Fv(:)d Ft(A)h Fo(!)g Ft(A)c Fo(\012)g Ft(H)t Fj(.)25 b(L)n(et)18 b Ft(\031)e Fv(:)f Ft(H)k Fo(!)p 841 561 V 15 w Ft(H)g Fv(=)d Ft(H)q(=H)t(K)1108 583 y Fs(+)1156 601 y Fj(b)n(e)j(the)g(quotient)g(map,)f(and)p 193 621 24 2 v 193 662 a Ft(\016)i Fv(=)e(\()p Fj(id)13 b Fo(\012)g Ft(\031)r Fv(\))p Ft(\016)19 b Fv(:)f Ft(A)g Fo(!)g Ft(A)13 b Fo(\012)p 761 622 45 2 v 12 w Ft(H)25 b Fj(the)20 b(induc)n(e)n(d)p 1087 622 V 20 w Ft(H)t Fj(-mo)n(dule)h(algebr)n(a)f(structur)n(e.)193 727 y(De\014ne)f Ft(B)d Fv(=)e Ft(A)489 709 y Fr(co)p 521 682 32 2 v(H)555 727 y Ft(:)243 785 y Fj(Then)19 b Ft(R)d Fo(\032)f Ft(B)21 b Fj(is)e(a)f(faithful)r(ly)h(\015at)g Ft(K)t Fj(-Galois)f(extension)j(wher)n(e)d(the)h(c)n(o)n(ac-)193 843 y(tion)24 b(isthe)g(r)n(estriction)f(of)h Ft(\016)r Fj(,)g(and)g Ft(B)j Fo(\032)e Ft(A)e Fj(is)h(a)f(faithful)r(ly)i (\015at)p 1491 803 45 2 v 23 w Ft(H)t Fj(-Galois)193 902 y(extension)19 b(with)f(c)n(o)n(action)p 705 861 24 2 v 18 w Ft(\016)q Fj(.)243 995 y Fv(One)e(ingredien)o(t)g(of)h(the) f(pro)q(of)i(of)f(Theorem)e(4.2)i(is)g(the)f(basic)h(Theorem)e(of)193 1053 y(Nic)o(hols)d(and)i(Zo)q(eller)e(\([M)o(,)h(3.1.5]\))g(whic)o(h)g (sa)o(ys)g(that)g Ft(H)18 b Fv(is)13 b(free)f(o)o(v)o(er)g(an)o(y)h (Hopf)193 1111 y(subalgebra.)243 1169 y(W)l(e)i(will)e(apply)i(Theorem) f(2.3)h(in)g(the)g(case)g(when)g Ft(H)k Fv(is)c(comm)o(utativ)n(e,)d (and)193 1227 y Ft(k)18 b Fv(has)f(c)o(haracteristic)e Ft(p)f(>)g Fv(0.)22 b(Then)793 1318 y Ft(H)837 1297 y Fs([1])890 1318 y Fv(:=)14 b Ft(k)r Fv([)p Ft(H)1041 1297 y Fr(p)1060 1318 y Fv(])p Ft(;)193 1407 y Fv(the)k(subalgebra)i (generated)e(b)o(y)g(all)g Ft(x)918 1389 y Fr(p)938 1407 y Ft(;)8 b(x)17 b Fo(2)h Ft(H)t Fv(,)h(is)f(a)h(Hopf)f(subalgebra,)i (and)193 1465 y Ft(H)q(=)p Fv(\()p Ft(x)305 1447 y Fr(p)339 1465 y Fo(j)14 b Ft(x)f Fo(2)i Ft(H)500 1447 y Fs(+)529 1465 y Fv(\))i(is)f(the)g(quotien)o(t)f(Hopf)h(algebra.)193 1559 y Fx(Remark)g(2.4.)24 b Fv(Let)11 b Ft(H)k Fv(b)q(e)c(comm)o (utativ)n(e,)e(and)i Ft(A)f Fv(a)i(comm)o(utativ)n(e)7 b Ft(H)t Fv(-como)q(dule)193 1617 y(algebra)18 b(with)f(structure)g (map)g Ft(\016)g Fv(:)e Ft(A)h Fo(!)f Ft(A)d Fo(\012)f Ft(H)t Fv(.)25 b(In)17 b(the)g(language)i(of)f(a\016ne)193 1675 y(sc)o(hemes)f(and)j(a\016ne)f(group)h(sc)o(hemes)d(\(see)i([DG],) g([W]\),)g Ft(H)24 b Fv(de\014nes)19 b(a)g(func-)193 1733 y(tor)g Ft(G)f Fv(from)g(comm)o(utativ)n(e)d Ft(k)r Fv(-algebras)k(to)g(groups,)h Ft(A)e Fv(de\014nes)g(a)h(functor)f Ft(X)193 1791 y Fv(from)e(comm)o(utativ)n(e)d Ft(k)r Fv(-algebras)18 b(to)f(sets)g(represen)o(ted)e(b)o(y)h Ft(H)21 b Fv(resp.)i Ft(A)p Fv(,)16 b(and)h Ft(\016)193 1849 y Fv(describ)q(es)f(an)h(action)767 1918 y Ft(\026)d Fv(:)f Ft(X)j Fo(\002)10 b Ft(G)15 b Fo(!)e Ft(X)q(;)193 1998 y Fv(that)j(is)g(for)g(all)f(comm)o(utativ)n(e)e(algebras)j Ft(T)22 b Fv(an)16 b(action)g(of)g(the)g(group)g Ft(G)p Fv(\()p Ft(T)7 b Fv(\))16 b(on)193 2056 y(the)g(set)g Ft(X)t Fv(\()p Ft(T)7 b Fv(\).)21 b(W)l(e)16 b(will)f(write)h Ft(\026)p Fv(\()p Ft(x;)8 b(g)r Fv(\))14 b(=)f Ft(xg)19 b Fv(for)d(all)g Ft(x)d Fo(2)h Ft(X)t Fv(\()p Ft(T)7 b Fv(\))p Ft(;)h(g)16 b Fo(2)e Ft(G)p Fv(\()p Ft(T)7 b Fv(\))p Ft(:)243 2114 y Fv(Recall)18 b(that)i(for)f(all)g(comm)o (utativ)n(e)d(algebras)k Ft(T)7 b Fv(,)19 b(the)g(group)i(structure)e (in)193 2172 y Ft(G)p Fv(\()p Ft(T)7 b Fv(\))22 b(=)g(Alg\()p Ft(H)q(;)8 b(T)f Fv(\))21 b(is)g(de\014ned)g(b)o(y)f(\()p Ft(f)g Fo(\001)14 b Ft(g)r Fv(\)\()p Ft(h)p Fv(\))23 b(=)f Ft(f)5 b Fv(\()p Ft(h)1279 2180 y Fs(\(1\))1327 2172 y Fv(\))p Ft(g)r Fv(\()p Ft(h)1418 2180 y Fs(\(2\))1465 2172 y Fv(\),)22 b(and)g(the)193 2230 y(action)g(is)h(giv)o(en)e(b)o(y) h(\()p Ft(xg)r Fv(\)\()p Ft(a)p Fv(\))h(=)i Ft(x)p Fv(\()p Ft(a)922 2238 y Fs(\(0\))968 2230 y Fv(\))p Ft(g)r Fv(\()p Ft(a)1057 2238 y Fs(\(1\))1104 2230 y Fv(\))p Ft(;)d Fv(for)g(all)g Ft(f)s(;)8 b(g)26 b Fo(2)f Ft(G)p Fv(\()p Ft(T)7 b Fv(\))p Ft(;)h(x)23 b Fo(2)193 2288 y Ft(X)t Fv(\()p Ft(T)7 b Fv(\))13 b(=)h(Alg\()p Ft(A;)8 b(T)f Fv(\))p Ft(;)h(t)k Fo(2)i Ft(T)7 b Fv(.)243 2346 y(Then)17 b(can)e(:)g Ft(A)c Fo(\012)575 2353 y Fr(R)615 2346 y Ft(A)k Fo(!)f Ft(A)d Fo(\012)h Ft(H)21 b Fv(is)c(surjectiv)o(e)e(if)h (and)i(only)f(if)f(the)h Fx(action)193 2404 y Fv(of)h Ft(G)h Fv(on)g Ft(X)j Fv(is)c(\(\014xed)g(p)q(oin)o(t\))g Fx(free)g Fv(in)g(the)f(sense)i(that)f(for)h(all)e(comm)o(utativ)o(e) 193 2462 y(algebras)j Ft(T)7 b Fv(,)19 b(and)h(all)f Ft(g)j Fo(2)d Ft(G)p Fv(\()p Ft(T)7 b Fv(\))p Ft(;)h(x)19 b Fo(2)g Ft(X)t Fv(\()p Ft(T)7 b Fv(\))20 b(the)f(equalit)o(y)f Ft(xg)j Fv(=)e Ft(x)g Fv(is)h(only)193 2521 y(p)q(ossible)c(for)h Ft(g)f Fv(=)e(1)p Ft(:)243 2579 y Fv(The)20 b(subgroup)i Ft(G)601 2586 y Fs(\(1\))670 2579 y Fv(of)f Ft(G)g Fv(represen)o(ted)e (b)o(y)h(the)h(quotien)o(t)e(Hopf)i(algebra)193 2637 y Ft(H)q(=)p Fv(\()p Ft(x)305 2619 y Fr(p)339 2637 y Fo(j)14 b Ft(x)f Fo(2)i Ft(H)500 2619 y Fs(+)529 2637 y Fv(\))i(is)f(called)f(the)h(\014rst)g(F)l(rob)q(enius)h(k)o(ernel)d (of)j Ft(G)p Fv(.)243 2695 y(Theorem)g(2.2)i(in)f(the)g(comm)o(utativ)o (e)d(case)j(is)h(the)f(fundamen)o(tal)f(result)h(on)193 2753 y(free)c(actions)h(of)h(\014nite)e(a\016ne)h(group)h(sc)o(hemes)d (on)i(a\016ne)g(sc)o(hemes)e(whic)o(h)h(go)q(es)193 2811 y(bac)o(k)i(to)g(Grothendiec)o(k.)p eop %%Page: 5 5 5 4 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)227 b(5)243 217 y Fv(Let)16 b(us)h(no)o(w)f(\014x)g (some)g Fx(notations)g Fv(and)h(assumptions.)243 275 y(In)e(this)g(pap)q(er)h(w)o(e)f(assume)g(that)g Ft(k)j Fv(is)d(a)h(\014eld)f(of)g(c)o(haracteristic)f Ft(p)h(>)e Fv(0,)j(and)193 333 y Ft(H)j Fv(will)14 b(alw)o(a)o(ys)h(b)q(e)h(a)f (comm)o(utativ)o(e)d(Hopf)j(algebra)g(with)g(underlying)g(algebra)332 415 y Ft(H)j Fv(=)13 b Ft(k)r Fv([)p Ft(X)522 422 y Fs(1)542 415 y Ft(;)8 b(:)g(:)g(:)g(;)g(X)692 422 y Fr(n)715 415 y Fv(])p Ft(=)p Fv(\()p Ft(X)816 392 y Fr(p)834 380 y Fq(s)849 387 y Fp(1)812 427 y Fs(1)871 415 y Ft(;)g(:)g(:)g(:)f(;)h(X) 1024 395 y Fr(p)1042 383 y Fq(s)1057 387 y(n)1020 427 y Fr(n)1082 415 y Fv(\))p Ft(;)g(n)14 b Fo(\025)f Fv(1)p Ft(;)8 b(s)1287 422 y Fs(1)1321 415 y Fo(\025)14 b(\001)8 b(\001)g(\001)14 b(\025)f Ft(s)1521 422 y Fr(n)1558 415 y Fo(\025)h Fv(1)p Ft(:)-1456 b Fv(\(2.1\))193 495 y(By)16 b(the)g(structure)h(theorem)e(of)i(in\014nitesimal,)c(that)k(is)g (\014nite)f(and)h(connected)193 554 y(group)24 b(sc)o(hemes)e(\(see)h ([W)o(,)i(14.4]\),)g(an)o(y)f(\014nite-dimensional,)e(comm)o(utativ)o (e)193 612 y(and)17 b(lo)q(cal)f(Hopf)g(algebra)h(o)o(v)o(er)e(a)i(p)q (erfect)e(\014eld)h(has)h(this)f(form.)243 670 y(Let)g Fg(A)31 b Fv(b)q(e)16 b(the)g(set)g(of)h(all)f(m)o(ulti-indice)o(s)e Ft(\013)g Fv(=)g(\()p Ft(\013)1169 677 y Fs(1)1189 670 y Ft(;)8 b(:)g(:)g(:)f(;)h(\013)1329 677 y Fr(n)1353 670 y Fv(\))16 b(with)g(0)f Fo(\024)e Ft(\013)1621 677 y Fr(i)1649 670 y Ft(<)193 728 y(p)217 710 y Fr(s)233 715 y Fq(i)249 728 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n:)243 786 y Fv(F)l(or)i Ft(\014)g Fv(=)e(\()p Ft(\014)473 793 y Fs(1)492 786 y Ft(;)8 b(:)g(:)g(:)g(;)g(\014)630 793 y Fr(n)653 786 y Fv(\))p Ft(;)g(\015)16 b Fv(=)e(\()p Ft(\015)831 793 y Fs(1)851 786 y Ft(;)8 b(:)g(:)g(:)f(;)h(\015)985 793 y Fr(n)1009 786 y Fv(\))14 b Fo(2)g Fg(N)1124 768 y Fr(n)1165 786 y Fv(w)o(e)h(de\014ne)345 866 y Ft(\014)f Fv(+)d Ft(\015)16 b Fv(=)e(\()p Ft(\014)576 873 y Fs(1)606 866 y Fv(+)d Ft(\015)680 873 y Fs(1)700 866 y Ft(;)d(:)g(:)g(:)g(;)g (\014)838 873 y Fr(n)872 866 y Fv(+)j Ft(\015)946 873 y Fr(n)970 866 y Fv(\))p Ft(;)24 b Fv(and)17 b Fo(j)p Ft(\014)s Fo(j)12 b Fv(=)i Ft(\014)1273 873 y Fs(1)1303 866 y Fv(+)d Fo(\001)d(\001)g(\001)k Fv(+)f Ft(\014)1499 873 y Fr(n)1522 866 y Ft(:)243 946 y Fv(F)l(or)16 b(all)g(i,)f(w)o(e)h (denote)g(the)g(residue)g(class)g(of)h Ft(X)1130 953 y Fr(i)1161 946 y Fv(in)e Ft(H)21 b Fv(b)o(y)16 b Ft(x)1374 953 y Fr(i)1387 946 y Fv(.)22 b(Then)684 1027 y Ft(x)712 1006 y Fr(\013)750 1027 y Fv(:=)14 b Ft(x)844 1005 y Fr(\013)867 1010 y Fp(1)844 1039 y Fs(1)894 1027 y Ft(:)8 b(:)g(:)f(x)987 1006 y Fr(\013)1010 1010 y Fq(n)987 1039 y Fr(n)1033 1027 y Ft(;)h(\013)14 b Fo(2)g Fg(A)d Ft(;)243 1107 y Fv(is)17 b(a)h Ft(k)r Fv(-basis)g(of)g Ft(H)t Fv(.)26 b(Note)17 b(that)h Ft(")p Fv(\()p Ft(x)936 1114 y Fr(i)949 1107 y Fv(\))e(=)g(0)i(for)g(all)f Ft(i)p Fv(,)g(where)g Ft(")f Fv(:)g Ft(H)k Fo(!)c Ft(k)k Fv(is)193 1165 y(the)d(counit)f(of)h Ft(H)t Fv(,)g(since)f Ft(H)21 b Fv(is)c(lo)q(cal)g(with)f(maximal)e(ideal)i(\()p Ft(x)1380 1172 y Fs(1)1400 1165 y Ft(;)8 b(:)g(:)g(:)f(;)h(x)1537 1172 y Fr(n)1560 1165 y Fv(\).)23 b(Let)193 1228 y(\003)e(:=)g Ft(x)349 1204 y Fr(p)367 1192 y Fq(s)382 1199 y Fp(1)400 1204 y Fm(\000)p Fs(1)349 1240 y(1)456 1228 y Ft(:)8 b(:)g(:)f(x)549 1210 y Fr(p)567 1198 y Fq(s)582 1202 y(n)604 1210 y Fm(\000)p Fs(1)549 1240 y Fr(n)652 1228 y Fv(.)34 b(Then)20 b(\003)h Fo(6)p Fv(=)h(0)f(and)g(\003)p Ft(x)1152 1235 y Fr(i)1187 1228 y Fv(=)g(0)g(for)f(all)g(1)i Fo(\024)f Ft(i)g Fo(\024)g Ft(n)p Fv(.)193 1286 y(Hence)15 b(\003)h(is)g(a)h(non-zero)g(in)o(tegral)e(in)h Ft(H)t Fv(.)243 1344 y(Let)c Ft(A)h Fv(b)q(e)f(an)h(algebra,)h(and)f Ft(\016)i Fv(:)e Ft(A)h Fo(!)f Ft(A)s Fo(\012)s Ft(H)t Fv(,)h(a)f(righ)o(t)f Ft(H)t Fv(-como)q(dule)g(algebra)193 1402 y(structure)k(on)h Ft(A)p Fv(.)j(W)l(e)c(will)g(alw)o(a)o(ys)g (write)544 1490 y Ft(\016)r Fv(\()p Ft(a)p Fv(\))d(=)697 1443 y Fn(X)698 1549 y Fr(\013)p Fm(2)p Ff(A)777 1490 y Ft(D)817 1497 y Fr(\013)842 1490 y Fv(\()p Ft(a)p Fv(\))e Fo(\012)g Ft(x)995 1470 y Fr(\013)1019 1490 y Ft(;)24 b Fv(for)17 b(all)f Ft(a)d Fo(2)h Ft(A:)243 1617 y Fv(Th)o(us)i(for)h (all)f Ft(\013)e Fo(2)g Fg(A)30 b Fv(and)17 b Ft(a;)8 b(b)13 b Fo(2)h Ft(A)p Fv(,)441 1705 y Ft(D)481 1712 y Fr(\013)506 1705 y Fv(\()p Ft(ab)p Fv(\))f(=)680 1658 y Fn(X)656 1764 y Fr(\014)r Fs(+)p Fr(\015)r Fs(=)p Fr(\013)666 1798 y(\014)r(;\015)r Fm(2)p Ff(A)783 1705 y Ft(D)823 1712 y Fr(\014)848 1705 y Fv(\()p Ft(a)p Fv(\))p Ft(D)952 1712 y Fr(\015)974 1705 y Fv(\()p Ft(b)p Fv(\))p Ft(;)24 b Fv(and)16 b Ft(D)1205 1713 y Fs(\(0)p Fr(;:::)o(;)p Fs(0\))1334 1705 y Fv(=)d(id)p Ft(:)193 1873 y Fv(F)l(or)18 b(all)g Ft(i)p Fv(,)f(let)h Ft(\016)495 1880 y Fr(i)525 1873 y Fv(=)f(\()p Ft(\016)621 1880 y Fr(ij)651 1873 y Fv(\))670 1880 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(n)799 1873 y Fo(2)g Fg(A)11 b Fv(,)21 b(where)d Ft(\016)1082 1880 y Fr(ij)1128 1873 y Fv(=)f(1,)h(if)g Ft(j)i Fv(=)d Ft(i)p Fv(,)g(and)i Ft(\016)1548 1880 y Fr(ij)1595 1873 y Fv(=)d(0,)193 1931 y(otherwise.)42 b(W)l(e)23 b(de\014ne)f Ft(D)728 1938 y Fr(i)769 1931 y Fv(=)j Ft(D)872 1938 y Fr(\016)888 1943 y Fq(i)904 1931 y Ft(;)8 b Fv(1)25 b Fo(\024)h Ft(i)f Fo(\024)g Ft(n:)e Fv(Th)o(us)h(the)f(linear)f(maps) 193 1989 y Ft(D)233 1996 y Fr(i)261 1989 y Fv(:)14 b Ft(A)f Fo(!)h Ft(A)g Fv(are)h(deriv)m(ations)f(of)h(the)g(algebra)g Ft(A)p Fv(,)f(and)h(for)g(all)f Ft(a)g Fo(2)g Ft(A)g Fv(w)o(e)g(ha)o(v)o(e)405 2077 y Ft(\016)r Fv(\()p Ft(a)p Fv(\))f(=)h Ft(a)c Fo(\012)h Fv(1)h(+)746 2030 y Fn(X)729 2135 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)843 2077 y Ft(D)883 2084 y Fr(i)898 2077 y Fv(\()p Ft(a)p Fv(\))e Fo(\012)h Ft(x)1050 2084 y Fr(i)1075 2077 y Fv(+)1132 2030 y Fn(X)1132 2136 y Fr(\013)p Fm(2)p Ff(A)1124 2167 y Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)1220 2077 y Ft(D)1260 2084 y Fr(\013)1285 2077 y Fv(\()p Ft(a)p Fv(\))f Fo(\012)h Ft(x)1437 2084 y Fr(\013)1462 2077 y Ft(:)-1283 b Fv(\(2.2\))243 2266 y(Let)21 b Ft(H)379 2248 y Fm(\003)421 2266 y Fv(b)q(e)g(the)h (dual)f(Hopf)g(algebra)h(of)g Ft(H)k Fv(and)c Fh(g)h Fv(=)f Ft(P)7 b Fv(\()p Ft(H)1431 2248 y Fm(\003)1451 2266 y Fv(\))22 b(the)f Ft(p)p Fv(-Lie)193 2324 y(algebra)e(of)f(the)g (primitiv)o(e)d(elemen)o(ts)g(in)j Ft(H)1024 2306 y Fm(\003)1044 2324 y Fv(.)27 b(Th)o(us)19 b Fh(g)f Fv(is)g(the)g(Lie)g(algebra)h(of) 193 2382 y(the)13 b(group)h(sc)o(heme)d(represen)o(ted)g(b)o(y)i Ft(H)t Fv(.)20 b(Recall)12 b(that)i(the)f(primitiv)n(e)d(elemen)o(ts) 193 2441 y(in)21 b Ft(H)299 2422 y Fm(\003)341 2441 y Fv(are)g(the)h Ft(")p Fv(-deriv)m(ations)f Ft(d)j Fv(:)e Ft(H)27 b Fo(!)c Ft(k)r Fv(,)g(that)f(is)f(the)h(linear)e(functions)193 2499 y(satisfying)14 b Ft(d)p Fv(\()p Ft(xy)r Fv(\))f(=)h Ft(")p Fv(\()p Ft(x)p Fv(\))p Ft(d)p Fv(\()p Ft(y)r Fv(\))6 b(+)g Ft(d)p Fv(\()p Ft(x)p Fv(\))p Ft(")p Fv(\()p Ft(y)r Fv(\))11 b(for)j(all)f Ft(x;)8 b(y)15 b Fo(2)f Ft(H)t Fv(.)21 b(F)l(or)13 b(all)g Ft(i)p Fv(,)h(let)f Ft(d)1646 2506 y Fr(i)1674 2499 y Fv(:)193 2557 y Ft(H)21 b Fo(!)c Ft(k)k Fv(b)q(e)d(the)g Ft(")p Fv(-deriv)m(ation)g(with)g Ft(d)929 2564 y Fr(i)944 2557 y Fv(\()p Ft(x)991 2564 y Fr(j)1008 2557 y Fv(\))g(=)f Ft(\016)1122 2564 y Fr(ij)1152 2557 y Ft(;)8 b Fv(1)17 b Fo(\024)g Ft(i;)8 b(j)20 b Fo(\024)d Ft(n)h Fv(\(Kronec)o(k)o(er)193 2615 y Ft(\016)r Fv(\).)i(Then)14 b Ft(d)420 2622 y Fs(1)440 2615 y Ft(;)8 b(:)g(:)g(:)g(;)g(d)575 2622 y Fr(n)613 2615 y Fv(is)13 b(a)i(basis)g(of)f Fh(g)p Fv(.)21 b(The)14 b(coaction)g Ft(\016)i Fv(:)d Ft(A)g Fo(!)h Ft(A)7 b Fo(\012)g Ft(H)17 b Fv(de\014nes)193 2673 y(a)g(left)f Ft(H)364 2655 y Fm(\003)384 2673 y Fv(-mo)q(dule)g(algebra)h(structure)f(on)h Ft(A)p Fv(,)f(and)h(the)g(action)g(of)g Ft(d)1496 2680 y Fr(i)1527 2673 y Fv(is)f(giv)o(en)193 2731 y(b)o(y)g(the)g(deriv)m (ation)g Ft(D)614 2738 y Fr(i)628 2731 y Fv(.)21 b(Th)o(us)568 2811 y Fh(g)14 b Fo(!)f Fv(Der\()p Ft(A;)8 b(A)p Fv(\))p Ft(;)g(d)929 2818 y Fr(i)956 2811 y Fo(7!)14 b Ft(D)1060 2818 y Fr(i)1074 2811 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)p eop %%Page: 6 6 6 5 bop 193 131 a Fk(6)182 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h(HANS-J) 1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)193 217 y Fv(is)g(a)h(map)e(of)i Ft(p)p Fv(-Lie)f(algebras.)22 b(In)16 b(particular,)548 308 y([)p Ft(D)602 315 y Fr(i)616 308 y Ft(;)8 b(D)678 315 y Fr(j)697 308 y Fv(])p Ft(;)g(D)774 284 y Fr(p)773 321 y(i)808 308 y Fo(2)872 261 y Fn(X)855 367 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)968 308 y Ft(k)r(D)1035 315 y Fr(l)1049 308 y Ft(;)g Fv(1)14 b Fo(\024)f Ft(i;)8 b(j)17 b Fo(\024)c Ft(n:)-1139 b Fv(\(2.3\))243 441 y(In)17 b(the)f(case)i(when)f Ft(s)645 448 y Fs(1)680 441 y Fv(=)e Fo(\001)8 b(\001)g(\001)15 b Fv(=)h Ft(s)883 448 y Fr(n)921 441 y Fv(=)g(1,)h Ft(H)1074 423 y Fm(\003)1111 441 y Fv(is)g(the)g(restricted)f(en)o(v)o(eloping) 193 499 y(algebra)g(of)f Fh(g)p Fv(,)g(and)h(the)e(action)i(is)e (determined)f(b)o(y)i(the)f(action)i(of)f Fh(g)p Fv(.)21 b(Hence)14 b(in)193 557 y(this)i(case)g Ft(A)428 539 y Fr(coH)508 557 y Fv(=)d Ft(A)596 539 y Fm(f)p Fr(D)643 544 y Fp(1)660 539 y Fr(;:::)o(;D)738 543 y Fq(n)759 539 y Fm(g)779 557 y Fv(,)j(where)442 641 y Ft(A)479 621 y Fm(f)p Fr(D)526 626 y Fp(1)543 621 y Fr(;:::)o(;D)621 625 y Fq(n)642 621 y Fm(g)675 641 y Fv(:=)e Fo(f)p Ft(a)f Fo(2)h Ft(A)g Fo(j)f Ft(D)970 648 y Fr(i)985 641 y Fv(\()p Ft(a)p Fv(\))g(=)h(0)j(for)f(all)g(1)e Fo(\024)g Ft(i)f Fo(\024)h Ft(n)p Fo(g)p Ft(:)-1346 b Fv(\(2.4\))659 753 y(3.)28 b Fu(The)18 b(additive)g(gr)o(oup)243 840 y Fv(The)e(Hopf)g (algebra)h Ft(H)673 847 y Fr(a)710 840 y Fv(with)g(underlying)e (algebra)319 925 y Ft(H)359 932 y Fr(a)394 925 y Fv(=)f Ft(k)r Fv([)p Ft(X)527 932 y Fs(1)547 925 y Ft(;)8 b(:)g(:)g(:)f(;)h(X) 696 932 y Fr(n)720 925 y Fv(])p Ft(=)p Fv(\()p Ft(X)821 902 y Fr(p)839 890 y Fq(s)854 897 y Fp(1)817 937 y Fs(1)875 925 y Ft(;)g(:)g(:)g(:)f(;)h(X)1028 905 y Fr(p)1046 893 y Fq(s)1061 897 y(n)1024 938 y Fr(n)1086 925 y Fv(\))p Ft(;)g(n)14 b Fo(\025)g Fv(1)p Ft(;)8 b(s)1292 932 y Fs(1)1325 925 y Fo(\025)14 b(\001)8 b(\001)g(\001)14 b(\025)f Ft(s)1525 932 y Fr(n)1563 925 y Fo(\025)g Fv(1)p Ft(;)-1460 b Fv(\(3.1\))193 1008 y(and)17 b(com)o(ultiplic)o(ation)d (giv)o(en)h(b)o(y)567 1091 y(\001\()p Ft(x)655 1098 y Fr(i)668 1091 y Fv(\))f(=)g Ft(x)781 1098 y Fr(i)806 1091 y Fo(\012)c Fv(1)i(+)f(1)g Fo(\012)g Ft(x)1053 1098 y Fr(i)1067 1091 y Ft(;)d Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)-1113 b Fv(\(3.2\))193 1177 y(will)21 b(b)q(e)i(called)e(the)i(Hopf)f(algebra)h(of)g(the)f(additiv)o(e)g (group)h(\(where)f Ft(n)h Fv(and)193 1235 y Ft(s)216 1242 y Fs(1)236 1235 y Ft(;)8 b(:)g(:)g(:)f(;)h(s)368 1242 y Fr(n)408 1235 y Fv(are)16 b(\014xed\).)243 1294 y(Note)i(that)h(for)h(an)o(y)e(comm)o(utativ)o(e)d(algebra)20 b Ft(T)7 b Fv(,)18 b Ft(G)1217 1301 y Fr(a)1238 1294 y Fv(\()p Ft(T)7 b Fv(\))18 b(:=)f(Alg\()p Ft(H)1533 1301 y Fr(a)1554 1294 y Ft(;)8 b(T)f Fv(\))18 b Fo(')193 1352 y(f)p Fv(\()p Ft(t)255 1359 y Fs(1)274 1352 y Ft(;)8 b(:)g(:)g(:)g(;)g(t)402 1359 y Fr(n)425 1352 y Fv(\))17 b Fo(2)h Ft(k)539 1334 y Fr(n)580 1352 y Fo(j)f Ft(t)629 1334 y Fr(p)647 1322 y Fq(s)662 1330 y(i)696 1352 y Fv(=)h(0)p Ft(;)8 b Fv(1)18 b Fo(\024)f Ft(i)h Fo(\024)f Ft(n)p Fo(g)p Fv(,)i(where)f(the)g(group)h(structure)g(is)193 1410 y(comp)q(onen)o(t)o(wise)c(addition.)243 1468 y(W)l(e)20 b(consider)h(coactions)g(of)g Ft(H)842 1475 y Fr(a)884 1468 y Fv(on)h(arbitrary)l(,)f(not)g(necessarily)f(comm)o(u-)193 1526 y(tativ)o(e)d(algebras)h Ft(A)p Fv(.)26 b(Ho)o(w)o(ev)o(er)16 b(in)h(all)g(the)h(later)f(results)h(ab)q(out)h Ft(H)t Fv(-como)q(dule)193 1584 y(algebras)f Ft(A)f Fv(for)h(our)g(general)f (Hopf)g(algebra)h Ft(H)k Fv(satisfying)c(\(2.1\))g(w)o(e)f(assume)193 1642 y(that)g Ft(A)e Fv(is)i(comm)o(utati)o(v)o(e.)193 1732 y Fx(Theorem)f(3.1.)24 b Fj(L)n(et)15 b Ft(H)664 1739 y Fr(a)701 1732 y Fj(b)n(e)h(the)h(Hopf)e(algebr)n(a)i(de\014ne)n (d)f(by)21 b Fv(\(3.1\))p Fj(,)16 b Fv(\(3.2\))p Fj(,)g(and)193 1790 y Ft(A)21 b Fj(a)h(right)g Ft(H)t Fj(-c)n(omo)n(dule)g(algebr)n(a) h(with)f(structur)n(e)g(map)f Ft(\016)j Fv(:)d Ft(A)h Fo(!)f Ft(A)14 b Fo(\012)g Ft(H)1651 1797 y Fr(a)1672 1790 y Fj(.)193 1848 y(De\014ne)19 b(the)f(derivations)g Ft(D)719 1855 y Fs(1)739 1848 y Ft(;)8 b(:)g(:)g(:)f(;)h(D)888 1855 y Fr(n)930 1848 y Fj(by)21 b Fv(\(2.2\))d Fj(and)g(let)g Ft(R)d Fv(=)e Ft(A)1418 1830 y Fr(coH)1479 1834 y Fq(a)1500 1848 y Fj(.)243 1906 y(Then)18 b(the)g(fol)r(lowing)h(ar)n(e)e(e)n (quivalent:)218 1977 y Fv(\(1\))k Ft(R)15 b Fo(\032)e Ft(A)k Fj(is)h(a)f(faithful)r(ly)i(\015at)e Ft(H)887 1984 y Fr(a)908 1977 y Fj(-Galois)h(extension.)218 2035 y Fv(\(2\))j Fj(Ther)n(e)i(ar)n(e)g Ft(y)560 2042 y Fs(1)580 2035 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)713 2042 y Fr(n)761 2035 y Fo(2)25 b Ft(A)e Fj(with)h Ft(\016)r Fv(\()p Ft(y)1058 2042 y Fr(i)1071 2035 y Fv(\))h(=)g Ft(y)1202 2042 y Fr(i)1231 2035 y Fo(\012)15 b Fv(1)h(+)g(1)f Fo(\012)h Ft(x)1501 2042 y Fr(i)1514 2035 y Ft(;)23 b Fj(for)g(al)r(l)301 2093 y Fv(1)15 b Fo(\024)e Ft(i)h Fo(\024)f Ft(n:)193 2164 y Fj(Supp)n(ose)g(\(2\))g(holds.)21 b(De\014ne)14 b Ft(y)770 2146 y Fr(\013)808 2164 y Fv(=)g Ft(y)886 2142 y Fr(\013)909 2147 y Fp(1)884 2176 y Fs(1)936 2164 y Fo(\001)8 b(\001)g(\001)g Ft(y)1028 2146 y Fr(\013)1051 2150 y Fq(n)1026 2176 y Fr(n)1074 2164 y Ft(;)g(\013)14 b Fv(=)g(\()p Ft(\013)1243 2171 y Fs(1)1262 2164 y Ft(;)8 b(:)g(:)g(:)g(;)g(\013)1403 2171 y Fr(n)1426 2164 y Fv(\))14 b Fo(2)g Fg(A)d Fj(.)24 b(Then)495 2247 y Ft(R)11 b Fo(\012)g Ft(H)633 2254 y Fr(a)668 2247 y Fo(!)j Ft(A;)8 b(r)k Fo(\012)f Ft(x)903 2226 y Fr(\013)941 2247 y Fo(7!)j Ft(r)q(y)1054 2226 y Fr(\013)1078 2247 y Ft(;)8 b(r)15 b Fo(2)f Ft(R;)8 b(\013)15 b Fo(2)f Fg(A)d Ft(;)193 2330 y Fj(is)k(a)g(left)h Ft(R)p Fj(-line)n(ar)h(and)e(right)g Ft(H)801 2337 y Fr(a)823 2330 y Fj(-c)n(oline)n(ar)g(isomorphism.)20 b(In)c(p)n(articular,)f(the)193 2388 y(elements)833 2451 y Ft(y)859 2430 y Fr(\013)884 2451 y Ft(;)8 b(\013)14 b Fo(2)g Fg(A)d Ft(;)193 2523 y Fj(form)17 b(an)g Ft(R)p Fj(-b)n(asis)h(of)g Ft(A)f Fj(as)g(a)g(left)i Ft(R)p Fj(-mo)n(dule.)193 2612 y(Pr)n(o)n(of.)g Fv(\(1\))14 b Fo(\))g Fv(\(2\):)21 b(By)13 b(\(1\),)h(A)g(is)g(an)g(injectiv)o(e)d Ft(H)1137 2619 y Fr(a)1159 2612 y Fv(-como)q(dule)i(\(see)g(for)i (exam-)193 2670 y(ple)j([S,)h(Theorem)f(I]\).)f(Hence)h(the)g(righ)o(t) h Ft(H)1034 2677 y Fr(a)1055 2670 y Fv(-colinear)g(map)f Ft(k)i Fo(!)e Ft(A;)8 b Fv(1)18 b Fo(7!)g Fv(1,)193 2728 y(can)e(b)q(e)h(extended)e(to)i(an)f Ft(H)723 2735 y Fr(a)745 2728 y Fv(-colinear)f(map)h Ft(\015)h Fv(:)c Ft(H)1160 2735 y Fr(a)1195 2728 y Fo(!)h Ft(A)p Fv(.)21 b(Then)16 b(for)g(all)g Ft(i)p Fv(,)449 2811 y Ft(\016)r Fv(\()p Ft(\015)s Fv(\()p Ft(x)567 2818 y Fr(i)580 2811 y Fv(\)\))e(=)g(\()p Ft(\015)g Fo(\012)d Fv(id)o(\)\001\()p Ft(x)939 2818 y Fr(i)953 2811 y Fv(\))j(=)f Ft(\015)s Fv(\()p Ft(x)1112 2818 y Fr(i)1126 2811 y Fv(\))e Fo(\012)g Fv(1)h(+)f(1)g Fo(\012)g Ft(x)1404 2818 y Fr(i)1418 2811 y Ft(;)p eop %%Page: 7 7 7 6 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)227 b(7)193 217 y Fv(since)18 b Ft(\015)j Fv(is)d Ft(H)452 224 y Fr(a)473 217 y Fv(-colinear)g(and)h Ft(\015)s Fv(\(1\))f(=)f(1)p Ft(;)i Fv(and)g(the)f(claim)e(follo)o(ws)i (with)h Ft(y)1605 224 y Fr(i)1636 217 y Fv(:=)193 275 y Ft(\015)s Fv(\()p Ft(x)268 282 y Fr(i)282 275 y Fv(\))d(for)g(all)g (1)e Fo(\024)g Ft(i)g Fo(\024)f Ft(n:)243 333 y Fv(Supp)q(ose)20 b(\(2\))f(holds.)29 b(Then)19 b(w)o(e)f(de\014ne)h(a)g Ft(k)r Fv(-linear)f(map)g Ft(\015)j Fv(:)d Ft(H)1473 340 y Fr(a)1513 333 y Fo(!)f Ft(A)i Fv(b)o(y)193 391 y Ft(\015)s Fv(\()p Ft(x)268 373 y Fr(\013)292 391 y Fv(\))14 b(:=)g Ft(y)417 373 y Fr(\013)457 391 y Fv(for)i(all)g Ft(\013)e Fo(2)g Fg(A)d Ft(:)18 b Fv(Since)e(\001)f(and)i Ft(\016)g Fv(are)f(algebra)h(maps,)e(and)i(for)f(all)193 449 y Ft(i)p Fv(,)435 514 y(\001\()p Ft(x)523 521 y Fr(i)536 514 y Fv(\))e(=)g Ft(x)649 521 y Fr(i)674 514 y Fo(\012)c Fv(1)i(+)f(1)g Fo(\012)g Ft(x)921 521 y Fr(i)935 514 y Ft(;)d(\016)r Fv(\()p Ft(y)1024 521 y Fr(i)1037 514 y Fv(\))14 b(=)g Ft(y)1146 521 y Fr(i)1171 514 y Fo(\012)c Fv(1)i(+)f(1)g Fo(\012)g Ft(x)1418 521 y Fr(i)1432 514 y Ft(;)193 589 y(\015)19 b Fv(is)d(righ)o(t)g Ft(H)444 596 y Fr(a)465 589 y Fv(-colinear.)243 647 y(Since)e Ft(H)409 654 y Fr(a)446 647 y Fv(as)i(an)g(algebra)g(is)g(generated)f (b)o(y)g(the)g(group-lik)o(e)g(elemen)o(t)e(1)j(and)193 706 y(the)e(primitiv)o(e)d(elemen)o(ts)h Ft(x)708 713 y Fr(i)722 706 y Fv(,)i Ft(k)j Fv(is)d(the)g(only)h(simple)d(sub)q (coalgebra)k(of)f Ft(H)1561 713 y Fr(a)1597 706 y Fv(\([M)o(,)193 764 y(5.5.1]\).)21 b(Therefore)15 b(the)g(map)g Ft(\015)j Fv(is)d(con)o(v)o(olution)g(in)o(v)o(ertible)e(\(cf.)20 b([M)o(,)c(5.2.10]\).)193 822 y(Th)o(us)h(the)f Ft(H)441 829 y Fr(a)462 822 y Fv(-extension)g Ft(R)e Fo(\032)g Ft(A)h Fv(is)i Ft(H)940 829 y Fr(a)961 822 y Fv(-cleft,)e(hence)g Ft(H)1273 829 y Fr(a)1294 822 y Fv(-Galois,)i(and)495 907 y Ft(R)11 b Fo(\012)g Ft(H)633 914 y Fr(a)668 907 y Fo(!)j Ft(A;)8 b(r)k Fo(\012)f Ft(x)903 887 y Fr(\013)941 907 y Fo(7!)j Ft(r)q(y)1054 887 y Fr(\013)1078 907 y Ft(;)8 b(r)15 b Fo(2)f Ft(R;)8 b(\013)15 b Fo(2)f Fg(A)d Ft(;)193 992 y Fv(is)16 b(bijectiv)o(e)e(\(see)i([M)o(,)g(8.2.4,)g (7.2.3]\).)p 1652 992 2 33 v 1654 961 30 2 v 1654 992 V 1683 992 2 33 v 193 1083 a Fx(Example)g(3.2.)24 b Fv(W)l(e)15 b(consider)f(the)h(Lie)g(algebra)g(case)g(of)h(the)f(additiv)o(e)e (group)576 1169 y Ft(H)616 1176 y Fr(a)651 1169 y Fv(=)h Ft(k)r Fv([)p Ft(X)784 1176 y Fs(1)804 1169 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)953 1176 y Fr(n)977 1169 y Fv(])p Ft(=)p Fv(\()p Ft(X)1078 1145 y Fr(p)1074 1181 y Fs(1)1098 1169 y Ft(;)g(:)g(:)g(:)g(;)g(X)1252 1148 y Fr(p)1248 1181 y(n)1272 1169 y Fv(\))p Ft(:)193 1254 y Fv(In)23 b(this)h(case)f(coactions)h(of)g Ft(H)796 1261 y Fr(a)841 1254 y Fv(are)f(giv)o(en)g(b)o(y)g(deriv)m(ations)g Ft(D)1434 1261 y Fs(1)1455 1254 y Ft(;)8 b(:)g(:)g(:)f(;)h(D)1604 1261 y Fr(n)1654 1254 y Fo(2)193 1312 y Fv(Der\()p Ft(A;)g(A)p Fv(\))15 b(with)510 1397 y Ft(D)550 1404 y Fr(i)565 1397 y Ft(D)605 1404 y Fr(j)637 1397 y Fv(=)f Ft(D)729 1404 y Fr(j)748 1397 y Ft(D)788 1404 y Fr(i)802 1397 y Ft(;)24 b Fv(and)17 b Ft(D)976 1374 y Fr(p)975 1410 y(i)1011 1397 y Fv(=)d(0)p Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i;)8 b(j)16 b Fo(\024)e Ft(n;)193 1483 y Fv(where)313 1584 y Ft(D)353 1591 y Fr(\013)392 1584 y Fv(=)449 1550 y Ft(D)490 1528 y Fr(\013)513 1533 y Fp(1)489 1562 y Fs(1)p 449 1572 84 2 v 458 1618 a Ft(\013)489 1625 y Fs(1)509 1618 y Fv(!)546 1584 y Fo(\001)8 b(\001)g(\001)617 1550 y Ft(D)658 1532 y Fr(\013)681 1536 y Fq(n)657 1562 y Fr(n)p 617 1572 88 2 v 627 1618 a Ft(\013)658 1625 y Fr(n)682 1618 y Fv(!)710 1584 y Ft(;)g(\013)14 b Fv(=)g(\()p Ft(\013)879 1591 y Fs(1)899 1584 y Ft(;)8 b(:)g(:)g(:)f(;)h(\013)1039 1591 y Fr(n)1062 1584 y Fv(\))p Ft(;)g Fv(0)14 b Fo(\024)g Ft(\013)1225 1591 y Fr(i)1253 1584 y Ft(<)g(p;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n:)193 1705 y Fj(Pr)n(o)n(of.)19 b Fv(This)d(follo)o(ws)f(easily)g(from)g(the)g(fact)h(that)g Ft(e)1168 1712 y Fr(\013)1206 1705 y Fv(=)1263 1685 y Fr(x)1283 1673 y Fq(\013)1303 1680 y Fp(1)p 1263 1693 60 2 v 1268 1722 a Fr(\013)1291 1727 y Fp(1)1308 1722 y Fs(!)1336 1705 y Fo(\001)8 b(\001)g(\001)1407 1685 y Fr(x)1427 1673 y Fq(\013)1447 1677 y(n)p 1407 1693 64 2 v 1412 1722 a Fr(\013)1435 1726 y Fq(n)1456 1722 y Fs(!)1475 1705 y Ft(;)g(\013)14 b Fo(2)g Fg(A)d Fv(,)19 b(is)193 1765 y(a)e Ft(k)r Fv(-basis)g(of)f Ft(H)492 1772 y Fr(a)530 1765 y Fv(with)698 1856 y(\001\()p Ft(e)781 1863 y Fr(\013)805 1856 y Fv(\))e(=)913 1809 y Fn(X)889 1915 y Fr(\014)r Fs(+)p Fr(\015)r Fs(=)p Fr(\013)900 1949 y(\014)r(;\015)r Fm(2)p Ff(A)1017 1856 y Ft(e)1040 1863 y Fr(\014)1074 1856 y Fo(\012)d Ft(e)1147 1863 y Fr(\015)1169 1856 y Ft(:)p 1652 2025 2 33 v 1654 1993 30 2 v 1654 2025 V 1683 2025 2 33 v 193 2116 a Fx(Corollary)19 b(3.3.)k Fj(Assume)c Ft(s)760 2123 y Fs(1)796 2116 y Fv(=)c Fo(\001)8 b(\001)g(\001)16 b Fv(=)g Ft(s)1000 2123 y Fr(n)1039 2116 y Fv(=)f(1)p Fj(.)26 b(L)n(et)18 b Ft(A)g Fj(b)n(e)h(an)g(algebr)n(a,)g(and)193 2174 y Ft(\016)c Fv(:)f Ft(A)f Fo(!)h Ft(A)8 b Fo(\012)g Ft(H)504 2181 y Fr(a)539 2174 y Fj(a)16 b(c)n(o)n(action.)22 b(De\014ne)c(the)e (derivations)g Ft(D)1311 2181 y Fs(1)1332 2174 y Ft(;)8 b(:)g(:)g(:)f(;)h(D)1481 2181 y Fr(n)1521 2174 y Fj(by)20 b Fv(\(2.2\))193 2232 y Fj(and)e(let)g Ft(R)c Fv(=)g Ft(A)496 2214 y Fr(coH)562 2232 y Fj(.)243 2290 y(Then)k(the)g(fol)r (lowing)h(ar)n(e)e(e)n(quivalent:)218 2362 y Fv(\(1\))k Ft(R)15 b Fo(\032)e Ft(A)k Fj(is)h(a)f(faithful)r(ly)i(\015at)e Ft(H)887 2369 y Fr(a)908 2362 y Fj(-Galois)h(extension.)218 2420 y Fv(\(2\))j Fj(Ther)n(e)d(ar)n(e)e Ft(y)548 2427 y Fs(1)568 2420 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)701 2427 y Fr(n)738 2420 y Fo(2)14 b Ft(A)j Fj(with)h Ft(D)985 2427 y Fr(i)1000 2420 y Fv(\()p Ft(y)1043 2427 y Fr(j)1061 2420 y Fv(\))c(=)f Ft(\016)1167 2427 y Fr(ij)1215 2420 y Fj(for)j(al)r(l)j Fv(1)14 b Fo(\024)g Ft(i;)8 b(j)16 b Fo(\024)e Ft(n:)193 2492 y Fj(Supp)n(ose)k(\(2\))f(holds.)22 b(Then)495 2577 y Ft(R)11 b Fo(\012)g Ft(H)633 2584 y Fr(a)668 2577 y Fo(!)j Ft(A;)8 b(r)k Fo(\012)f Ft(x)903 2556 y Fr(\013)941 2577 y Fo(7!)j Ft(r)q(y)1054 2556 y Fr(\013)1078 2577 y Ft(;)8 b(r)15 b Fo(2)f Ft(R;)8 b(\013)15 b Fo(2)f Fg(A)d Ft(;)193 2662 y Fj(is)17 b(a)h(left)g Ft(R)p Fj(-line)n(ar)h(and)e(right)h Ft(H)815 2669 y Fr(a)836 2662 y Fj(-c)n(oline)n(ar)g(isomorphism.)193 2753 y(Pr)n(o)n(of.)h Fv(This)d(follo)o(ws)h(from)e(Theorem)g(3.1)h (and)h(Example)e(3.2)p 1652 2811 V 1654 2780 30 2 v 1654 2811 V 1683 2811 2 33 v eop %%Page: 8 8 8 7 bop 193 131 a Fk(8)182 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h(HANS-J) 1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)243 217 y Fv(The)i(next)f(Lemma)f(sho)o(ws)j(that)f(under)g(certain)f (conditions)h(the)g(coin)o(v)m(ari-)193 275 y(an)o(t)c(elemen)o(ts)d (in)j(the)g(Lie)f(algebra)i(case)f(are)g(the)g(coin)o(v)m(arian)o(t)f (elemen)o(ts)e(under)193 333 y(some)g(action)h(of)g(the)g(additiv)o(e)e (group.)21 b(The)12 b(idea)g(of)g(this)g(Lemma)d(app)q(ears)k(sev-)193 391 y(eral)j(times)e(in)i(the)h(literature)e(\(see)h([L])g(resp.)21 b([RM)o(])16 b(when)h(the)f(c)o(haracteristic)193 449 y(of)h(the)f(\014eld)f(is)h(0)h(resp.)k Ft(p)14 b(>)g Fv(0.\))22 b(W)l(e)16 b(include)f(the)h(pro)q(of)h(for)g(completeness.) 193 537 y Fx(Lemma)f(3.4.)23 b Fj(L)n(et)18 b Ft(A)h Fj(b)n(e)g(a)f(c)n(ommutative)i(algebr)n(a,)f Ft(H)k Fj(a)c(Hopf)g(algebr)n(a)g(sat-)193 595 y(isfying)j Fv(\(2.1\))17 b Fj(with)g Ft(s)598 602 y Fs(1)632 595 y Fv(=)c Fo(\001)8 b(\001)g(\001)14 b Fv(=)g Ft(s)830 602 y Fr(n)867 595 y Fv(=)g(1)p Fj(,)k(and)f Ft(\016)h Fj(a)f(right)g Ft(H)t Fj(-c)n(omo)n(dule)h(algebr)n(a)193 653 y(structur)n(e)f(on)h Ft(A)f Fj(with)h(derivations)g Ft(D)919 660 y Fs(1)939 653 y Ft(;)8 b(:)g(:)g(:)g(;)g(D)1089 660 y Fr(n)1130 653 y Fj(de\014ne)n(d)18 b(by)k Fv(\(2.2\))p Fj(.)243 711 y(Assume)15 b Ft(y)446 718 y Fs(1)466 711 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)599 718 y Fr(n)636 711 y Fo(2)14 b Ft(A)h Fj(such)h(that)f(the)h(matrix)f Fv(\()p Ft(D)1233 718 y Fr(i)1248 711 y Fv(\()p Ft(y)1291 718 y Fr(j)1309 711 y Fv(\)\))1347 718 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)1496 711 y Fj(is)g(invert-)193 769 y(ible.)243 827 y(Then)j(ther)n(e)f(ar)n(e)g(derivations)h Ft(@)851 834 y Fs(1)870 827 y Ft(;)8 b(:)g(:)g(:)g(;)g(@)1006 834 y Fr(n)1043 827 y Fo(2)14 b Fj(Der)p Fv(\()p Ft(A;)8 b(A)p Fv(\))16 b Fj(such)i(that)227 910 y Fv([)p Ft(@)267 917 y Fr(i)281 910 y Ft(;)8 b(@)329 917 y Fr(j)346 910 y Fv(])14 b(=)g(0)p Ft(;)8 b(@)501 887 y Fr(p)498 923 y(i)534 910 y Fv(=)14 b(0)p Ft(;)8 b(@)658 917 y Fr(i)672 910 y Fv(\()p Ft(y)715 917 y Fr(j)733 910 y Fv(\))14 b(=)f Ft(\016)839 917 y Fr(ij)869 910 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i;)8 b(j)16 b Fo(\024)e Ft(n;)25 b Fj(and)18 b Ft(A)1310 890 y Fr(coH)1389 910 y Fv(=)c Ft(A)1478 890 y Fm(f)p Fr(@)1515 895 y Fp(1)1531 890 y Fr(;:::)o(;@)1599 894 y Fq(n)1620 890 y Fm(g)1640 910 y Ft(:)193 990 y Fj(In)k(p)n(articular,)e(the)i(elements)590 1069 y Ft(y)616 1047 y Fr(\013)639 1052 y Fp(1)614 1081 y Fs(1)666 1069 y Fo(\001)8 b(\001)g(\001)g Ft(y)758 1048 y Fr(\013)781 1052 y Fq(n)756 1081 y Fr(n)804 1069 y Ft(;)g Fv(0)14 b Fo(\024)g Ft(\013)948 1076 y Fr(i)976 1069 y Ft(<)f(p;)8 b Fv(1)15 b Fo(\024)e Ft(i)h Fo(\024)g Ft(n;)193 1148 y Fj(form)j(a)g(b)n(asis)g(of)g Ft(A)g Fj(over)h Ft(R)p Fj(.)193 1235 y(Pr)n(o)n(of.)h Fv(Let)d(\()p Ft(a)474 1242 y Fr(ij)504 1235 y Fv(\))g(b)q(e)h(the)f(in)o(v)o(erse)e(matrix)h (of)h(\()p Ft(D)1124 1242 y Fr(i)1139 1235 y Fv(\()p Ft(y)1182 1242 y Fr(j)1200 1235 y Fv(\)\))p Ft(:)g Fv(De\014ne)652 1326 y Ft(@)678 1333 y Fr(i)705 1326 y Fv(=)776 1279 y Fn(X)757 1384 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(n)876 1326 y Ft(a)902 1333 y Fr(ij)931 1326 y Ft(D)971 1333 y Fr(j)990 1326 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n:)193 1458 y Fv(Then)k Ft(@)348 1465 y Fr(i)362 1458 y Fv(\()p Ft(y)405 1465 y Fr(l)418 1458 y Fv(\))e(=)h Ft(\016)530 1465 y Fr(il)572 1458 y Fv(for)h(all)g(1)f Fo(\024)f Ft(i;)8 b(l)17 b Fo(\024)f Ft(n)p Fv(,)i(and)1097 1421 y Fn(P)1150 1473 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)1266 1458 y Ft(AD)1343 1465 y Fr(i)1374 1458 y Fv(=)1428 1421 y Fn(P)1481 1473 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)1597 1458 y Ft(A@)1660 1465 y Fr(i)1674 1458 y Ft(:)193 1520 y Fv(In)e(particular,)f(using)j(\(2.4\),)608 1603 y Ft(A)645 1582 y Fm(f)p Fr(@)682 1587 y Fp(1)698 1582 y Fr(;:::)o(;@)766 1586 y Fq(n)786 1582 y Fm(g)820 1603 y Fv(=)c Ft(A)909 1582 y Fm(f)p Fr(D)956 1587 y Fp(1)972 1582 y Fr(;:::)o(;D)1050 1586 y Fq(n)1071 1582 y Fm(g)1105 1603 y Fv(=)g Ft(A)1194 1582 y Fr(coH)1259 1603 y Ft(:)193 1682 y Fv(Since)h Ft(A)h Fv(is)g(comm)o(utativ)o(e)c(it)k(is)g(easy)h (to)f(see)g(b)o(y)g(\(2.3\))h(that)606 1772 y([)p Ft(@)646 1779 y Fr(i)659 1772 y Ft(;)8 b(@)707 1779 y Fr(j)725 1772 y Fv(])13 b Fo(2)816 1724 y Fn(X)799 1830 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)914 1772 y Ft(AD)991 1779 y Fr(i)1019 1772 y Fv(=)1087 1724 y Fn(X)1070 1830 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)1185 1772 y Ft(A@)1248 1779 y Fr(i)1261 1772 y Ft(:)193 1903 y Fv(By)j(Ho)q(c)o (hsc)o(hild's)g(form)o(ula)f([Ma,)i(25.5],)g(for)g(an)o(y)g Ft(a)e Fo(2)g Ft(A)i Fv(and)h(an)o(y)f(deriv)m(ation)193 1961 y Ft(D)f Fo(2)e Fv(Der\()p Ft(A;)8 b(A)p Fv(\),)14 b(\()p Ft(aD)q Fv(\))641 1943 y Fr(p)676 1961 y Fv(is)i(an)g Ft(A)p Fv(-linear)e(com)o(bination)g(of)i Ft(D)1352 1943 y Fr(p)1388 1961 y Fv(and)g Ft(D)q Fv(.)22 b(Hence)193 2019 y(for)16 b(all)g Ft(i)g Fv(b)o(y)h(\(2.3\),)648 2088 y Ft(@)677 2065 y Fr(p)674 2101 y(i)710 2088 y Fo(2)774 2041 y Fn(X)757 2146 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)871 2088 y Ft(AD)948 2095 y Fr(i)976 2088 y Fv(=)1045 2041 y Fn(X)1028 2146 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)1142 2088 y Ft(A@)1205 2095 y Fr(i)1219 2088 y Ft(:)193 2206 y Fv(Since)f(an)h Ft(A)p Fv(-linear)f(com)o(bination)g(of)h Ft(@)939 2213 y Fs(1)958 2206 y Ft(;)8 b(:)g(:)g(:)g(;)g(@)1094 2213 y Fr(n)1134 2206 y Fv(is)16 b(zero)h(if)f(its)h(ev)m(aluation)g (on)193 2265 y(all)e Ft(y)284 2272 y Fr(j)302 2265 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(j)j Fo(\024)c Ft(n)p Fv(,)i(v)m(anishes,)h(w)o(e)f(see)g(that)h([)p Ft(@)1061 2272 y Fr(i)1075 2265 y Ft(;)8 b(@)1123 2272 y Fr(j)1140 2265 y Fv(])14 b(=)g(0)p Ft(;)8 b(@)1295 2241 y Fr(p)1292 2277 y(i)1328 2265 y Fv(=)14 b(0)p Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i;)8 b(j)16 b Fo(\024)e Ft(n:)243 2323 y Fv(Finally)l(,)e(the)i(statemen)o(t)f(ab)q(out)i(the)f Ft(R)p Fv(-basis)h(of)g Ft(A)e Fv(follo)o(ws)h(from)f(Corollary)193 2381 y(3.3)k(and)f(Example)f(3.2.)p 1652 2381 2 33 v 1654 2349 30 2 v 1654 2381 V 1683 2381 2 33 v 445 2481 a(4.)28 b Fu(A)18 b(Ja)o(cobi-criterion)g(f)o(or)h(free)f(a)o(ctions) 193 2568 y Fx(Theorem)e(4.1.)24 b Fj(L)n(et)14 b Ft(H)19 b Fj(b)n(e)c(a)f(Hopf)g(algebr)n(a)h(satisfying)k Fv(\(2.1\))p Fj(,)c(and)g Ft(A)f Fj(a)g(c)n(om-)193 2627 y(mutative)j(right)g Ft(H)t Fj(-c)n(omo)n(dule)g(algebr)n(a)f(with)h(structur)n(e)f(map)g Ft(\016)g Fv(:)d Ft(A)g Fo(!)h Ft(A)8 b Fo(\012)g Ft(H)t Fj(.)193 2685 y(De\014ne)19 b(the)f(derivations)g Ft(D)719 2692 y Fs(1)739 2685 y Ft(;)8 b(:)g(:)g(:)f(;)h(D)888 2692 y Fr(n)930 2685 y Fj(by)21 b Fv(\(2.2\))d Fj(and)g(let)g Ft(R)d Fv(=)e Ft(A)1418 2667 y Fr(coH)1484 2685 y Fj(.)243 2743 y(Then)18 b(the)g(fol)r(lowing)h(ar)n(e)e(e)n(quivalent:)218 2811 y Fv(\(1\))k Ft(R)15 b Fo(\032)e Ft(A)k Fj(is)h(an)f Ft(H)t Fj(-Galois)h(extension.)p eop %%Page: 9 9 9 8 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)227 b(9)218 217 y Fv(\(2\))21 b Fj(Ther)n(e)14 b(ar)n(e)f Ft(N)19 b Fo(2)14 b Fg(N)g Fj(and)g Ft(y)786 224 y Fs(1)805 217 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)939 224 y Fr(N)986 217 y Fo(2)14 b Ft(A)f Fj(such)h(that)f(the)i(matrix)e Fv(\()p Ft(D)1574 224 y Fr(i)1588 217 y Fv(\()p Ft(y)1631 224 y Fr(j)1649 217 y Fv(\)\))301 275 y Fj(is)k(right)h(invertible,)h (that)e(is)g(ther)n(e)g(ar)n(e)g Ft(a)1079 282 y Fr(j)r(l)1121 275 y Fo(2)d Ft(A;)8 b Fv(1)14 b Fo(\024)g Ft(j)i Fo(\024)e Ft(N)r(;)8 b Fv(1)15 b Fo(\024)e Ft(l)i Fo(\024)e Ft(n)p Fj(,)301 333 y(with)407 296 y Fn(P)460 348 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(N)591 333 y Ft(D)631 340 y Fr(i)645 333 y Fv(\()p Ft(y)688 340 y Fr(j)706 333 y Fv(\))p Ft(a)751 340 y Fr(j)r(l)794 333 y Fv(=)g Ft(\016)867 340 y Fr(il)892 333 y Ft(;)k Fj(for)g(al)r(l)i Fv(1)14 b Fo(\024)f Ft(i;)8 b(l)14 b Fo(\024)g Ft(n:)218 396 y Fv(\(3\))21 b Fj(If)14 b Ft(A)f Fj(is)g(lo)n(c)n(al:)21 b(Ther)n(e)13 b(ar)n(e)g Ft(y)814 403 y Fs(1)833 396 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)967 403 y Fr(n)1004 396 y Fo(2)14 b Ft(A)f Fj(such)h(that)g(for)f(al)r(l)h Fv(1)h Fo(\024)e Ft(m)h Fo(\024)f Ft(n)p Fj(,)301 454 y(the)18 b Ft(m)11 b Fo(\002)g Ft(m)17 b Fj(matrix)g Fv(\()p Ft(D)761 461 y Fr(i)776 454 y Fv(\()p Ft(y)819 461 y Fr(j)837 454 y Fv(\)\))875 461 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)1036 454 y Fj(over)h Ft(A)f Fj(is)g (invertible.)193 535 y(Pr)n(o)n(of.)i Fv(\(1\))f Fo(\))f Fv(\(2\):)23 b(By)17 b(\(1\),)g(1)12 b Fo(\012)g Ft(x)874 542 y Fr(l)904 535 y Fv(is)17 b(in)g(the)g(image)f(of)h(the)g (canonical)g(map)193 593 y Ft(A)c Fo(\012)282 600 y Fr(R)324 593 y Ft(A)18 b Fo(!)h Ft(A)13 b Fo(\012)g Ft(H)24 b Fv(for)19 b(all)g(1)h Fo(\024)f Ft(l)g Fo(\024)g Ft(n)p Fv(.)31 b(Hence)18 b(there)h(are)g(elemen)o(ts)e Ft(y)1617 600 y Fr(j)1654 593 y Fo(2)193 651 y Ft(A;)8 b Fv(1)18 b Fo(\024)f Ft(j)k Fo(\024)d Ft(N)5 b Fv(,)19 b(for)g(some)f Ft(N)23 b Fo(2)18 b Fg(N)i Fv(suc)o(h)e(that)h(there)g(are)f Ft(a)1352 658 y Fr(j)r(l)1399 651 y Fo(2)g Ft(A;)8 b Fv(1)18 b Fo(\024)g Ft(j)j Fo(\024)193 709 y Ft(N)r(;)8 b Fv(1)15 b Fo(\024)e Ft(l)i Fo(\024)e Ft(n)p Fv(,)j(with)529 748 y Fn(X)505 854 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(N)530 888 y(\013)p Fm(2)p Ff(A)633 796 y Ft(a)659 803 y Fr(j)r(l)688 796 y Ft(D)728 803 y Fr(\013)753 796 y Fv(\()p Ft(y)796 803 y Fr(j)814 796 y Fv(\))11 b Fo(\012)g Ft(x)922 775 y Fr(\013)960 796 y Fv(=)j(1)e Fo(\012)f Ft(x)1126 803 y Fr(l)1138 796 y Ft(;)d Fv(1)14 b Fo(\024)g Ft(l)g Fo(\024)g Ft(n:)193 957 y Fv(In)i(particular,)f(for)i(all)f(1)e Fo(\024)f Ft(i;)8 b(l)14 b Fo(\024)g Ft(n)p Fv(,)905 919 y Fn(P)957 971 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(N)1088 957 y Ft(a)1114 964 y Fr(j)r(l)1143 957 y Ft(D)1183 964 y Fr(i)1197 957 y Fv(\()p Ft(y)1240 964 y Fr(j)1258 957 y Fv(\))g(=)g Ft(\016)1365 964 y Fr(il)1390 957 y Ft(:)243 1019 y Fv(\(2\))20 b Fo(\))f Fv(\(1\):)28 b(By)18 b(Theorem)g(2.2)i(it)f(is)g(enough)h(to)g(sho)o(w)g(that)g(the)f(in)o (tegral)193 1082 y(\003)c(:=)f Ft(x)336 1058 y Fr(p)354 1047 y Fq(s)369 1054 y Fp(1)388 1058 y Fm(\000)p Fs(1)336 1094 y(1)443 1082 y Ft(:)8 b(:)g(:)g(x)537 1064 y Fr(p)555 1052 y Fq(s)570 1056 y(n)592 1064 y Fm(\000)p Fs(1)537 1094 y Fr(n)656 1082 y Fv(is)17 b(in)g(the)f(image)g(of)h(the)g (canonical)g(map)f(can)f(:)g Ft(A)c Fo(\012)1659 1089 y Fr(R)193 1140 y Ft(A)i Fo(!)h Ft(A)d Fo(\012)f Ft(H)t Fv(.)243 1198 y(F)l(or)16 b(all)g(1)e Fo(\024)g Ft(l)g Fo(\024)g Ft(n)p Fv(,)i(using)h(\(2.2\))f(and)h(\(2\))g(w)o(e)f (compute:)193 1287 y(can\()285 1240 y Fn(X)313 1345 y Fr(j)365 1287 y Ft(a)391 1294 y Fr(j)r(l)431 1287 y Fo(\012)11 b Ft(y)505 1294 y Fr(j)523 1287 y Fv(\))j(=)602 1240 y Fn(X)630 1345 y Fr(j)682 1287 y Ft(a)708 1294 y Fr(j)r(l)737 1287 y Ft(y)761 1294 y Fr(j)790 1287 y Fo(\012)d Fv(1)h(+)925 1240 y Fn(X)942 1345 y Fr(j;i)1005 1287 y Ft(a)1031 1294 y Fr(j)r(l)1060 1287 y Ft(D)1100 1294 y Fr(i)1114 1287 y Fv(\()p Ft(y)1157 1294 y Fr(j)1175 1287 y Fv(\))f Fo(\012)g Ft(x)1283 1294 y Fr(i)605 1427 y Fv(+)674 1380 y Fn(X)654 1488 y Fr(j;)p Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)774 1427 y Ft(a)800 1434 y Fr(j)r(l)828 1427 y Ft(D)868 1434 y Fr(\013)894 1427 y Fv(\()p Ft(y)937 1434 y Fr(j)955 1427 y Fv(\))g Fo(\012)f Ft(x)1062 1407 y Fr(\013)556 1572 y Fv(=)602 1525 y Fn(X)630 1630 y Fr(j)682 1572 y Ft(a)708 1579 y Fr(j)r(l)737 1572 y Ft(y)761 1579 y Fr(j)790 1572 y Fo(\012)h Fv(1)h(+)f(1)g Fo(\012)g Ft(x)1038 1579 y Fr(l)1062 1572 y Fv(+)1130 1525 y Fn(X)1111 1633 y Fr(j;)p Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)1230 1572 y Ft(a)1256 1579 y Fr(j)r(l)1285 1572 y Ft(D)1325 1579 y Fr(\013)1350 1572 y Fv(\()p Ft(y)1393 1579 y Fr(j)1411 1572 y Fv(\))g Fo(\012)g Ft(x)1519 1552 y Fr(\013)1544 1572 y Ft(:)193 1708 y Fv(De\014ning)k Ft(z)409 1715 y Fr(l)436 1708 y Fv(=)487 1671 y Fn(P)540 1723 y Fr(j)566 1708 y Ft(a)592 1715 y Fr(j)r(l)630 1708 y Fo(\012)8 b Ft(y)701 1715 y Fr(j)727 1708 y Fo(\000)774 1671 y Fn(P)827 1723 y Fr(j)854 1708 y Ft(a)880 1715 y Fr(j)r(l)908 1708 y Ft(y)932 1715 y Fr(j)959 1708 y Fo(\012)g Fv(1,)15 b(and)h Ft(a)1179 1690 y Fr(\013)1179 1721 y(l)1217 1708 y Fv(=)1269 1671 y Fn(P)1321 1723 y Fr(j)1348 1708 y Ft(a)1374 1715 y Fr(j)r(l)1402 1708 y Ft(D)1442 1715 y Fr(\013)1468 1708 y Fv(\()p Ft(y)1511 1715 y Fr(j)1529 1708 y Fv(\))f(for)g(all)193 1771 y(1)f Fo(\024)g Ft(l)g Fo(\024)g Ft(n)i Fv(and)h Ft(\013)d Fo(2)g Fg(A)d Ft(;)d Fo(j)p Ft(\013)p Fo(j)17 b(\025)d Fv(2,)i(w)o(e)g(obtain)493 1860 y(can\()p Ft(z)608 1867 y Fr(l)620 1860 y Fv(\))e(=)g(1)d Fo(\012)g Ft(x)818 1867 y Fr(l)842 1860 y Fv(+)899 1813 y Fn(X)891 1920 y Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)987 1860 y Ft(a)1013 1839 y Fr(\013)1013 1872 y(l)1048 1860 y Fo(\012)g Ft(x)1126 1839 y Fr(\013)1150 1860 y Ft(;)d Fv(1)14 b Fo(\024)g Ft(l)g Fo(\024)g Ft(n:)193 1993 y Fv(W)l(e)i(claim)e(that)622 2051 y(can)q(\()p Ft(z)740 2027 y Fr(p)758 2016 y Fq(s)773 2023 y Fp(1)791 2027 y Fm(\000)p Fs(1)738 2063 y(1)847 2051 y Fo(\001)8 b(\001)g(\001)g Ft(z)938 2030 y Fr(p)956 2019 y Fq(s)971 2023 y(n)994 2030 y Fm(\000)p Fs(1)936 2063 y Fr(n)1041 2051 y Fv(\))13 b(=)h(1)e Fo(\012)f Fv(\003)p Ft(:)193 2119 y Fv(Since)k(the)h(canonical)g(map)g(is)g(an)h (algebra)f(map,)f(w)o(e)h(ha)o(v)o(e)193 2199 y(can\()p Ft(z)310 2176 y Fr(p)328 2164 y Fq(s)343 2171 y Fp(1)362 2176 y Fm(\000)p Fs(1)308 2212 y(1)417 2199 y Fo(\001)8 b(\001)g(\001)h Ft(z)509 2179 y Fr(p)527 2167 y Fq(s)542 2171 y(n)564 2179 y Fm(\000)p Fs(1)507 2212 y Fr(n)611 2199 y Fv(\))14 b(=)g(can\()p Ft(z)811 2206 y Fs(1)830 2199 y Fv(\))849 2179 y Fr(p)867 2167 y Fq(s)882 2174 y Fp(1)901 2179 y Fm(\000)p Fs(1)956 2199 y Fo(\001)8 b(\001)g(\001)h Fv(can\()p Ft(z)1138 2206 y Fr(n)1161 2199 y Fv(\))1180 2179 y Fr(p)1198 2167 y Fq(s)1213 2171 y(n)1236 2179 y Fm(\000)p Fs(1)644 2305 y Fv(=)716 2257 y Fn(Y)696 2364 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)809 2250 y Fn(\020)839 2305 y Fv(1)i Fo(\012)g Ft(x)952 2312 y Fr(l)976 2305 y Fv(+)1032 2257 y Fn(X)1025 2365 y Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)1120 2305 y Ft(a)1146 2284 y Fr(\013)1146 2317 y(l)1182 2305 y Fo(\012)g Ft(x)1260 2284 y Fr(\013)1284 2250 y Fn(\021)1314 2261 y Fr(p)1332 2248 y Fq(s)1347 2257 y(l)1360 2261 y Fm(\000)p Fs(1)1407 2305 y Ft(:)193 2438 y Fv(W)l(e)16 b(see)g(that)g(this)h(pro)q(duct)f (is)h(a)f(sum)f(of)i(terms)e(of)h(the)g(form)254 2492 y Fn(Y)233 2598 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)338 2539 y Fv(\(1)c Fo(\012)e Ft(x)470 2546 y Fr(l)483 2539 y Fv(\))502 2519 y Fr(c)517 2525 y Fq(l)532 2484 y Fn(\020)577 2492 y(X)570 2600 y Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)665 2539 y Ft(a)691 2519 y Fr(\013)691 2551 y(l)727 2539 y Fo(\012)g Ft(x)804 2519 y Fr(\013)829 2484 y Fn(\021)859 2495 y Fr(d)877 2501 y Fq(l)891 2539 y Ft(;)e(c)934 2546 y Fr(l)958 2539 y Fv(+)j Ft(d)1032 2546 y Fr(l)1059 2539 y Fv(=)j Ft(p)1135 2519 y Fr(s)1151 2525 y Fq(l)1177 2539 y Fo(\000)d Fv(1)p Ft(;)24 b Fv(for)17 b(all)e(1)g Fo(\024)e Ft(l)i Fo(\024)e Ft(n:)193 2672 y Fv(The)j(general)g(term)f (of)h(this)g(sum)g(can)g(b)q(e)g(written)g(as)206 2759 y Ft(a)10 b Fo(\012)h Ft(x)320 2738 y Fr(\015)342 2759 y Ft(;)d(a)13 b Fo(2)h Ft(A;)8 b(\015)17 b Fo(2)d Fg(N)633 2738 y Fr(n)657 2759 y Ft(;)8 b Fo(j)p Ft(\015)s Fo(j)14 b(\025)818 2711 y Fn(X)801 2817 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)906 2759 y Fv(\()p Ft(c)946 2766 y Fr(l)970 2759 y Fv(+)d(2)p Ft(d)1068 2766 y Fr(l)1082 2759 y Fv(\))j(=)1183 2711 y Fn(X)1166 2817 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)1271 2759 y Fv(\()p Ft(p)1314 2738 y Fr(s)1330 2744 y Fq(l)1356 2759 y Fo(\000)d Fv(1\))h(+)1526 2711 y Fn(X)1510 2817 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)1623 2759 y Ft(d)1648 2766 y Fr(l)1661 2759 y Ft(:)p eop %%Page: 10 10 10 9 bop 193 131 a Fk(10)163 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h (HANS-J)1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)193 217 y Fv(Only)d(the)h(terms)e Ft(a)6 b Fo(\012)g Ft(x)631 199 y Fr(\015)666 217 y Fv(with)13 b Ft(\015)k Fv(=)d(\()p Ft(\015)912 224 y Fs(1)932 217 y Ft(;)8 b(:)g(:)g(:)f(;)h(\015)1066 224 y Fr(n)1090 217 y Fv(\))14 b(and)g Ft(\015)1240 224 y Fr(l)1267 217 y Fo(\024)g Ft(p)1344 199 y Fr(s)1360 205 y Fq(l)1381 217 y Fo(\000)6 b Fv(1)p Ft(;)i Fv(1)14 b Fo(\024)g Ft(l)g Fo(\024)g Ft(n)p Fv(,)193 275 y(can)i(b)q(e)f (non-zero.)22 b(Therefore)15 b(w)o(e)g(only)g(ha)o(v)o(e)f(to)i (consider)f Ft(a)9 b Fo(\012)g Ft(x)1431 257 y Fr(\015)1469 275 y Fv(with)15 b Fo(j)p Ft(\015)s Fo(j)f(\024)193 296 y Fn(P)246 348 y Fs(1)p Fm(\024)p Fr(l)p Fm(\024)p Fr(n)352 333 y Fv(\()p Ft(p)395 315 y Fr(s)411 321 y Fq(l)437 333 y Fo(\000)d Fv(1\),)16 b(that)h(is)f(with)g Ft(d)851 340 y Fr(l)878 333 y Fv(=)e(0)j(for)f(all)g Ft(l)q Fv(.)21 b(This)16 b(pro)o(v)o(es)g(our)h(claim.)243 395 y(\(2\))f Fo(\))g Fv(\(3\))h(follo)o(ws)f(from)f(the)h(next)g(Lemma.)p 1652 453 2 33 v 1654 422 30 2 v 1654 453 V 1683 453 2 33 v 193 543 a Fx(Lemma)g(4.2.)23 b Fj(L)n(et)e Ft(A)f Fj(b)n(e)h(a)g(c)n(ommutative)h(lo)n(c)n(al)f(ring,)h Fv(1)e Fo(\024)g Ft(n;)8 b(N)26 b Fo(2)20 b Fg(N)i Fj(and)193 601 y Ft(d)218 608 y Fr(ij)263 601 y Fo(2)15 b Ft(A;)8 b Fv(1)14 b Fo(\024)h Ft(i)f Fo(\024)g Ft(n;)8 b Fv(1)15 b Fo(\024)f Ft(j)k Fo(\024)c Ft(N)5 b Fj(.)24 b(Assume)18 b(that)h(ther)n(e)e(ar)n(e)h Ft(a)1377 608 y Fr(j)r(l)1420 601 y Fo(2)d Ft(A;)8 b Fv(1)14 b Fo(\024)g Ft(l)i Fo(\024)193 659 y Ft(n;)8 b Fv(1)14 b Fo(\024)g Ft(j)j Fo(\024)c Ft(N)23 b Fj(with)592 622 y Fn(P)644 673 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(N)775 659 y Ft(d)800 666 y Fr(ij)831 659 y Ft(a)857 666 y Fr(j)r(l)899 659 y Fv(=)14 b Ft(\016)973 666 y Fr(il)998 659 y Ft(;)j Fj(for)f(al)r(l)j Fv(1)14 b Fo(\024)g Ft(i;)8 b(l)14 b Fo(\024)f Ft(n)p Fj(.)243 719 y(Then)23 b(ther)n(e)h(ar)n(e)e Fv(1)j Fo(\024)f Ft(j)723 726 y Fs(1)767 719 y Ft(<)h Fo(\001)8 b(\001)g(\001)24 b Ft(<)g(j)994 726 y Fr(n)1042 719 y Fo(\024)g Ft(N)29 b Fj(and)23 b(a)g(p)n(ermutation)g Ft(\033)j Fo(2)193 777 y Ft(S)223 784 y Fr(n)267 777 y Fj(such)c(that)f(for)f(al)r(l)i Fv(1)f Fo(\024)f Ft(m)g Fo(\024)g Ft(n)p Fj(,)h(the)h Ft(m)13 b Fo(\002)g Ft(m)21 b Fj(matrix)g Fv(\()p Ft(d)1391 784 y Fr(ij)1417 791 y Fq(\033)q Fp(\()p Fq(l)p Fp(\))1474 777 y Fv(\))1493 784 y Fs(1)p Fm(\024)p Fr(i;l)p Fm(\024)p Fr(m)1652 777 y Fj(is)193 836 y(invertible.)193 926 y(Pr)n(o)n(of.)e Fv(Since)e(a)h(square)g(matrix)e(o)o(v)o(er)h Ft(A)g Fv(is)h(in)o(v)o(ertible)d(if)i(and)h(only)g(if)f(its)g(de-)193 984 y(terminan)o(t)f(is)i Fo(6\021)f Fv(0)33 b(mo)q(d)17 b Fh(m)p Fv(,)h Fh(m)g Fv(the)g(maximal)d(ideal)j(of)g Ft(A)p Fv(,)g(w)o(e)f(ma)o(y)g(assume)193 1042 y(that)g Ft(A)e Fv(is)i(a)f(\014eld.)243 1101 y(Since)g(the)h(ro)o(ws)g(of)h(\() p Ft(d)670 1108 y Fr(ij)700 1101 y Fv(\))f(are)g(linearly)f(indep)q (enden)o(t,)g(there)g(is)h(a)g(sequence)193 1159 y(1)d Fo(\024)g Ft(j)304 1166 y Fs(1)337 1159 y Ft(<)g Fo(\001)8 b(\001)g(\001)14 b Ft(<)g(j)533 1166 y Fr(n)570 1159 y Fo(\024)g Ft(N)22 b Fv(suc)o(h)16 b(that)g(\()p Ft(d)943 1166 y Fr(ij)969 1172 y Fq(l)984 1159 y Fv(\))1003 1166 y Fs(1)p Fm(\024)p Fr(i;l)p Fm(\024)p Fr(n)1148 1159 y Fv(is)g(in)o(v)o(ertible.)243 1217 y(Hence)g(it)h(su\016ces)g(to)g (sho)o(w)h(that)g(for)f(an)o(y)h(in)o(v)o(ertible)c(square)k(matrix)d Ft(B)j Fv(=)193 1275 y(\()p Ft(b)233 1282 y Fr(ij)263 1275 y Fv(\))282 1282 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)435 1275 y Fv(o)o(v)o(er)g(a)i(\014eld)e(there)h(is)g(a)g(p)q(erm) o(utation)f Ft(\033)j Fo(2)e Ft(S)1330 1282 y Fr(n)1372 1275 y Fv(suc)o(h)g(that)h(eac)o(h)193 1333 y(submatrix)g(\()p Ft(b)470 1340 y Fr(ij)500 1333 y Fv(\))519 1340 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)663 1333 y Fv(,)i(1)h Fo(\024)g Ft(m)f Fo(\024)h Ft(n)p Fv(,)f(is)f(in)o(v)o(ertible.)34 b(W)l(e)21 b(pro)o(v)o(e)g(this)g(re-)193 1391 y(sult)g(b)o(y)f (induction)h(on)g Ft(n)p Fv(.)35 b(Since)20 b(the)h(\014rst)g Ft(n)15 b Fo(\000)f Fv(1)21 b(ro)o(ws)g(of)g Ft(B)j Fv(are)d(linearly) 193 1449 y(indep)q(enden)o(t,)d(there)h(are)g(column)e(indices)h(1)h Fo(\024)f Ft(j)1154 1456 y Fs(1)1192 1449 y Ft(<)g Fo(\001)8 b(\001)g(\001)19 b Ft(<)f(j)1401 1456 y Fr(n)p Fm(\000)p Fs(1)1488 1449 y Fo(\024)h Ft(n)g Fv(suc)o(h)193 1507 y(that)12 b(\()p Ft(b)334 1514 y Fr(ij)360 1520 y Fq(l)374 1507 y Fv(\))393 1514 y Fs(1)p Fm(\024)p Fr(i;l)p Fm(\024)p Fr(n)p Fm(\000)p Fs(1)578 1507 y Fv(is)g(in)o(v)o(ertible.)k(By)11 b(induction)g(there)g(is)g(a)h(p)q(erm)o(utation)f Ft(\034)19 b Fo(2)193 1565 y Ft(S)223 1572 y Fr(n)p Fm(\000)p Fs(1)304 1565 y Fv(suc)o(h)12 b(that)h(all)e(submatrices)g(\()p Ft(b)880 1572 y Fr(ij)906 1579 y Fq(\034)s Fp(\()p Fq(l)p Fp(\))962 1565 y Fv(\))981 1572 y Fs(1)p Fm(\024)p Fr(i;l)p Fm(\024)p Fr(m)1119 1565 y Fv(,)i(1)h Fo(\024)g Ft(m)f Fo(\024)h Ft(n)s Fo(\000)s Fv(1,)e(are)h(in)o(v)o(ert-)193 1628 y(ible.)20 b(Let)14 b Ft(j)409 1635 y Fr(n)448 1628 y Fv(b)q(e)h(the)f(column)f(index)h(not)h(con)o(tained)f(in)h Fo(f)p Ft(j)1299 1635 y Fs(1)1319 1628 y Ft(;)8 b(:)g(:)g(:)f(;)h(j) 1448 1635 y Fr(n)p Fm(\000)p Fs(1)1517 1628 y Fo(g)p Fv(.)20 b(Then)193 1686 y(\()p Ft(j)232 1694 y Fr(\034)t Fs(\(1\))299 1686 y Ft(;)8 b(:)g(:)g(:)f(;)h(j)428 1694 y Fr(\034)t Fs(\()p Fr(n)p Fm(\000)p Fs(1\))544 1686 y Ft(;)g(j)586 1693 y Fr(n)609 1686 y Fv(\))16 b(is)g(the)g(p)q(erm)o (utation)g(of)g(the)g(columns)f(w)o(e)h(w)o(an)o(t.)p 1652 1686 V 1654 1655 30 2 v 1654 1686 V 1683 1686 2 33 v 193 1776 a Fx(Theorem)g(4.3.)24 b Fj(L)n(et)c Ft(H)t Fj(,)h Ft(\016)f Fv(:)e Ft(A)h Fo(!)f Ft(A)13 b Fo(\012)g Ft(H)24 b Fj(and)c Ft(R)h Fj(b)n(e)f(as)g(in)h(The)n(or)n(em)e(4.1)193 1834 y(and)c(assume)g Ft(y)477 1841 y Fs(1)496 1834 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)630 1841 y Fr(n)667 1834 y Fo(2)14 b Ft(A)g Fj(such)h(that)g(for)f(al)r(l)i Fv(1)e Fo(\024)g Ft(m)f Fo(\024)h Ft(n)p Fj(,)h Fv(\()p Ft(D)1430 1841 y Fr(i)1445 1834 y Fv(\()p Ft(y)1488 1841 y Fr(j)1506 1834 y Fv(\)\))1544 1841 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)193 1892 y Fj(is)i(invertible.)243 1950 y(Then)h(the)g(elements)545 2033 y Ft(y)571 2013 y Fr(\013)610 2033 y Fv(:=)13 b Ft(y)701 2011 y Fr(\013)724 2016 y Fp(1)699 2045 y Fs(1)751 2033 y Fo(\001)8 b(\001)g(\001)g Ft(y)843 2013 y Fr(\013)866 2017 y Fq(n)841 2045 y Fr(n)889 2033 y Ft(;)g(\013)14 b Fv(=)g(\()p Ft(\013)1058 2040 y Fs(1)1086 2033 y Ft(:)8 b(:)g(:)f(\013)1182 2040 y Fr(n)1206 2033 y Fv(\))13 b Fo(2)h Fg(A)d Ft(;)193 2116 y Fj(form)17 b(an)g Ft(R)p Fj(-b)n(asis)h(of)g Ft(A)p Fj(.)193 2206 y(Pr)n(o)n(of.)h Fv(W)l(e)d(pro)q(ceed)g(b)o(y)g (induction)g(on)g(the)g(biggest)h(exp)q(onen)o(t)f Ft(s)1441 2213 y Fs(1)1461 2206 y Fv(.)243 2264 y(If)i Ft(s)317 2271 y Fs(1)356 2264 y Fv(=)h(1,)h(w)o(e)e(are)h(in)g(the)g(Lie)g (algebra)h(case,)f(and)h(the)f(claim)e(follo)o(ws)i(b)o(y)193 2322 y(Lemma)14 b(3.4.)22 b(F)l(or)16 b(the)g(induction)g(step)g(supp)q (ose)308 2405 y Ft(s)331 2412 y Fs(1)365 2405 y Fo(\025)e Ft(s)441 2412 y Fs(2)469 2405 y Fo(\001)8 b(\001)g(\001)13 b(\025)h Ft(s)616 2412 y Fr(m)663 2405 y Ft(>)g Fv(1)p Ft(;)8 b(s)784 2412 y Fr(m)p Fs(+1)876 2405 y Fv(=)14 b Fo(\001)8 b(\001)g(\001)14 b Fv(=)g Ft(s)1075 2412 y Fs(1)1095 2405 y Ft(;)24 b Fv(for)16 b(some)f(1)g Fo(\024)e Ft(m)h Fo(\024)f Ft(n:)193 2488 y Fv(Recall)f(from)h(Section)g(2)h (that)g Ft(H)803 2470 y Fs([1])856 2488 y Fv(=)g Ft(k)r Fv([)p Ft(H)993 2470 y Fr(p)1013 2488 y Fv(])f(is)g(a)h(Hopf)f (subalgebra)i(of)f Ft(H)k Fv(with)193 2549 y(quotien)o(t)d(Hopf)i (algebra)p 675 2509 45 2 v 16 w Ft(H)720 2557 y Fs(\(1\))781 2549 y Fv(=)c Ft(H)q(=)p Fv(\()p Ft(x)944 2531 y Fr(p)979 2549 y Fo(j)g Ft(x)h Fo(2)g Ft(H)1139 2531 y Fs(+)1169 2549 y Fv(\))p Ft(:)h Fv(Then)255 2645 y Ft(k)r Fv([)p Ft(X)336 2652 y Fs(1)356 2645 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)505 2652 y Fr(m)539 2645 y Fv(])p Ft(=)p Fv(\()p Ft(X)640 2622 y Fr(p)658 2610 y Fq(s)673 2617 y Fp(1)690 2610 y Fe(\000)p Fp(1)636 2658 y Fs(1)733 2645 y Ft(;)g(:)g(:)g(:)g(;)g(X) 887 2625 y Fr(p)905 2613 y Fq(s)920 2617 y(m)948 2613 y Fe(\000)p Fp(1)883 2658 y Fr(m)992 2645 y Fv(\))14 b Fo(!)f Ft(H)1132 2625 y Fs([1])1172 2645 y Ft(;)p 1194 2605 55 2 v 8 w(X)1234 2652 y Fr(i)1262 2645 y Fo(7!)h Ft(x)1354 2622 y Fr(p)1354 2658 y(i)1373 2645 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(m;)193 2728 y Fv(and)345 2811 y Ft(k)r Fv([)p Ft(X)426 2818 y Fs(1)446 2811 y Ft(;)8 b(:)g(:)g(:)f(;)h(X)595 2818 y Fr(n)619 2811 y Fv(])p Ft(=)p Fv(\()p Ft(X)720 2788 y Fr(p)716 2824 y Fs(1)740 2811 y Ft(;)g(:)g(:)g(:)f(;)h(X)893 2791 y Fr(p)889 2824 y(n)913 2811 y Fv(\))14 b Fo(!)p 1010 2771 45 2 v 14 w Ft(H)1054 2819 y Fs(\(1\))1101 2811 y Ft(;)p 1123 2771 55 2 v 8 w(X)1163 2818 y Fr(i)1192 2811 y Fo(7!)p 1255 2784 42 2 v 13 w Ft(x)1283 2818 y Fr(i)1297 2811 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)p eop %%Page: 11 11 11 10 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)208 b(11)193 217 y Fv(are)16 b(isomorphisms.)243 280 y(Let)g Ft(B)h Fv(=)d Ft(A)473 261 y Fr(co)p 505 234 32 2 v(H)536 268 y Fp(\(1\))580 280 y Fv(.)21 b(By)16 b(Theorem)f(4.1,)h Ft(R)e Fo(\032)g Ft(A)i Fv(is)g(an)h Ft(H)t Fv(-Galois)g(extension.)193 338 y(Hence)f(b)o(y)i(Theorem)e (2.3,)i Ft(R)f Fo(\032)g Ft(B)j Fv(is)e(an)g Ft(H)1041 320 y Fs([1])1081 338 y Fv(-Galois)g(and)h Ft(B)g Fo(\032)d Ft(A)h Fv(an)p 1580 298 45 2 v 19 w Ft(H)1624 346 y Fs(\(1\))1671 338 y Fv(-)193 396 y(Galois)h(extension.)25 b(The)17 b(coaction)h(of)g Ft(B)i Fv(is)e(the)f(restriction)g(of)h Ft(\016)r Fv(.)25 b(Therefore)193 454 y(w)o(e)16 b(obtain)h(from)e(the) h(form)o(ula)f(\(2.2\))h(for)h(the)f(coaction)g(for)g(all)g Ft(b)e Fo(2)g Ft(B)s Fv(,)592 543 y Ft(\016)r Fv(\()p Ft(b)p Fv(\))f(=)h Ft(b)d Fo(\012)f Fv(1)i(+)914 496 y Fn(X)906 602 y Fr(p\013)p Fm(2)p Ff(A)1002 543 y Ft(D)1042 550 y Fr(p\013)1085 543 y Fv(\()p Ft(b)p Fv(\))f Fo(\012)g Ft(x)1233 523 y Fr(p\013)1275 543 y Ft(:)193 675 y Fv(Here)k(w)o(e)h (use)g(the)g(notation)h Ft(p\013)e Fv(=)f(\()p Ft(p\013)939 682 y Fs(1)959 675 y Ft(;)8 b(:)g(:)g(:)f(;)h(p\013)1123 682 y Fr(n)1147 675 y Fv(\))16 b(for)h(all)e Ft(\013)g Fv(=)e(\()p Ft(\013)1471 682 y Fs(1)1491 675 y Ft(;)8 b(:)g(:)g(:)f(;)h(\013)1631 682 y Fr(n)1655 675 y Fv(\).)243 733 y(On)16 b(the)h(other)f(hand,)h(b)o(y)f(taking)h Ft(p)p Fv(-th)g(p)q(o)o(w)o(ers)g(it)f(follo)o(ws)h(from)e(\(2.2\))i (that)193 792 y(for)f(all)g Ft(a)e Fo(2)g Ft(A)p Fv(,)354 878 y Ft(\016)r Fv(\()p Ft(a)423 858 y Fr(p)442 878 y Fv(\))f(=)h Ft(a)552 858 y Fr(p)583 878 y Fo(\012)c Fv(1)i(+)734 831 y Fn(X)717 936 y Fs(1)p Fm(\024)p Fr(i)p Fm(\024)p Fr(n)831 878 y Ft(D)871 885 y Fr(i)886 878 y Fv(\()p Ft(a)p Fv(\))950 858 y Fr(p)980 878 y Fo(\012)f Ft(x)1058 854 y Fr(p)1058 891 y(i)1089 878 y Fv(+)1145 831 y Fn(X)1146 937 y Fr(\013)p Fm(2)p Ff(A)1138 968 y Fm(j)p Fr(\013)p Fm(j\025)p Fs(2)1233 878 y Ft(D)1273 885 y Fr(\013)1299 878 y Fv(\()p Ft(a)p Fv(\))1363 858 y Fr(p)1393 878 y Fo(\012)g Ft(x)1471 885 y Fr(p\013)1513 878 y Ft(:)193 1043 y Fv(In)16 b(particular,)f Ft(D)532 1050 y Fr(i)547 1043 y Fv(\()p Ft(a)p Fv(\))611 1025 y Fr(p)644 1043 y Fv(=)f Ft(D)736 1050 y Fr(p\016)770 1055 y Fq(i)785 1043 y Fv(\()p Ft(a)830 1025 y Fr(p)849 1043 y Fv(\).)243 1101 y(By)e(assumption,)h(the)g(matrix)e(\()p Ft(D)880 1108 y Fr(i)895 1101 y Fv(\()p Ft(y)938 1108 y Fr(j)956 1101 y Fv(\)\))994 1108 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(t)1132 1101 y Fv(is)i(in)o(v)o(ertible)d(for)k(all)e(1)j Fo(\024)e Ft(t)h Fo(\024)193 1160 y Ft(m)p Fv(.)32 b(Hence)19 b(also)i(\()p Ft(D)592 1167 y Fr(i)607 1160 y Fv(\()p Ft(y)650 1167 y Fr(j)668 1160 y Fv(\))687 1141 y Fr(p)706 1160 y Fv(\))725 1167 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(t)871 1160 y Fv(=)f(\()p Ft(D)988 1167 y Fr(p\016)1022 1172 y Fq(i)1038 1160 y Fv(\()p Ft(y)1083 1136 y Fr(p)1081 1172 y(j)1102 1160 y Fv(\)\))1140 1167 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(t)1285 1160 y Fv(is)g(in)o(v)o(ertible)d(for)k (all)193 1218 y(1)14 b Fo(\024)g Ft(t)f Fo(\024)h Ft(m)p Fv(.)21 b(Therefore)16 b(it)f(follo)o(ws)h(b)o(y)g(induction)g(that)523 1296 y Ft(y)549 1273 y Fr(p\013)590 1278 y Fp(1)547 1308 y Fs(1)617 1296 y Fo(\001)8 b(\001)g(\001)g Ft(y)709 1276 y Fr(p\013)750 1280 y Fq(m)707 1308 y Fr(m)781 1296 y Ft(;)g Fv(0)14 b Fo(\024)f Ft(\013)924 1303 y Fr(i)952 1296 y Ft(<)h(p)1028 1276 y Fr(s)1044 1281 y Fq(i)1058 1276 y Fm(\000)p Fs(1)1105 1296 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(m;)193 1375 y Fv(is)i(an)h Ft(R)p Fv(-basis)g(of)g Ft(B)s Fv(.)243 1433 y(Moreo)o(v)o(er)e(w)o(e)h (kno)o(w)g(from)f(the)h(Lie)g(algebra)h(case)f(that)596 1514 y Ft(y)622 1490 y Fr(\014)642 1495 y Fp(1)620 1526 y Fs(1)669 1514 y Fo(\001)8 b(\001)g(\001)g Ft(y)761 1493 y Fr(\014)781 1497 y Fq(n)759 1526 y Fr(n)804 1514 y Ft(;)g Fv(0)14 b Fo(\024)g Ft(\014)945 1521 y Fr(l)971 1514 y Ft(<)g(p;)8 b Fv(1)14 b Fo(\024)g Ft(l)g Fo(\024)g Ft(n;)193 1592 y Fv(is)i(a)h Ft(B)s Fv(-basis)f(of)h Ft(A)p Fv(.)243 1650 y(Hence)701 1708 y Ft(y)727 1685 y Fr(p\013)768 1690 y Fp(1)725 1721 y Fs(1)794 1708 y Fo(\001)8 b(\001)g(\001)h Ft(y)887 1688 y Fr(p\013)928 1692 y Fq(m)885 1721 y Fr(m)958 1708 y Ft(y)984 1685 y Fr(\014)1004 1690 y Fp(1)982 1721 y Fs(1)1031 1708 y Fo(\001)f(\001)g(\001)h Ft(y)1124 1688 y Fr(\014)1144 1692 y Fq(n)1122 1721 y Fr(n)1166 1708 y Ft(;)429 1777 y Fv(0)15 b Fo(\024)e Ft(\013)551 1784 y Fr(i)579 1777 y Ft(<)h(p)655 1757 y Fr(s)671 1762 y Fq(i)685 1757 y Fm(\000)p Fs(1)732 1777 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(m;)8 b Fv(0)13 b Fo(\024)h Ft(\014)1111 1784 y Fr(l)1137 1777 y Ft(<)g(p;)8 b Fv(1)15 b Fo(\024)e Ft(l)i Fo(\024)e Ft(n;)193 1846 y Fv(is)j(an)h Ft(R)p Fv(-basis)g(of)g Ft(A)p Fv(,)e(and)i(w)o(e)f(are)g(done.)p 1652 1846 2 33 v 1654 1814 30 2 v 1654 1846 V 1683 1846 2 33 v 243 1929 a(Note)23 b(that)i(an)o(y)f Ft(H)t Fv(-Galois)h (extension)f(of)g(comm)o(utativ)n(e)d(algebras)k Ft(R)j Fo(\032)193 1987 y Ft(A)p Fv(,)19 b Ft(H)24 b Fv(a)19 b(\014nite-dimensional)e(Hopf)j(algebra,)g(is)f(\014nite)f(b)o(y)h (Theorem)f(2.2.)31 b(In)193 2045 y(particular,)15 b(if)h Ft(A)g Fv(is)g(lo)q(cal,)g(then)g(also)h Ft(R)f Fv(is)g(lo)q(cal.)193 2132 y Fx(Corollary)j(4.4.)k Fj(L)n(et)18 b Ft(H)t Fj(,)i Ft(\016)d Fv(:)f Ft(A)f Fo(!)h Ft(A)11 b Fo(\012)h Ft(H)23 b Fj(and)c Ft(R)g Fj(b)n(e)g(as)f(in)h(The)n(or)n(em)f(4.1,)193 2190 y(and)f(assume)g(that)h Ft(A)e Fj(is)h(lo)n(c)n(al)g(with)h (maximal)f(ide)n(al)g Fh(m)1225 2197 y Fr(A)1254 2190 y Fj(.)22 b(Then)c(the)f(fol)r(lowing)193 2248 y(ar)n(e)g(e)n (quivalent:)218 2317 y Fv(\(1\))k Ft(R)15 b Fo(\032)e Ft(A)j Fj(is)h(an)f Ft(H)t Fj(-Galois)h(extension,)i(and)d Ft(R)9 b Fv(+)g Fh(m)1245 2324 y Fr(A)1288 2317 y Fv(=)k Ft(A)p Fj(,)j(that)h(is)f Ft(A)g Fj(and)301 2375 y Ft(R)i Fj(have)h(the)e(same)h(r)n(esidue)f(\014eld.)218 2433 y Fv(\(2\))k Fj(Ther)n(e)e(ar)n(e)e Ft(y)550 2440 y Fs(1)570 2433 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)703 2440 y Fr(n)742 2433 y Fo(2)16 b Fh(m)829 2440 y Fr(A)876 2433 y Fj(such)j(that)f(the)h (matrix)g Fv(\()p Ft(D)1387 2440 y Fr(i)1401 2433 y Fv(\()p Ft(y)1444 2440 y Fr(j)1462 2433 y Fv(\)\))1500 2440 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)1652 2433 y Fj(is)301 2491 y(invertible.)193 2578 y(Pr)n(o)n(of.)g Fv(\(1\))f Fo(\))g Fv(\(2\):)25 b(By)17 b(Theorem)f(4.1)i(there)g(are)g Ft(y)1190 2585 y Fs(1)1209 2578 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)1342 2585 y Fr(n)1382 2578 y Fo(2)17 b Ft(A)g Fv(suc)o(h)h(that)193 2636 y(\()p Ft(D)252 2643 y Fr(i)267 2636 y Fv(\()p Ft(y)310 2643 y Fr(j)328 2636 y Fv(\)\))366 2643 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)517 2636 y Fv(is)f(in)o(v)o(ertible.)22 b(By)17 b(assumption)g(w)o(e)g(can)g(write)g Ft(y)1455 2643 y Fr(i)1485 2636 y Fv(=)e Ft(r)1560 2643 y Fr(i)1586 2636 y Fv(+)f Fn(e)-30 b Ft(y)1660 2643 y Fr(i)1674 2636 y Ft(;)193 2694 y Fv(where)18 b Ft(r)358 2701 y Fr(i)389 2694 y Fo(2)f Ft(R)i Fv(and)h Fn(e)-30 b Ft(y)615 2701 y Fr(i)646 2694 y Fo(2)17 b Fh(m)734 2701 y Fr(A)781 2694 y Fv(for)h(all)g(1)f Fo(\024)g Ft(i)g Fo(\024)f Ft(n)p Fv(.)27 b(Then)18 b Ft(D)1352 2701 y Fr(i)1367 2694 y Fv(\()p Ft(r)1408 2701 y Fr(j)1426 2694 y Fv(\))f(=)g(0)i(for)f (all)193 2753 y(1)e Fo(\024)f Ft(i;)8 b(j)18 b Fo(\024)d Ft(n)i Fv(since)g Ft(R)e Fv(=)h Ft(A)728 2735 y Fr(coH)793 2753 y Fv(.)24 b(Th)o(us)18 b(\()p Ft(D)1015 2760 y Fr(i)1029 2753 y Fv(\()r Fn(e)-30 b Ft(y)1072 2760 y Fr(j)1090 2753 y Fv(\)\))1128 2760 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)1278 2753 y Fv(=)15 b(\()p Ft(D)1390 2760 y Fr(i)1404 2753 y Fv(\()p Ft(y)1447 2760 y Fr(j)1465 2753 y Fv(\)\))1503 2760 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)1655 2753 y Fv(is)193 2811 y(in)o(v)o(ertible.)p eop %%Page: 12 12 12 11 bop 193 131 a Fk(12)163 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h (HANS-J)1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)243 217 y Fv(\(2\))g Fo(\))g Fv(\(1\):)22 b(By)15 b(Theorem)g(4.1,)h Ft(R)f Fo(\032)e Ft(A)j Fv(is)g Ft(H)t Fv(-Galois,)h(and)f(b)o(y)g (Lemma)e(4.2)193 275 y(w)o(e)j(can)h(c)o(ho)q(ose)g(the)f Ft(y)622 282 y Fr(i)653 275 y Fv(in)h Fh(m)750 282 y Fr(A)796 275 y Fv(suc)o(h)f(that)h(condition)f(\(3\))h(in)f(Theorem)g (4.1)h(is)193 333 y(satis\014ed.)i(Then)14 b(the)f(equalit)o(y)e Ft(R)5 b Fv(+)g Fh(m)909 340 y Fr(A)951 333 y Fv(=)14 b Ft(A)f Fv(follo)o(ws)g(from)f(Theorem)g(4.3.)p 1652 333 2 33 v 1654 302 30 2 v 1654 333 V 1683 333 2 33 v 193 429 a Fx(Corollary)19 b(4.5.)k Fj(L)n(et)c Ft(H)t Fj(,)h Ft(\016)f Fv(:)e Ft(A)h Fo(!)f Ft(A)12 b Fo(\012)g Ft(H)24 b Fj(and)c Ft(R)f Fj(b)n(e)h(as)f(in)h(The)n(or)n(em)f(4.1)193 487 y(and)c(assume)g Ft(y)477 494 y Fs(1)496 487 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)630 494 y Fr(n)667 487 y Fo(2)14 b Ft(A)g Fj(such)h(that)g(for)f(al)r(l)i Fv(1)e Fo(\024)g Ft(m)f Fo(\024)h Ft(n)p Fj(,)h Fv(\()p Ft(D)1430 494 y Fr(i)1445 487 y Fv(\()p Ft(y)1488 494 y Fr(j)1506 487 y Fv(\)\))1544 494 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)193 545 y Fj(is)i(invertible.)218 620 y Fv(\(1\))k Fj(If)d Ft(s)376 627 y Fs(1)409 620 y Fv(=)c Fo(\001)8 b(\001)g(\001)14 b Fv(=)g Ft(s)608 627 y Fr(n)645 620 y Fv(=:)f Ft(s;)k Fj(then)i Ft(y)899 597 y Fr(p)917 585 y Fq(s)897 633 y Fr(i)948 620 y Fo(2)14 b Ft(R)k Fj(for)f(al)r(l)i Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n)p Fj(.)218 683 y Fv(\(2\))21 b Fj(If)d Ft(y)379 660 y Fr(p)397 648 y Fq(s)412 656 y(i)377 696 y Fr(i)442 683 y Fo(2)c Ft(R)k Fj(for)f(al)r(l)h Fv(1)c Fo(\024)g Ft(i)g Fo(\024)f Ft(n)p Fj(,)18 b(then)g(the)g(algebr)n(a)g(map)218 781 y Ft(R)p Fv([)p Ft(T)298 788 y Fs(1)317 781 y Ft(;)8 b(:)g(:)g(:)g(;)g(T)456 788 y Fr(n)479 781 y Fv(])p Ft(=)p Fv(\()p Ft(T)572 757 y Fr(p)590 746 y Fq(s)605 753 y Fp(1)565 793 y Fs(1)636 781 y Fo(\000)i Ft(y)711 757 y Fr(p)729 746 y Fq(s)744 753 y Fp(1)709 793 y Fs(1)765 781 y Ft(;)e(:)g(:)g(:)f(;)h(T)910 760 y Fr(p)928 749 y Fq(s)943 753 y(n)903 793 y Fr(n)978 781 y Fo(\000)j Ft(y)1054 760 y Fr(p)1072 749 y Fq(s)1087 753 y(n)1052 793 y Fr(n)1111 781 y Fv(\))j Fo(!)g Ft(A;)p 1267 741 43 2 v 8 w(T)1296 788 y Fr(i)1323 781 y Fo(7!)f Ft(y)1410 788 y Fr(i)1424 781 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)301 874 y Fj(wher)n(e)j Ft(R)p Fv([)p Ft(T)518 881 y Fs(1)538 874 y Ft(;)8 b(:)g(:)g(:)f(;)h(T)676 881 y Fr(n)699 874 y Fv(])16 b Fj(is)h(the)g(p)n(olynomial)g(ring)g(in) g Ft(n)g Fj(variables,)h(and)f(the)301 932 y(c)n(anonic)n(al)i(map)566 1027 y Ft(R=)p Fv(\()p Ft(y)672 1003 y Fr(p)690 991 y Fq(s)705 998 y Fp(1)670 1039 y Fs(1)726 1027 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)862 1006 y Fr(p)880 994 y Fq(s)895 998 y(n)860 1039 y Fr(n)919 1027 y Fv(\))14 b Fo(!)f Ft(A=)p Fv(\()p Ft(y)1119 1034 y Fs(1)1139 1027 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)1272 1034 y Fr(n)1295 1027 y Fv(\))301 1120 y Fj(ar)n(e)17 b(bije)n(ctive.)193 1215 y(Pr)n(o)n(of.)i Fv(\(1\))g(follo)o(ws)f(b)o(y)g(raising)i(\(2.2\))f(to)g(the)f Ft(p)1107 1197 y Fr(s)1126 1215 y Fv(-th)h(p)q(o)o(w)o(er,)f(and)i (\(2\))e(follo)o(ws)193 1273 y(from)d(4.3.)p 1652 1273 2 33 v 1654 1242 30 2 v 1654 1273 V 1683 1273 2 33 v 243 1374 a(The)g(examples)d(b)q(elo)o(w)j(sho)o(w)h(that)f(Theorem)e (4.3)j(do)q(es)f(not)h(hold)e(if)h(w)o(e)f(just)193 1432 y(assume)c(that)i(the)f(matrix)e(\()p Ft(D)748 1439 y Fr(i)763 1432 y Fv(\()p Ft(y)806 1439 y Fr(j)824 1432 y Fv(\)\))862 1439 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)1007 1432 y Fv(is)i(in)o(v)o(ertible;)f(also)h(in)g(the)g (situation)193 1496 y(of)17 b(Corollary)f(4.5)h(it)e(is)h(not)h(alw)o (a)o(ys)f(true)g(that)h Ft(y)1118 1473 y Fr(p)1136 1461 y Fq(s)1151 1469 y(i)1116 1509 y Fr(i)1181 1496 y Fo(2)d Ft(R)j Fv(for)f(all)g(1)e Fo(\024)g Ft(i)f Fo(\024)h Ft(n)p Fv(.)193 1592 y Fx(Example)i(4.6.)24 b Fv(W)l(e)16 b(consider)g(the)g(Hopf)g(algebra)490 1692 y Ft(H)j Fv(=)13 b Ft(H)640 1699 y Fr(a)675 1692 y Fv(=)h Ft(k)r Fv([)p Ft(X)808 1699 y Fs(1)828 1692 y Ft(;)8 b(X)890 1699 y Fs(2)910 1692 y Fv(])p Ft(=)p Fv(\()p Ft(X)1011 1668 y Fr(p)1029 1656 y Fp(2)1007 1704 y Fs(1)1048 1692 y Ft(;)g(X)1114 1668 y Fr(p)1110 1704 y Fs(2)1134 1692 y Fv(\))14 b(=)g Ft(k)r Fv([)p Ft(x)1288 1699 y Fs(1)1307 1692 y Ft(;)8 b(x)1357 1699 y Fs(2)1376 1692 y Fv(])193 1785 y(of)k(the)g(additiv)o(e)e(group)j(with)f Ft(n)i Fv(=)f(2)g(and)f Ft(s)991 1792 y Fs(1)1025 1785 y Fv(=)h(2)p Ft(;)8 b(s)1145 1792 y Fs(2)1179 1785 y Fv(=)14 b(1,)f(and)f(the)g(p)q (olynomial)193 1843 y(algebra)17 b(in)f(t)o(w)o(o)g(v)m(ariables)g Ft(A)d Fv(=)h Ft(k)r Fv([)p Ft(T)887 1850 y Fs(1)906 1843 y Ft(;)8 b(T)957 1850 y Fs(2)976 1843 y Fv(])p Ft(:)243 1901 y Fv(1\))16 b(The)h(algebra)f(map)g Ft(\016)f Fv(:)e Ft(A)h Fo(!)f Ft(A)e Fo(\012)g Ft(H)q(;)16 b Fv(de\014ned)g(b)o(y)324 1994 y Ft(\016)r Fv(\()p Ft(T)396 2001 y Fs(1)415 1994 y Fv(\))e(=)g Ft(T)529 2001 y Fs(1)559 1994 y Fo(\012)d Fv(1)g(+)g(1)h Fo(\012)f Ft(x)807 2001 y Fs(2)826 1994 y Ft(;)d(\016)r Fv(\()p Ft(T)920 2001 y Fs(2)939 1994 y Fv(\))13 b(=)h Ft(T)1052 2001 y Fs(2)1083 1994 y Fo(\012)c Fv(1)i(+)f(1)g Fo(\012)g Ft(x)1330 2001 y Fs(1)1361 1994 y Fv(+)g(1)g Fo(\012)g Ft(x)1523 2001 y Fs(2)1542 1994 y Ft(;)193 2087 y Fv(is)21 b(an)h Ft(H)t Fv(-como)q(dule)e(algebra)i (structure)f(on)g Ft(A)p Fv(.)36 b(Let)21 b Ft(y)1268 2094 y Fr(i)1304 2087 y Fv(:=)h Ft(T)1407 2094 y Fr(i)1420 2087 y Ft(;)8 b Fv(1)23 b Fo(\024)f Ft(i)f Fo(\024)h Fv(2)p Ft(:)193 2145 y Fv(Then)16 b(the)g(matrix)f(\()p Ft(D)622 2152 y Fr(i)636 2145 y Fv(\()p Ft(y)679 2152 y Fr(j)697 2145 y Fv(\)\))735 2152 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fs(2)882 2145 y Fv(is)h(in)o(v)o(ertible,)d(but)j (the)g(elemen)o(ts)627 2241 y Ft(y)653 2220 y Fr(i)665 2225 y Fp(1)651 2254 y Fs(1)684 2241 y Ft(y)710 2220 y Fr(i)722 2225 y Fp(2)708 2254 y Fs(2)741 2241 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)871 2248 y Fs(1)904 2241 y Ft(<)g(p)980 2221 y Fs(2)1000 2241 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)1130 2248 y Fs(2)1163 2241 y Ft(<)g(p;)193 2334 y Fv(do)j(not)f(form)g(a)g(basis)h(of)g Ft(A)e Fv(o)o(v)o(er)h Ft(R)e Fv(=)g Ft(A)976 2316 y Fr(coH)1041 2334 y Ft(:)243 2392 y Fv(2\))i(The)h(algebra)f(map)g Ft(\016)f Fv(:)e Ft(A)h Fo(!)f Ft(A)e Fo(\012)g Ft(H)q(;)16 b Fv(de\014ned)g(b)o(y)324 2485 y Ft(\016)r Fv(\()p Ft(T)396 2492 y Fs(1)415 2485 y Fv(\))e(=)g Ft(T)529 2492 y Fs(1)559 2485 y Fo(\012)d Fv(1)g(+)g(1)h Fo(\012)f Ft(x)807 2492 y Fs(1)826 2485 y Ft(;)d(\016)r Fv(\()p Ft(T)920 2492 y Fs(2)939 2485 y Fv(\))13 b(=)h Ft(T)1052 2492 y Fs(2)1083 2485 y Fo(\012)c Fv(1)i(+)f(1)g Fo(\012)g Ft(x)1330 2492 y Fs(1)1361 2485 y Fv(+)g(1)g Fo(\012)g Ft(x)1523 2492 y Fs(2)1542 2485 y Ft(;)193 2578 y Fv(is)21 b(an)h Ft(H)t Fv(-como)q(dule)e(algebra)i (structure)f(on)g Ft(A)p Fv(.)36 b(Let)21 b Ft(y)1268 2585 y Fr(i)1304 2578 y Fv(:=)h Ft(T)1407 2585 y Fr(i)1420 2578 y Ft(;)8 b Fv(1)23 b Fo(\024)f Ft(i)f Fo(\024)h Fv(2)p Ft(:)193 2636 y Fv(Then)e(the)f(matrices)f(\()p Ft(D)669 2643 y Fr(i)684 2636 y Fv(\()p Ft(y)727 2643 y Fr(j)745 2636 y Fv(\)\))783 2643 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(m)946 2636 y Fv(are)i(in)o(v)o(ertible)d(for)j(1) g Fo(\024)f Ft(m)h Fo(\024)f Fv(2)p Ft(;)h Fv(and)193 2695 y Ft(y)219 2672 y Fr(p)217 2708 y(i)252 2695 y Fo(62)14 b Ft(R)h Fv(=)e Ft(A)439 2677 y Fr(coH)505 2695 y Ft(;)8 b Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Fv(2)p Ft(:)243 2753 y Fv(3\))21 b(In)g(b)q(oth)i(examples)c(1\))j(and)f(2\),)i Ft(\016)g Fv(induces)d(an)i Ft(H)t Fv(-como)q(dule)f(algebra)193 2811 y(structure)16 b(on)h(the)f(lo)q(calization)f Ft(A)849 2818 y Fi(m)877 2811 y Ft(;)8 b Fh(m)14 b Fv(:=)g(\()p Ft(T)1065 2818 y Fs(1)1084 2811 y Ft(;)8 b(T)1135 2818 y Fs(2)1154 2811 y Fv(\))p Ft(:)p eop %%Page: 13 13 13 12 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)208 b(13)193 217 y Fj(Pr)n(o)n(of.)19 b Fv(1\))25 b(It)g(is)g(easy)f(to)i(c)o(hec)o(k)d(that)i Ft(\016)i Fv(is)d(coasso)q(ciativ.)48 b(By)24 b(de\014nition,)193 275 y Ft(D)233 282 y Fs(1)253 275 y Fv(\()p Ft(T)301 282 y Fs(1)320 275 y Fv(\))14 b(=)g(0)p Ft(;)8 b(D)491 282 y Fs(1)511 275 y Fv(\()p Ft(T)559 282 y Fs(2)579 275 y Fv(\))14 b(=)f(1)p Ft(;)8 b(D)749 282 y Fs(2)770 275 y Fv(\()p Ft(T)818 282 y Fs(1)837 275 y Fv(\))14 b(=)g(1)p Ft(;)8 b(D)1008 282 y Fs(2)1028 275 y Fv(\()p Ft(T)1076 282 y Fs(2)1095 275 y Fv(\))14 b(=)g(1)p Ft(:)i Fv(Hence)f(the)h(elemen)o(ts)618 364 y Ft(T)654 342 y Fr(i)666 347 y Fp(1)647 376 y Fs(2)684 364 y Ft(T)720 342 y Fr(i)732 347 y Fp(2)713 376 y Fs(1)751 364 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)881 371 y Fs(1)914 364 y Ft(<)g(p)990 343 y Fs(2)1010 364 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)1140 371 y Fs(2)1173 364 y Ft(<)g(p;)193 451 y Fv(are)i(an)h Ft(R)p Fv(-basis)h(of)e Ft(A)g Fv(b)o(y)g(Theorem)f(4.3.)21 b(In)16 b(particular,)g Ft(A)f Fv(is)h(free)g(o)o(v)o(er)f Ft(R)i Fv(of)193 509 y(rank)f Ft(p)329 491 y Fs(3)350 509 y Ft(:)f Fv(Therefore)h(the)g(elemen)o(ts)618 596 y Ft(T)654 574 y Fr(i)666 579 y Fp(1)647 608 y Fs(1)684 596 y Ft(T)720 574 y Fr(i)732 579 y Fp(2)713 608 y Fs(2)751 596 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)881 603 y Fs(1)914 596 y Ft(<)g(p)990 575 y Fs(2)1010 596 y Ft(;)8 b Fv(0)14 b Fo(\024)g Ft(i)1140 603 y Fs(2)1173 596 y Ft(<)g(p;)193 683 y Fv(do)21 b(not)h(generate)e(the)h Ft(R)p Fv(-mo)q(dule)g Ft(A;)f Fv(since)g Ft(T)1107 659 y Fr(p)1100 695 y Fs(1)1148 683 y Fo(2)i Ft(R;)f Fv(although)h(the)e(2)15 b Fo(\002)f Fv(2)193 741 y(matrix)g(\()p Ft(D)410 748 y Fr(i)425 741 y Fv(\()p Ft(T)473 748 y Fr(j)491 741 y Fv(\)\))529 748 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fs(2)675 741 y Fv(is)i(in)o(v)o(ertible.)243 799 y(2\))i(is)g(easy)f(to)i(c)o(hec)o (k.)24 b(T)l(o)18 b(pro)o(v)o(e)f(3\))h(note)g(that)g(for)g(an)o(y)g (comm)o(utativ)n(e)d Ft(H)t Fv(-)193 857 y(como)q(dule)g(algebra)i (structure)f Ft(\016)f Fv(:)f Ft(A)f Fo(!)h Ft(A)d Fo(\012)g Ft(H)t Fv(,)16 b Ft(H)k Fv(comm)o(utativ)o(e)13 b(and)j(lo)q(cal)193 915 y(with)k(\()p Fh(m)365 922 y Fr(H)399 915 y Fv(\))418 897 y Fr(p)436 885 y Fq(s)475 915 y Fv(=)h(0,)g(and)g(for)g(all)f Ft(a)g Fo(2)h Ft(A)p Fv(,)f Ft(\016)r Fv(\()p Ft(a)p Fv(\))g Fo(2)h Ft(a)14 b Fo(\012)f Fv(1)i(+)e Ft(A)h Fo(\012)f Fh(m)1499 922 y Fr(H)1533 915 y Fv(,)21 b(hence)193 973 y(\()p Ft(\016)r Fv(\()p Ft(a)p Fv(\)\))319 955 y Fr(p)337 943 y Fq(s)368 973 y Fv(=)14 b Ft(a)446 955 y Fr(p)464 943 y Fq(s)489 973 y Fo(\012)7 b Fv(1)p Ft(:)14 b Fv(Th)o(us)g Ft(\016)i Fv(induces)e(an)h Ft(H)t Fv(-como)q(dule)e (algebra)i(structure)f(on)193 1031 y(eac)o(h)i(lo)q(calization)f(of)i Ft(A)p Fv(.)p 1652 1089 2 33 v 1654 1058 30 2 v 1654 1089 V 1683 1089 2 33 v 717 1212 a(5.)28 b Fu(An)18 b(applica)m(tion) 243 1299 y Fv(W)l(e)f(\014nally)h(apply)g(the)f(previous)h(results)g (to)g(the)g(case)g(when)g Ft(A)g Fv(is)f(regular)193 1357 y(lo)q(cal)f(or)h(regular)f(lo)q(cal)g(and)h(complete.)193 1449 y Fx(Theorem)f(5.1.)24 b Fj(L)n(et)14 b Ft(H)19 b Fj(b)n(e)c(a)f(Hopf)g(algebr)n(a)h(satisfying)k Fv(\(2.1\))p Fj(,)c(and)g Ft(A)f Fj(a)g(c)n(om-)193 1507 y(mutative)j(right)g Ft(H)t Fj(-c)n(omo)n(dule)g(algebr)n(a)f(with)h(structur)n(e)f(map)g Ft(\016)g Fv(:)d Ft(A)g Fo(!)h Ft(A)8 b Fo(\012)g Ft(H)t Fj(.)193 1565 y(De\014ne)19 b(the)f(derivations)g Ft(D)719 1572 y Fs(1)739 1565 y Ft(;)8 b(:)g(:)g(:)f(;)h(D)888 1572 y Fr(n)930 1565 y Fj(by)21 b Fv(\(2.2\))d Fj(and)g(let)g Ft(R)d Fv(=)e Ft(A)1418 1547 y Fr(coH)1484 1565 y Fj(.)243 1623 y(Assume)19 b(that)h Ft(A)f Fj(is)g(a)g(r)n(e)n(gular)f(lo)n(c)n (al)i(ring)f(of)g(dimension)h Ft(d)f Fj(with)h(maximal)193 1682 y(ide)n(al)g Fh(m)348 1689 y Fr(A)396 1682 y Fj(and)g(ther)n(e)g (ar)n(e)f Ft(y)726 1689 y Fs(1)746 1682 y Ft(;)8 b(:)g(:)g(:)f(;)h(y) 879 1689 y Fr(n)921 1682 y Fo(2)18 b Fh(m)1010 1689 y Fr(A)1058 1682 y Fj(such)j(that)f(for)f(al)r(l)i Fv(1)e Fo(\024)e Ft(m)h Fo(\024)g Ft(n)p Fj(,)193 1740 y Fv(\()p Ft(D)252 1747 y Fr(i)267 1740 y Fv(\()p Ft(y)310 1747 y Fr(j)328 1740 y Fv(\)\))366 1747 y Fs(1)p Fm(\024)p Fr(i;j)r Fm(\024)p Fr(n)517 1740 y Fj(is)f(invertible.)243 1798 y(Then)h Ft(R)g Fj(is)f(r)n(e)n(gular)g(lo)n(c)n(al.)243 1856 y(\(1\))g(Assume)i(mor)n(e)n(over)e(that)h Ft(s)842 1863 y Fs(1)876 1856 y Fv(=)d Fo(\001)8 b(\001)g(\001)15 b Fv(=)f Ft(s)1077 1863 y Fr(n)1101 1856 y Fj(,)k(or)f(mor)n(e)g(gener) n(al)r(ly)j(that)e(for)193 1914 y(al)r(l)h Fv(1)14 b Fo(\024)g Ft(i)f Fo(\024)h Ft(n;)8 b(y)514 1896 y Fr(p)532 1884 y Fq(s)547 1892 y(i)577 1914 y Fo(2)14 b Ft(R)p Fj(.)243 1972 y(Then)k(ther)n(e)f(ar)n(e)g Ft(z)597 1979 y Fr(n)p Fs(+1)665 1972 y Ft(;)8 b(:)g(:)g(:)g(;)g(z)798 1979 y Fr(d)831 1972 y Fo(2)14 b Ft(R)k Fj(such)g(that)232 2059 y Ft(y)256 2066 y Fs(1)276 2059 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)409 2066 y Fr(n)433 2059 y Ft(;)g(z)478 2066 y Fr(n)p Fs(+1)546 2059 y Ft(;)g(:)g(:)g(:)f(;)h(z)678 2066 y Fr(d)715 2059 y Fj(is)17 b(a)h(r)n(e)n(gular)f(system)g(of)g(p)n(ar)n(ameters)f(of)i Ft(A;)25 b Fj(and)248 2157 y Ft(y)274 2133 y Fr(p)292 2122 y Fq(s)307 2129 y Fp(1)272 2169 y Fs(1)327 2157 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)462 2137 y Fr(p)480 2125 y Fq(s)495 2129 y(n)460 2169 y Fr(n)519 2157 y Ft(;)g(z)564 2164 y Fr(n)p Fs(+1)633 2157 y Ft(;)g(:)g(:)g(:)f(;)h(z)765 2164 y Fr(d)802 2157 y Fj(is)17 b(a)h(r)n(e)n(gular)f(system)g(of)g(p)n (ar)n(ameters)f(of)i Ft(R:)243 2234 y Fj(\(2\))h(Assume)g(in)g (addition)h(to)f(\(1\))g(that)g(the)g(r)n(e)n(gular)g(lo)n(c)n(al)g (ring)g Fv(\()p Ft(A;)8 b Fh(m)1586 2241 y Fr(A)1614 2234 y Fv(\))19 b Fj(is)193 2292 y(c)n(omplete,)h(and)f Ft(k)f Fo(\032)e Ft(A=)p Fh(m)703 2299 y Fr(A)751 2292 y Fj(is)i(a)h(sep)n(ar)n(able)f(\014eld)i(extension.)28 b(Then)20 b(ther)n(e)e(is)193 2350 y(a)f(sub\014eld)i Ft(k)d Fo(\032)e Ft(F)20 b Fo(\032)14 b Ft(R)k Fj(of)f Ft(R)h Fj(such)g(that)701 2442 y Ft(F)768 2414 y Fm(')754 2442 y Fo(\000)-32 b(!)13 b Ft(R=)p Fh(m)923 2449 y Fr(R)982 2414 y Fm(')967 2442 y Fo(\000)-32 b(!)14 b Ft(A=)p Fh(m)1137 2449 y Fr(A)1165 2442 y Ft(;)193 2529 y Fj(and)k(an)f(algebr)n(a)h (isomorphism)583 2620 y Ft(F)7 b Fv([[)p Ft(Y)678 2627 y Fs(1)696 2620 y Ft(;)h(:)g(:)g(:)g(:Y)826 2627 y Fr(n)849 2620 y Ft(;)g(Z)904 2627 y Fr(n)p Fs(+1)973 2620 y Ft(;)g(:)g(:)g(:)f (;)h(Z)1115 2627 y Fr(d)1136 2620 y Fv(]])1191 2592 y Fm(')1177 2620 y Fo(\000)-32 b(!)13 b Ft(A;)193 2707 y Fj(inducing)19 b(an)f(algebr)n(a)g(isomorphism)537 2799 y Ft(F)7 b Fv([[)p Ft(Y)642 2775 y Fr(p)660 2764 y Fq(s)675 2771 y Fp(1)632 2811 y Fs(1)696 2799 y Ft(;)h(:)g(:)g(:)f (:Y)836 2778 y Fr(p)854 2767 y Fq(s)869 2771 y(n)825 2811 y Fr(n)894 2799 y Ft(;)h(Z)949 2806 y Fr(n)p Fs(+1)1017 2799 y Ft(;)g(:)g(:)g(:)g(;)g(Z)1160 2806 y Fr(d)1180 2799 y Fv(]])1236 2771 y Fm(')1221 2799 y Fo(\000)-32 b(!)14 b Ft(R:)p eop %%Page: 14 14 14 13 bop 193 131 a Fk(14)163 b(GAET)m(ANA)16 b(RESTUCCIA)g(AND)h (HANS-J)1104 122 y(\177)1099 131 y(UR)o(GEN)f(SCHNEIDER)193 217 y Fj(Pr)n(o)n(of.)j Fv(By)13 b(Theorem)f(4.1,)i Ft(R)g Fo(\032)f Ft(A)g Fv(is)h Ft(H)t Fv(-Galois,)g(in)f(particular)g Ft(A)g Fv(is)g(a)h(\014nitely)193 275 y(generated)f(free)g Ft(R)p Fv(-mo)q(dule.)20 b(Hence)12 b Ft(R)h Fv(is)h(lo)q(cal,)f(no)q (etherian)h(and)f(regular)h(\(see)193 333 y([Ma,)h(23.7]\).)243 391 y(Assume)i(\(1\).)30 b(F)l(ollo)o(wing)19 b(the)g(argumen)o(t)e(in) i([RM)o(,)h(Theorem)d(5],)i(w)o(e)g(\014rst)193 449 y(note)j(that)g (the)f(residue)g(classes)h(of)g Ft(y)924 456 y Fs(1)943 449 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)1077 456 y Fr(n)1122 449 y Fv(in)21 b Fh(m)1222 456 y Fr(A)1251 449 y Ft(=)p Fh(m)1313 431 y Fs(2)1313 462 y Fr(A)1364 449 y Fv(are)g(linearly)f (in-)193 507 y(dep)q(enden)o(t)f(o)o(v)o(er)f Ft(A=)p Fh(m)638 514 y Fr(A)666 507 y Fv(.)30 b(Supp)q(ose)905 470 y Fn(P)957 522 y Fs(1)p Fm(\024)p Fr(j)r Fm(\024)p Fr(n)1078 507 y Ft(a)1104 514 y Fr(j)1122 507 y Ft(y)1146 514 y Fr(j)1182 507 y Fo(2)19 b Fh(m)1272 489 y Fs(2)1272 520 y Fr(A)1320 507 y Fv(for)h(some)e Ft(a)1549 514 y Fr(j)1585 507 y Fo(2)h Ft(A)p Fv(.)193 567 y(By)e(applying)g(the)g (deriv)m(ations)h Ft(@)832 574 y Fr(i)863 567 y Fv(of)g(the)f(pro)q(of) h(of)g(Lemma)d(3.4)j(w)o(e)f(see)g(that)193 625 y Ft(a)219 632 y Fr(i)246 625 y Fo(2)e Fh(m)332 632 y Fr(A)376 625 y Fv(for)i(all)f Ft(i)p Fv(.)243 686 y(Th)o(us)j(b)o(y)f(4.5)h Ft(R=)p Fv(\()p Ft(y)626 663 y Fr(p)644 651 y Fq(s)659 658 y Fp(1)624 698 y Fs(1)680 686 y Ft(;)8 b(:)g(:)g(:)f(;)h(y)815 668 y Fr(p)833 656 y Fq(s)848 660 y(n)813 699 y Fr(n)872 686 y Fv(\))924 658 y Fm(')909 686 y Fo(\000)-32 b(!)18 b Ft(A=)p Fv(\()p Ft(y)1088 693 y Fs(1)1107 686 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)1241 693 y Fr(n)1264 686 y Fv(\))18 b(is)h(regular,)g(and)g(w)o(e)193 749 y(can)g(extend)e(the)i Ft(y)558 725 y Fr(p)576 714 y Fq(s)591 721 y(i)556 762 y Fr(i)625 749 y Fv(b)o(y)f Ft(z)718 756 y Fr(n)p Fs(+1)787 749 y Ft(;)8 b(:)g(:)g(:)f(;)h(z)919 756 y Fr(d)957 749 y Fv(to)19 b(a)g(regular)g(system)e(of)h(parameters)193 807 y(of)e Ft(R)p Fv(.)21 b(Then)16 b Ft(y)471 814 y Fs(1)490 807 y Ft(;)8 b(:)g(:)g(:)g(;)g(y)624 814 y Fr(n)647 807 y Ft(;)g(z)692 814 y Fr(n)p Fs(+1)760 807 y Ft(;)g(:)g(:)g(:)f(;)h (z)892 814 y Fr(d)928 807 y Fv(is)15 b(a)h(regular)g(system)e(of)i (parameters)e(of)193 865 y Ft(A)i Fv(since)f Ft(A)h Fv(and)h Ft(R)g Fv(ha)o(v)o(e)e(the)h(same)f(residue)h(\014eld)g(b)o(y)f(4.4.) 243 923 y(T)l(o)g(pro)o(v)o(e)e(\(2\),)i(w)o(e)e(note)i(that)f Ft(R)h Fv(is)f(also)h(complete)d(lo)q(cal)i(since)g Ft(A)g Fv(is)g(\014nitely)193 981 y(generated)j(and)h(free)e(o)o(v)o(er)g Ft(A)h Fv(\(The)g Fh(m)923 988 y Fr(R)952 981 y Fv(-adic)g(top)q(ology) h(of)f Ft(A)g Fv(is)g(the)g Fh(m)1556 988 y Fr(A)1584 981 y Fv(-adic)193 1047 y(top)q(ology)l(,)i(since)f Ft(A=)p Fh(m)626 1054 y Fr(R)655 1047 y Ft(A)g Fv(is)g(artinian.)27 b(Hence)17 b Ft(R)h Fo(\032)1237 1034 y Fn(b)1228 1047 y Ft(R)g Fo(\032)f Ft(A)p Fv(,)g(and)i Ft(A)f Fv(is)g(free)193 1112 y(o)o(v)o(er)d Ft(R)i Fv(and)455 1099 y Fn(b)446 1112 y Ft(R)g Fv(of)f(the)g(same)f(rank.)22 b(Th)o(us)16 b Ft(R)f Fv(=)1129 1099 y Fn(b)1120 1112 y Ft(R)h Fv(b)o(y)g(Nak)m(a)o (y)o(ama\).)243 1178 y(Since)g Ft(R=)p Fh(m)470 1185 y Fr(R)531 1149 y Fm(')516 1178 y Fo(\000)-32 b(!)16 b Ft(A=)p Fh(m)688 1185 y Fr(A)735 1178 y Fv(is)h(separable)h(o)o(v)o (er)e Ft(k)k Fv(there)d(is)g(a)h(co)q(e\016cien)o(t)e(\014eld)193 1236 y Ft(k)29 b Fo(\032)e Ft(F)34 b Fo(\032)27 b Ft(R)e Fv(b)o(y)f([Ma)o(,)i(28.3].)45 b(The)25 b(claim)d(follo)o(ws)i(from)f (\(1\))h(and)h([Ma,)193 1294 y(29.7].)p 1652 1294 2 33 v 1654 1263 30 2 v 1654 1294 V 1683 1294 2 33 v 795 1479 a Fu(References)193 1558 y Fw([DG])19 b(M.)13 b(Demazure,)h(P)m(.)f (Gabriel,)f(Group)q(es)j(Alg)o(\023)-20 b(ebriques)14 b(I,)f(North-Holland,)g(1970.)193 1607 y([L])35 b(J.)12 b(Lipman,)e(F)m(ree)j(deriv)n(ations)e(mo)q(dules)g(on)h(algebraic)g(v) n(arieties,)g Fl(A)o(mer.)g(J.)g(Math.)h Fd(87)278 1657 y Fw(\(1965\),)g(874{898.)193 1707 y([Ma])20 b(H.)c(Matsum)o(ura,)g (Comm)o(utati)o(v)o(e)e(Ring)i(Theory)m(,)h(Cam)o(bridge)e(studies)j (in)e(adv)n(anced)278 1757 y(mathematics)c(8,)h(Cam)o(bridge)f(Univ)o (ersit)o(y)i(Press,)h(1986.)193 1807 y([M])23 b(S.)11 b(Mon)o(tgomery)m(,)d(Hopf)j(algebras)f(and)h(their)g(actions)g(on)f (rings,)h(CBMS)g(Lecture)i(Notes)278 1856 y(82,)g(Amer.)g(Math.)g(So)q (c.,)h(1993.)193 1906 y([MS])20 b(S.)13 b(Mon)o(tgomery)g(and)h(H.-J.)f (Sc)o(hneider,)i(Prime)e(ideals)g(in)h(Hopf)f(Galois)g(extensions,)278 1956 y Fl(Isr)n(ael)h(J.)h(Math.)f Fd(112)f Fw(\(1999\),)g(187{235.)193 2006 y([RM])19 b(G.)c(Restuccia)i(and)e(H.)h(Matsum)o(ura,)e(In)o (tegrable)i(deriv)n(ations)f(I)q(I,)h(A)o(tti)f(Accademia)278 2056 y(P)o(eloritana)e(dei)h(P)o(ericolan)o(ti)g(Classe)g(Sc,)g(Fis.)f (Mat.)g(Natur.)h(LXX)g(\(1992\),)f(153{172.)193 2105 y([R)o(U])19 b(G.)c(Restuccia)h(and)f(R.)g(Utano,)g Fc(n)p Fw(-dimensional)e(actions)i(of)g(\014nite)h(ab)q(elian)e(Hopf)h(al-)278 2155 y(gebras)i(in)e(c)o(haracteristic)i Fc(p)e(>)g Fw(0,)h Fl(R)n(ev.)h(R)n(oumaine)g(Math.)g(Pur)n(es)g(Appl.)e Fd(43)h Fw(\(1998\),)278 2205 y(881{895.)193 2255 y([R)m(T])j(G.)c (Restuccia)i(and)f(A.)g(T)o(yc,)g(Regularit)o(y)f(of)h(the)g(ring)g(of) g(in)o(v)n(arian)o(ts)f(under)i(certain)278 2305 y(actions)10 b(of)f(\014nite)g(ab)q(elian)g(Hopf)g(algebras)g(in)g(c)o (haracteristic)i Fc(p)p Fw(,)f Fl(J.)g(A)o(lgebr)n(a)f Fd(159)g Fw(\(1993\),)278 2355 y(347{357.)193 2404 y([S])38 b(H.-J.)19 b(Sc)o(hneider,)j(Principal)c(homogeneous)h(spaces)i(for)e (arbitrary)g(Hopf)g(algebras,)278 2454 y Fl(Isr)n(ael)14 b(J.)h(Math.)f Fd(72)f Fw(\(1990\),)g(167{195.)193 2504 y([W])19 b(W.)d(C.)f(W)m(aterhouse,)i(In)o(tro)q(duction)f(to)g (a\016ne)g(group)g(sc)o(hemes,)h(Graduate)f(T)m(exts)h(in)278 2554 y(Mathematics)c(66,)g(Springer,)h(1979.)243 2712 y Fb(Universit)446 2709 y(\022)445 2712 y(a)j(di)g(Messina,)i(Dip)m(ar) m(timento)f(di)e(Ma)m(tema)m(tica,,)j(Contrad)o(a)f(P)l(a-)193 2762 y(p)m(ardo,)f(salit)m(a)f(Sper)o(one)i(31,,)e(98166)f(Sant'A)o(ga) m(t)m(a)i(\(ME\),)f(It)m(al)m(y)243 2811 y Fl(E-mail)e(addr)n(ess)s Fw(:)19 b Fa(grest@dipmat.un)o(ime.i)o(t)p eop %%Page: 15 15 15 14 bop 438 122 a Fk(ON)17 b(A)o(CTIONS)e(OF)i(INFINITESIMAL)f(GR)o (OUP)g(SCHEMES)208 b(15)243 217 y Fb(Ma)m(thema)m(tisches)24 b(Institut,)i(Universit)1021 214 y(\177)1020 217 y(at)d(M)1135 214 y(\177)1134 217 y(unchen,)28 b(Theresienstra\031e)193 267 y(39,)16 b(D-80333)f(M)490 264 y(\177)489 267 y(unchen,)j(Germany) 243 317 y Fl(E-mail)c(addr)n(ess)s Fw(:)19 b Fa(hanssch@rz.math)o (emati)o(k.uni)o(-muen)o(chen.)o(de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF