Inductive Theorems and the Structure of Projective Modules over Crossed Group Rings

Transcription

1 saqartvelos mecnerebata erovnul akadems moambe, t,, 8 BULLETIN OF THE GEOGIAN NATIONAL ACADEMY OF CIENCE, vol, no, 8 Mathematcs Inductve Theorems and the tructure of Projectve Modules over Crossed Group ngs Gorg akvashvl Ila tate Unversty, Tbls, Georga (Presented by Academy Member Hvedr Inassardze) ABTACT It s proved that the Grothendeck functor and the wan-gersten hgher algebrac K- functors of a crossed group rng [,, ] are Frobenus modules As the corollares an nducton theorem for ths functors and a reducton theorem for fntely generated [,, ] -projectve modules (f s a dscrete normalzaton rng) are proved Under some restrctons on n ( :) t s shown that fntely generated [,, ] -projectve modules are decomposed nto the drect sum of left deals of the rng [,, ] More stronger results are proved, when d 8 Bull Georg Natl Acad c Key words: crossed group rng, Frobenus modules, nducton theorem, algebrac K-functors In 96 G wan proved [] that a Grothendeck functor G ( [ ]) of a group rng [ ] s a Frobenus functor As a consequence, he proved that for a Dedeknd doman of characterstc and a fnte group any fntely generated projectve [ ]-module decomposes as a drect sum of left deals of [ ], f no prme dvder of s nvertble n He also proved that ths drect sum may be replaced by the drect sum of a free [ ] module and an deal of [ ] wan's results were based on two theorems havng an ndependent value: on the nducton theorem for the functors G ( ) and K ( [ ]) and on the "reducton" theorem In 968 TY Lam [] proved that K ( [ ]) functor s a Frobenus module over G ( [ ]) and that an nducton theorem s vald for K ( [ ]) In 973 AI Nemytov [3] proved that wan-gersten algebrac K-functors Kn( [ ]), n are Frobenus modules over G ( [ ]) and nducton theorems are vald for these functors ([4, 5]) Inducton theorems for some knds of algebrac K- functors of group rngs were obtaned n 986 by K Kawakubo [6] and n 5 [7] by A Bartels and W Luck [7] In ths paper we prove (Theorem) that wan-gersten algebrac K-functors Km( [,, ]) are Frobenus modules and generalze an nducton theorem for ths functors (Theorem ), where [,, ] s a crossed group rng Wth the help of nducton theorem for K ( [,, ]) a "reducton" theorem for fntely generated projectve [,, ] -modules s proved, f s a dscrete valuaton rng (Theorem 3) 8 Bull Georg Natl Acad c

2 Inductve Theorems and the tructure of Projectve Modules over Crossed Group ngs 7 and the theorems on the structure of fntely generated projectve [,, ] - and [, ] -modules are obtaned, whch generalze the above mentoned wan's theorems Let be a commutatve rng wth dentty, a group, : Aut a morphsm of groups, U ( ) a set of nvertble elements of and : U ( ) such a mappng that (, y) ( y, z) ( y, z) (, yz) Then a crossed group rng [,, ] ([8, 9]) s a free -module wth the set of free generates and wth multplcaton r r rr (, ), where s an mage of va a mappng [,, ] and r, r If ( ) d and ~ (e (, y) ( ) ( y) ( y), : U ( )), then [,, ] [ ] Further all modules are the left modules, M( A ) and PA ( ) denotes respectvely categores of fntely generated A-modules and fntely generated projectve A-modules; M ( [,, ]) s a category of fntely generated -projectve [,, ] -modules; ( [,, ]) G s a Grothendeck group of the category M ( [,, ]) and wll be always a fnte group Man results of the paper are Theorems 6 and 7 These theorems were proved by the author n [-] The partcular case when ~ was announced n [] and ts proof was a subject of the author s doctoral thess n 98 Ths theorems are smlar to the results of Kawakubo [6], whch were obtaned later n 986 for some knds of algebrac K-functors of group rngs and partcular cases of crossed group rngs Let G be a category, ngs - a category of rngs, G : G ngs - a contravarant functor, G( ) : G( ) G( ) If to each morphsm : n G corresponds such a morphsm : G( ) G( ) n ngs that Id Id and ( j) j (whenever j makes sense), then a functor G s called a Frobenus functor [] f t satsfes the Frobenus recprocty formula ( a b) a b Let Ab be a category of commutatve groups A contravarant functor K : G Ab s called a Frobenus module [] on the Frobenus Functor G f t satsfes the followng condtons: () K( ) s a module over G( ); () For each morphsm of groups : a morphsm : K( ) K( ) ests (whenever j makes sense) that ( j) j ; () ( y ( a)) ( y) a, ( () b) (b) Here K(), G( ), y G( ), a K( ), b G( ) Let G( ) denote a category whose objects are all subgroups and morphsms monomorphsms : Then the functors G ( [ ]) and K ( [,, ]) are contravarant functors from the category G( ) to the categores ngs and Ab, respectvely It s known [] that G ( [ ]) s a Frobenus functor Let us denote { r ( ) r r} Theorem Let be an algebra over a commutatve rng wth dentty Then G ( [,, ]) and ( [,, ]) Km, m,, functors are Frobenus modules on the Frobenus functor G ( [ ]) If s an algebra over, then s a -algebra Let us construct a morphsm of rngs : [,, ] [ ] [,, ], ( r) r Then for any [ ]-module M and [,, ] -module P the module M P s a [,, ] -module by the acton r( m p) ( r)( m p) m rp Let us denote such a module by M P Proposton If [ ]-module M s -projectve and [,, ] -module P s [,, ] - projectve, then M P s [,, ] -projectve Proposton Let be an algebra over, - a subgroup, M Mod, M Mod, P [,, ] Mod and P [,, ] Mod Then [,, ] M P ( [,, ] M ) P ' ' [,, ] [,, ] m Bull Georg Natl Acad c, vol, no, 8

3 8 Gorg akvashvl as [,, ] -modules Theorem follows from Propostons and and results from [3] Let M be some famly of objects from G Let us denote for G K ( M ) {Im( : K( ) K( )) :, M}, Let A B be abelan groups and n an nde of A n B, e nb A From Theorem follows the nducton theorem: Theorem Let c( ) be a set of all cyclc subgroups of group Then Km( [,, ]) c( ) and G ( [,, ]) c( ) have an nde n n m( [,, ]) K and G ( [,, ]) respectvely for all m If s an algebra over a feld, then n may be replaced by n From the theorems above follows the reducton theorem for [,, ] -projectve modules: Theorem 3 Let be a dscrete valued rng wth the quotent feld K; P, Q P( [,, ]) and K P K Q as K[,, ] -modules Then P Q as [,, ] -modules emark K[,, ] acts on K P as ( p) p Ths theorem was proved by wan [] for group rngs To prove Theorem 3 t suffces to prove the followng Theorem 4 Let k be a feld Then Cartan homomorphs : K( k[,, ]) G( k[,, ]) s njectve Theorem 4 tself reduces to the case when the group s cyclc; for cyclc groups Theorem 4 follows from Theorem 5 Let A be a (noncommutatve) prncpal deal doman, n whch each deal s bounded Let I A two sded deal, K ( A / I ) and G ( A/ I ) - Grothendeck groups of the categores P( A / I ) and M( A / I ) respectvely Then Cartan homomorphsm : K( A/ I) G( A/ I) s njectve Now we can study the projectve [,, ] -modules f s a Dedeknd doman Let Ker( : Aut( )) If ( ) d, we denote [,, ]: [, ] Theorem 6 Let a Dedeknd doman char uppose no one prme dvder of n s nvertble n and () s -projectve; () f p spec( ), p ( n), then ( )( p) p; () f p s a prme dvder of the number n, p p spec( ) and p s a ylov p- subgroup of, then p acts trvally on / p ; (v) ( ) Then any fntely generated projectve [,, ] -module splts n drect sum of left deals of the rng [,, ] In a partcular case when ( ) d, we may prove a stronger result Theorem 7 Let be a Dedeknd doman char If no one prme dvder of n ( :) s nvertble n, then any fntely generated projectve [, ] -module s the drect sum of a free [, ] -module and a left deal I [, ] For any non-zero deal j an deal I can be chosen n such a way that I and j would be coprme deals Proof of Theorem 6 It est such an mbeddng of a module P n a free [,, ] -module F that P : F n, ( P : F) n Let a, a,, a k be [,, ] -bass of F Let us consder a morphsm of [,, ] -modules : F [,, ], a nce rf P r [,, ] I, thus ( P : F) ( I : [,, ]) Therefore from ( P : F) ( n) follows that ( I : [,, ]) ( n) Then the deal I s [,, ] -projectve : P I s surjectve, therefore P P I The theorem s easy to prove by mathematcal nducton on rkk P Proof of Theorem 7 Under the condtons of Theorem 7 any module P s somorphc to a drect sum I of deals of [, ] ; n addton for any nonzero deal J the deals I can be chosen n such a Bull Georg Natl Acad c, vol, no, 8

4 Inductve Theorems and the tructure of Projectve Modules over Crossed Group ngs 9 way that ( I : [, ]) J for all We may suppose K I K[, ] Then t s suffcent to prove the followng: let I, I [, ] be such a projectve deals that ( I : [, ]) and ( I : [, ]) are coprme to J and K I K I K[, ] ; then I I [, ] I, where I [, ] s a left deal and ( I : [, ]) J Let J ( I: [, ]) It est I [, ] such that I I and ( I : [, ]) JJ Let us replace I wth I Therefore, we may assume that there est b ( I: [, ]) and b ( I : [, ]) such that b b Let F be the free [, ] -module wth two free generators e, e and V Ie Ie F Then A I I and ( V : F) J It s clear that the elements e eb eb and e e e are also free generators of F, because e e be, e e be But e V because b I, b I Consequently V [, ] e Ie where I { a [, ] re V} It s clear also that ( I : [, ]) J because ( I : [, ]) ( V : F) მათემატიკა ინდუქციური თეორემები და ჯვარედინ ჯგუფურ რგოლზე პროექციული მოდულების სტრუქტურა გ რაქვიაშვილი ილიას სახელმწიფო უნივერსიტეტი, თბილისი, საქართველო (წარმოდგენილია აკადემიის წევრის ხ ინასარიძის მიერ) დამტკიცებულია, რომ [,, ] ჯვარედინი ჯგუფური რგოლის გროტენდიკის ფუნქტორი და სუონ-გერსტენის უმაღლესი K-ფუნქტორები ფრობენიუსის მოდულებია შედეგად დამტკიცებულია ამ ფუნქტორებისათვის ინდუქციურობის და რედუქციულობის თეორემები სასრულად წარმოქმნილი [,, ] -პროექციული მოდულებისათვის, როცა არის დისკრეტულად ნორმირებული რგოლი როდესაც არის დედეკინდის რგოლი, ჯგუფის რიგზე გარკვეული შეზღუდვებით დამტკი- ცებულია, რომ სასრულად წარმოქმნილი [,, ] -პროექციული მოდულები იშლება [,, ] რგოლის მარცხენა იდეალების პირდაპირ ჯამად უფრო ძლიერი შედეგებია მიღებული, როდესაც d Bull Georg Natl Acad c, vol, no, 8

5 Gorg akvashvl EFEENCE wan G (96) Induced representatons and projectve modules Ann of Math, 7: Lam T Y (968) Inducton theorems for Grothendeck groups and Whtehead groups of fnte groups Ann cent Ec Norm uper, : Nemytov A I (973) Kn( ) kak frobenusovy modul nad funktorom, G ( ) Uspekh Matematcheskkh Nauk, 8:87-88 (n ussan) 4 wan G (97) Nonabelan homologcal algebra and K-theory Proc ymp Pure Math (AM), 7: Gersten M (97) On the functor K J of Algebra, 7:-37 6 Kawakubo K (986) Inducton theorems for equvarant K-theory and J-theory J Math oc Japan, 38: Bartels A, Luck W (7) Inducton theorems and somorphsm conjectures for K- and L-Theory Forum Mathematcum, 9:-8 8 Jacobson N (943) The theory of rngs, New-York 9 Auslander M, Goldman O (96) The Brauer group of a commutatve rng Trans AM 97: akvashvl G (979) Obobshchene teoremy Artna dla poluprostykh algebr nduktsonnye teoremy dla poradkov skreshchennykh gruppovykh kolets Bull of the Acad cences of the Georgan 96:5-8 (n ussan) akvashvl G (98) O stroen proektvnykh module nad skreshchennym gruppovym kol tsam Bull of the Acad cences of the Georgan : (n ussan) akvashvl G (98) Induktsonnye teoremy proektvnye modul nad skreshchennym gruppovym Kol tsam Trudy Tblsskogo Matematcheskogo Insttuta A M azmadze Cbornk rabot po algebre, 3:9-7 (n ussan) 3 Qullen D (97) Hgher algebrac K-theory, I Lect Notes n Math 34:85-47 eceved December, 7 Bull Georg Natl Acad c, vol, no, 8