A lifting theorem for modular representations

A liftin g th e o re m fo r m o d u la r r e p r e s e n ta tio n s B y J . A . Gr e e n T h e U n iv e r s ity , M a n c h e s te r ( C o m m u nica ted b y M . H . A .N e w m a n , F .R .8 . R e c e iv e d Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 I k is the residue-class field o/p of a ring 0 of p-adic integers. Sufficient conditions are found, n representation p foa group O over k m ay be lifted to a representation r o f O over I o, particularly in the case where it is assumed that such a lifting exists for the restriction of p to a given subgroup H of G. The conditions involve certain homological invariants of p . jl-1 . L e t 1. In t r o d u c t io n Oeb a g r o u p , o a c o m m u ta tiv e r in g w ith i d e n t i t y e le m e n t, a n d U a (fee le f t o -m o d u le . A r e p r e s e n t a t i o n o f G o n is a m a p p in g of in t o t h e r in g U) c o -e n d o m o rp h is m s o f U , s u c h t h a t r(x )r(y) = r(xy) Id ( x , y € .G ) r ( l ) = t, (1*1) (1*2) shere 1 is t h e i d e n t i t y e le m e n t o f G ,and i is t h e id e n tic a l m |e in f a c t o - a u to m o r p h is m s o f U , f o r b y (1*1) a n d (1*2), r ( x ) h a s t h e tw o - s id e d Iv e rs e r ( x ~ x) e(x G ). I B y a n (o, G )-m odulesi h e r e m e a n t a p a i r w h e re is a f r e e le f t o -m o d u le ijid rsi a r e p r e s e n t a t i o n o f G o n U . I t is c u s to m a r y t o t h i n k o f t h e e le m e n ts o f G * o p e r a to r s o n U , a n d t o w r ite ux= .ur{ x ) (u U , a is n o t a t i o n s u p p re s s e s e x p lic it r e fe re n c e t o r , w h ic h m u s t b e in f e r r e d f r o m t h e jp n te x t. S im ila r ly , ( U,r) i ts e l f m a y b e d e s c r ib e d in f o r m a lly a s t h e (o, G )-m o 1 1-2. I n t h e r e s t o f t h e p a p e r o w ill b e t h e r in g o f P - a d i c in te g e r s , w h e r e is a ti m e id e a l in so m e a lg e b r a ic n u m b e r fie ld . W r ite p = f o r t h e u n iq u e m a x im a l l e a l o f o, t h e n « Cl P n = 71=0 (0 ) a id o is c o m p le te in t h e m e tr ic to p o lo g y d e fin e d b y t a k i n g t h e id e a ls p w a s a b a s is f t h e n e ig h b o u rh o o d s o f z e ro in o. L e t k d e n o te t h e re s id u e - c la s s fie ld o /p . | I f U is a fr e e le f t o -m o d u le , t h e n A = U f p U c a n b e r e g a r d e d a s a fc-space (le ft ■-m odule) b y t h e n a t u r a l r u l e : i f u — u + pa n d i . u = d u = a u + p U . C o n v e rse ly , a n y g iv e n & -space A is is o m o rp h ic w i t h t h e Q u o tien t U j p U , w h e re U is a fre e le f t o -m o d u le s u c h t h a t r a n k U = d im A . T h e r e lu ll b e n o lo ss o f g e n e r a lity , th e r e f o r e , i f t h e fc-space A w h ic h a p p e a r s in t h e s e q u e l is id e n tifie d w ith U / p U . I T h e rin g E ( U ) o f o -e n d o m o rp h is m s o f is its e l f a n o -m o d u le , d e fin in g a£ , f o r ;iv en a £ o, E, e E ( U ) , t o b e t h e m a p p in g u -> {a u ) £ € U ) . T h e s e ts p nE { U ) = n nE ( U ) [ 135 ] = a 136 J . A. Green a re id e a ls o f E ( TJ) ( n = 0 ,1 , 2 , . . . ) . W r ite £ = £ + p E { U ) (£ e E { U ) a u n iq u e fc-en d o m o rp h ism (also d e n o te d b y f ) o f th e & -space A = U /p U , b y uE, = uE,U ), a n d c o n v erse ly , e v e r y ^ -e n d o m o rp h ism o f A h a s th e fo rm £ fo r so m e £ e E ( U ) . Ii th is w a y th e rin g E ( A ) o f ^ -e n d o m o rp h ism s o f A is id e n tifie d w ith th e q u o tie n t ring Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 E (U )lp E (U ). 1*3. I f r: G -> E (U )si a r e p r e s e n ta tio n o f G o n U , i t is c le a r t h a t t r : G -> E ( A) d e fin e d b y r ( x ) — r (x ) ( x € (? )is a r e p r e s e n ta p r o b le m ’ is th e c o n v e rse : g iv e n a &-space A ( = U / p U ) . a n d a re p re s e n ta tio n p \ G -> E ( A) o f G o n A , to fin d a re p r e s e n ta tio n o f on such th a t — th such th at H x)= p {x) (lJ I f th is c a n b e d o n e, i t w ill b e s a id t h a t p , o r e q u a lly th e (?)-m odule A = (A, c a n b e l i f t e d to o, a n d t h a t r is a l i f t i n g o f p . j R . B ra u e r h a s p ro v e d th e fo llo w in g f u n d a m e n ta l liftin g th e o re m fo r a fin ite g ro u p G (see A rtin , N e s b itt & T h r a ll 1948, p . 115). L e t V b e th e g ro u p a lg e b ra o f G o v e r k , re g a rd e d as (k , (?)-m odu le in th e u s u a l w a y , i.e. a s r ig h t r e g u la r repre-! s e n ta tio n m o d u le fo r G. T h e n a n y d ir e c t s u m m a n d A c a n be li f t e d to 0. T h e d ire c t s u m m a n d s A o f T in th is sense ( th a t is, th e d ire c t r ig h t id e a l su m m an d s! o f T) a re p ro je c tiv e (k, (?)-m o d u les, in th e sen se o f h o m o lo g ical a lg e b ra . I t follows! t h a t th e h o m o lo g y g ro u p s e x t n (A, M) (C a rta n & E ile n b e rg 1956, see a lso §2) a re zero, fo r a ll n >0 a n d a ll (lc, (?)-m o d u les M. T h e fo llo w in g th e o re m , w h ic h i in § 3*3, is th e re fo re a g e n e ra liz a tio n o f B r a u e r ’s. Th e o r e m 1. T h e n A c a n be lifte d to 0. L e t Gbe a n y g r o u p , a n d A a n y (k, G f m o d u l e s u c h th a t e x t2 ( T h e p u rp o s e o f th is p a p e r is to o b ta in re s u lts o f a s im ila r k in d fo r th e ‘r e la tiv e ’ liftin g p ro b le m . L e t Heb a s u b g ro u p o f r e s tric tio n o f pot H , a n d A H th e (k, 77)-m odule ( A , p H ). S u p p o se t h a t p H lif te d to a re p re s e n ta tio n r H o f H o n U . T h e n th e p ro b le m is to e x te n d r H to a liftin g r o f G o n U , in o th e r w o rd s, to fin d a r e p re s e n ta tio n r o f G o n U w h ic h sh a ll s a tisfy , in a d d itio n to (1-3), also * ,, r(h ) = r H (h) (1.4) S om e su fficien t c o n d itio n s fo r th is a re g iv e n in th e o re m 2 (§ 3-5). T h e fo llo w in g ! c o ro lla ry to th e o re m 2 m a y h a v e so m e in te re s t. L e t k be a f i e l d o f c h a ra c te ristic p , a n d G a f i n i t e g r o u p s u c h t h a t e v e r y p a i r o f d i s t i n c t S y l o w p - s u b g r o u p s o f G m e e t i n {1}. L e t H be the n o r m a l i z e r i n p - s u b g r o u p Gp o f G. T h e n a {k, G )-m o d u le A c a n be lifte d to 0 i f a n d o n l y l ifte d to 0. fix e d S y lo w c a n be T h is re s u lt h o ld s, in p a rtic u la r, w h e n th e o rd e r o f G is d iv isib le b y p to th e first p o w e r o n ly . 2. R e l a t i v e h o mo l o g y g r o u ps 2-1. I t w ill b e u sefu l to re v ie w h e re b rie fly c e rta in h o m o lo g ical in v a r ia n ts o f (k , (?)-m odules, w h ic h w ill e n te r in to th e s u b s e q u e n t w o rk . F o r a fu ll d iscu ssio n o f th e se , see H o c h sc h ild 1956. A lifting theorem for modular representations 137 L e t A , M b e a n y tw o ( k , ( ^ - m o d u le s , a n d l e t - h o m &(A , M ) d e n o te t h e s p a c e f a l l lin e a r m a p p in g s o f A in to M , th e s e b e in g r e g a r d e d a s ^ -s p a c e s s im p ly . F o r a n y e F , x e G d e fin e x f a n d fxeF t o b e t h e m a p p in g s A -> (A x)f an d A -> (A /) x (A e A ), e sp e c tiv e ly . I n t h is w a y F b e c o m e s a tw o - s id e d (7 -m o d u le. A n n - c o c h a i n ( n > 0) fo r t h e p a i r (A, M ) is a f u n c tio n / w i t h n a r g u m e n ts ro m G, w h o se v a lu e s f ( x l i . . . , x n ) lie in condition .. . , x n) = 0 i f a n y x t = 1. F na d w h ic h (2*1) Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 fh e c o b o u n d a r y d f o f / is t h e ( n + l) - c o c h a in g iv e n b y n d f ( x 1, . . . , x n+1) = x j ( x i , . . . , x n ) + 2 ( - l 1=1 + ( - l ) n +1f ( x 1, . . . , x n ) x n+1 («!, (2*2) (A 0 -c o c h a in is s im p ly a ‘c o n s t a n t ’ / e F , and d = xf N o w le t Heb a s u b g r o u p o f G . A n n -c o c h a in r e l a t i v e to H is a n n - c o c h a i n / s u c h t h a t h f { x x, . . . , x n ) f{ x i , ..., x ^ _ x h , x $ ,.. . , = xn ) (2*3) fix>i " • y X n h ) — f(x-±, . . . , x n ) h , an d for e d l h e H , x 1, . . . , x n e G a n d i = 2 , . . . , n . T h e s e t o f a ll r e la tiv d e n o te d b y <7^(A ,M ), t h e s e t o f a ll r e la tiv e n -c o c y c le s , t h a t is, r e la tiv e n - c o c h a in s / su c h t h a t d f = 0, b y Z % { A>M ), a n d t h e s e t o f r e la tiv e n - c o b o u n d a rie e le m e n ts o f d C jr * ( A , M ), b y B % { A, M ) (in t h e c a se n = 0, t h is l a s t g to b e zero ). T h e n B % {A , M) < A , M ) ^ C%{ A , M ), a n d th e r e la tiv e ‘e x t g ro u p ’ is d e fin e d b y f e x t ” (A ,M ) = Z & (A ,M )/J3& (A ,M ). I n p a r tic u la r i f H = {1}, t h e w o rd s ‘r e la tiv e t o H ’ a n d t h e s u b s c r ip t H w o m itte d , a n d t h e n e x t n (A, M ) is t h e u s u a l e x t g ro u p , a s d e fin e d in C a r ta n & E ile n b e rg (1 9 5 6 )4 2*2. I t is u s e fu l to n o tic e t h a t (2*1) a n d (2*3) to g e th e r im p ly f ( x v ..., x n ) = 0 if a n y (2-4) C o n v ersely , a n y n - c o c y c l e f w h i c h s a tis f ie s (2*4) i s a n n -c o c y c le r e la t iv e to H . F o r i t is g iv e n t h a t d f — 0, a n d th e n b y p u t t i n g e a c h o f x x, x n+1 in (2-2) e q u a l t o a n e le m e n t h e H , in tu r n , e q u a tio n s (2-3) a r e s e e n to fo llo w fro m (2*4). I n p a r tic u la r th e c o n d itio n fo r a 2 -co cy cle f e Z A \ ,M ) t o b e lo n g t o , x)= 0 H ). (2-5) t In H ochschild (1956, p. 255) th e cochains / are n ot norm alized, i.e. do n ot satisfy (2*1), but it is easy to show th a t this condition does n ot change th e group e x t if (A, M). % The present n otation differs slightly from th a t o f H ochschild (1956), w ho w ould denote the ‘ab solu te’ e x t groups b y ext^, and the relative groups b y e x t (jB>8), where i? =]?((?), $ = r ( ff) are th e group algebras over k o f G , H , respectively. A ,M ) 138 J. A. Green 2- 3. L e t r(6 r), T ( H ) d e n o te th e g ro u p a lg e b ra s o v e r of respectively* (k, )Gm - o d u le A w ill b e c alled H -p ro jec ( m ) , r ( H ))-p ro je c tiv e in th e sense o f H o c h sc h ild (1956). I n p a r tic u la r , a (k,G m m o d u le is { l} -p ro je ctiv e i f a n d o n ly i f i t is p ro je c tiv e in th e u s u a l sense. I n th e g e n e ra l th e o r y i t is sh o w n t h a t th e g ro u p s (A, M ) c a n b e d efin ed in| te rm s o f a ( r ( £ ) , T (II))-p ro je c tiv e re so lu tio n (H o c h sc h ild , p . 252) o f A, a n d it| follow s t h a t A 0 a n d a ll G )-m o d u le s M H ig m a n (1954? P- 371) h a s p ro v e d a th e o re m w h ic h im p lie s a t o n ce t h a t If A i s H - p r o je c tiv e , t h e n e x t£ (A, M ) = 0 Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 I f the i n d e x ( G :H ) i s f i n i t e a n d p r i m e to the c h a ra c te ristic p o f k , m o d u le i s H -p r o je c tiv e . I n p a rtic u la r, if G is fin ite a n d h a s o rd e r p rim e to p , then e v e ry ( k (r)-m , odule is p ro je c tiv e . 3. Th e l i f t i n g s e q u e n c e 3- 1. T h e to p o lo g y o n th e rin g 0 c an be tra n s fe rre d to E ( U ) , ta k in g fo r a b a s i o f n e ig h b o u rh o o d s o f zero in E ( U ) th e id e als p nE ( U T h is gives a m e tric to p o lo g y o n E ( U ) w h ic h re sp e c ts th e s tr u c tu r e o f E ( U ) as ring ! a n d o-m od ule. I t follow s fro m th e co m p le ten e ss o f 0 t h a t E ( U ) is also com plete!! in th e sense t h a t a seq u en ce (gn )o f e le m e n ts o f E ( U ) c o £ = lim £ w in E ( U ) , p ro v id e d a n y p o sitiv e in te g e r (gn )satisfies th e C au c h y co n v erg en ce c rite rio n i s ,there e x ists a n in te g e r N im - i n = 0 su ch t h a t, fo r a m o d p sE ( U ) A n im m e d ia te consequence o f th is is th e follow ing le m m a . L e mma 1. I f £ (£ €. E { U j ) i s a k - a u t o m o r p h i s m o f A, t h e n £ o f U. P ro o f. L e t an 0 ij (ey E ( U ) ) be th e in v e rse o f f . T h e n — Vi m 1 m od hence g y = i + a , fo r som e o c e p E ( U ) . B y th e co m p leten ess o f E ( U ) , th e series I 1 - a + a 8 - . , . h a s a su m crc f f l j U ),a n d clea rly (l + a )o- = t. T h e re fo re h a s a rig h t I in v e rse y<y\ sim ila rly g h a s a le ft in v erse, a n d so g is a n a u to m o rp h ism . N o ta tio n . I f g , y eE(U), w rite g m y m o d n n to m e a n g = y m o d 3-2. I n all t h a t follows, th e (k, d )-m o d u le A = (A th e s u b g ro u p of and | th e fittin g rH of p aHre a ssu m e d g iv en , as in § 1-3. I t is re q u ir s e n ta tio n roiG onU w h ic h satisfies (1-3) a n d (1-4). O ne w a y o f a tte m p tin g th is i s l b y c o n stru c tin g a c o n v erg en t seq uence o f a p p ro x im a tio n s to r, as follow s L e t n I be a n y p o sitiv e in te g e r, th e n a m a p p in g -> w ill b e called a n n t h a p p r o x i M m a t i o n (w ith re sp e c t to p a n d r H ) if ( x € G), r =P{X) rn{x ) r n (y) = rn (x y ) and rrSJl ) = r h (^) m od 7 Tn( (h € (3 -lJ x, y e(?) (3-2.) ( 3 '3 J 139 A lifting theorem for modular representations p (3 -ln ) a n d le m m a 1, r n ( x ) is a n a u to m o r p h is m o f f o r a ll mation r x c a n a lw a y s b e f o u n d : t a k e r x(h) = r H (h ) f o r A f ir s t a p p r o x i­ a n d fo r It r x(x) b e a n y e le m e n t o f E ( U ) s u c h t h a t r x{x) = p , l i f t i n g s e q u e n c e is a s e q u e n c e {rn} w h e re r n is a n n t h a p p r o x im a tio n a n d r n (x m ) °d rn+l(x) = € f-.. n — l ? 2 , . . . . E v i d e n t l y t h e e x is te n c e o f a lif tin g s e q u e n c e im p lie s t h e e x is te n c e « a m a p p in g r = l i m r n o f G in to E {X J ), f o r b y (3-4) t h e s e q u e n c e {rn (x )} is C a u c h y e n v e rg e n t a n d th e r e f o r e r ( x ) = l i m r n (a;) e x is ts f o r e a c h F r o m (3*1), (3-2) g d (3-3) fo llo w , r e s p e c tiv e ly , (1*3), (1-1) a n d (1-4), w h ic h s h o w t h a t r h a s t h e Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 g o p erties r e q u ir e d . i 3-3. L e t r n \ -G> )U (E be a n y m a p p in g w h ic h s a tisfie s ( 3 - ln ), (3-2n ) a n d r B ? r n (y ) = rn (x) r n (x y ) + & so m e 4>{x,y) e E ( U ) . S in c e r n ( l ) = | e f u n c tio n / g iv e n b y i, <f>(x,y) = 0 i f e i _____ f(x, y) <f>(x ’ y) = (*» y € #)> m ich h a s v a lu e s in E ( A) = h o m fc (A , A ), is a 2 -c o c h a in f o r t h e m o d u le p a i r 1 f a c t / is a 2 -c o c y cle , fo r u s in g (3*5) t o c o m p u te , m o d n n+1, t h e tw o sid e s o f t h e K (a ? ) r n (y )} r n (z ) = r n (x ) {rn (y ) r n (z)} cie v e rifie s^ t h a t d f = 0, i.e . / e Z 2(A , A ). C a ll/ t h e 2 -c o c y c le ^ L e m m a 2. L e t r n : G -> E ( U ) be a n y n t h a p p r o x i m a t i o n , a n d let f be th e 2 -c o c y c le Mien a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r th e e x i s t e n c e o f a n ( + 1 a p p ro x im a tio n L w h ic h s(atife3-4n) /= (3-6) dc kr som e c e C'1(A, A ) s u c h t h a t c (h ) = 0 f o r a l l h I P r o o f . S u p p o s e f ir s t t h a t (3*6) h o ld s . S in c e = 0 fo r h e H , o n e m a y d e fin e lr each x e G a n e le m e n t y ( x ) e E ( U ) in s u c h a w a y t h a t y ( x ) = c(x) Snd a lso e G ), y ( h ) — 0 fo r h e H . T h e n (3*6) c a n b e w r itt e n <j>{x,y) = r j x ) y { y ) - y { x y ) - > r y { x ) r n {y) d e fin e r n+x b y r n+1(x ) = r n ( x ) - n ny ( x ) m od7r ( ) . ( 3 - 8 ) (3-9) (xeG). E v id e n tly r n+x sa tisfie s (3*ln+1), (3*3n+1) a n d (3-4n), m o re o v e r r n+x(x ) r n + i(y ) = {r n(x ) ~ = r n(x y ) + = r n ( x y ) — u ny ( x y ) = n ny ( x ) } [rn (y ) - n n{f>(x, y ) - y ( x ) r n (y ) m o d 7 rn+1, b y (3-8) r n+i(x y ) ‘ lO ius r n+1 a lso sa tisfie s (3-2n+1). | f In m aking th is calculation, th e reader w ill observe th a t if / = she definitions (§2-1) o f x f and fx rae such th a t x f = p ( x ) f = r n (x ) <t> and f x = f p ( x ) — <J>rn( x ). e E { U ) ) and x e G , 140 J. A. Green C o n v erse ly if rn+1is a fu n c tio n w h ic h satisfies (3*2n+1), (3-3n+ (3*9) to define y . C le arly y ( h ) = 0 fo r a ll h e H , th e re fo re (3*7) defines a functic c e 0 1(A, A ) su c h t h a t c(h) = 0 fo r a ll u s e d to e s ta b lis h (3*8), h e n c e / = dc a s re q u ire d . Co r o l l a r y . A s u ffic ie n t c o n d i t i o n f o r th e e x is te n c e o f r n+1 i s t h a t f o r som b y (2-4). I t is n o w e a s y to p ro v e th e o re m 1 (§ 1*3), u sin g th e case w h e re H = {1} a n d ki is, th e re fo re , g iv e n b y r H { \ ) = i. C o n d itio n (3-3n) re d u c e s to r n ( \ ) = i. S u p p o se tha r v . . . , r n h a v e b e e n d efin ed to s a tis fy (3 -lm), (3-2TO), (3-3m) (1 ^ m a n d (3-4m (1 ^ m ^ n — 1). L e t / b e th e 2-cocycle o f rn , so t h a t / e A , A ). S in ce Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 c e (7]j(A, A ). F o r i f c € (7]y(A, A ), th e n c(h) = 0 fo r a ll e x t2 (A, A ) = 0, th e r e e x ists c e C x(A , A) = C # (A , A ) su c h t h a t / = dc. B y le m m a 2 e x ists a n ( w + l ) s t a p p ro x im a tio n rn+1 w h ic h satisfies (3-4J . Since th is h o ld s for any n ^ 1, a n d sin ce a first a p p ro x im a tio n r x c a n c e r ta in ly b e d efin ed , i t is possib to defin e (recu rsiv ely ) a liftin g seq u e n c e {rM}. T h is p ro v e s th e o re m 1. 3*4. W h e n H is a n a r b itr a r y s u b g ro u p o f th e sim p le c o n d itio n e x t # (A, A ) = 0is n o t en o u g h to e n a b le r H to b e e x te n d e d to a liftin g o f T h is is b e c a u se the! 2 -c o c y c le / o f a n n th . a p p ro x im a tio n r n is n o t, in g e n e ra l, a 2-cocycle re la tiv e to H : H o w e v e r if c e rta in f u r th e r c o n d itio n s a re im p o se d o n A , i t w ill b e sh o w n t h a t each! n t h a p p ro x im a tio n rn is c o n g ru e n t m od7rn to a n n t h a p p ro x im a tio n r * whose c o c y c le /* does b elo n g to Z ff(A , A ). T h ese n e w c o n d itio n s in v o lv e c e rta in 1-cocycles, w h ic h w ill n o w b e d efin ed . Let' vb e a fix su b g ro u p H n v H v - 1, a n d w rite a v = v ~ xa v , a ~ v a e H v, th e n b o th a a n d a v h e in H . B y (3 - 2 J a n d ( 3 - 3 J rn b e a n n t h a p p ro x im a tio n , a n d le t 3 h en ce mod??* (a e H J, rn (v)~x r H ( a ) r n (v) = r f f (a*) + {a e H v) (3-10); fo r som e £,(«) e E ( U ) . P u t zv(a) = £v(a). I t is c le a r t h (3-10) to c o m p u te , m od7rw+1, th e tw o sides o f th e e q u a lity K th e re re s u lts I n o th e r w o rd s, th e re p re s e n ta tio n o f (v )~l r H ^ ) r n ^ ) } { r n ( v ) - 1 r H {b )rn (v)} = «%(&) + zv(a) V* » zv(ab) ( a , b e H v). zv e Z x( A V, A V)w , h ere A , is th e Hv o n A defined b y p v(a) = p ( a v) (a e m o d u le (A ). I n th e case v e H,rn (v) = r H (v) b y (3-3n), a n d i t follow s fro m (3-10) t h a t = S up p o se n o w t h a t v is a n y g iv e n e le m e n t o f G, a n d t h a t zv e A„). T h i4 I m e an s th e re is a 0-cochain, i.e. a n e le m e n t m v e A ), su ch t h a t z v(a) = a vm v —m v a v ( e H v). 141 A lifting theorem for modular representations b e a n y e le m e n t o f E ( U ) s u c h t h a t ]TV = m v , t h e n £ J a ) ^ r J a ? ) p v - p vm rJa?) o d ( a e H v). r Z ( v y ~ r n ( v ) ( t - n np v)(. 3-11) m e t n a s im p le c a lc u la tio n s h o w s t h a t r* (v )“ 1 r f f (a )r* (t?) = r H ( a v) "+ 1 ( (3-12) &e m m a 3. L e t r n be a n y n t h a p p r o x i m a t i o n , a n d le t V be a s e t o f r e p r e s e n t a t i v e s o f R o u b l e c o se ts H v H o f H i n G , s u c h t h a t Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 v =j= l 1g T h e n i f zv B (A^j A^,) J each , a n n t h a p p r o x i m a t i o n r * c a n be f o u n d w h i c h h a s th e p r o p e r t i e s r * ( x ) s r n {x ) m o d 77w th e (3-13) 2,-cocycle f*of r * b e l o n g s to Z % ( fro o f. E a c h A ). (3-14) eh Ga s a f a c t o r i z a t i o n o f t h e f o r m x x = h x v h 2( h x, I e le m e n t v in (3-15) is u n iq u e ly d e te r m in e d b y x , b u t a r e n o t , i n g e n e r a l, gque. F o r e a c h x e cGh o o se a fix e d f a c to r iz a ti o n (3* 15), t o b e c a lle d t h e ‘ s t a n d lo riz a tio n 5 o f x,an d m a k e t h i s c h o ic e i n s u c h a w a y t h a t t h e s t a n d a r d f a c to r li o n o f a n e le m e n t plow d e fin e r * ( l ) t o b e i, a n d f o r U cG p u t v e V is v = .v\ 1 . v e V , v =j= 1, d e fin e r * ( x ) = r u ( h 1) r * ( v ) r H ( h , ) , b re x — h x v h 2is t h e s t a n d a r d f a c t o r i z a t i o n o f p i f h e H h a s s t a n d a r d f a c to r iz a ti o n I t is c le a r t h a t (3-13) h o ld s ; h —t h e n r*(h) = r h Q1!) rh Q1^) = r n(h)‘ bse f a c ts s h o w t h a t L et x b e a n y e le m e n t o f G , a n d x = (any e le m e n t o f H . T h e n , _ , r * is a n n t b a p p r o x im a tio n . h x v h i2t s s t a n d a r OC/lb --- j f a c to r iz a ti o n o f x h , a lth o u g h n o t n e c e s s a r ily t h e s t a n d a r d o n e . I f xh = i h e s t a n d a r d f a c to r iz a tio n o f x h , o n e fin d s b y c o m p a r is o n aere le re fo re kx — k 2 — a ~ vh 2h , h x x k x= a = r*{xh) 1,1 e H r\ W_1 = H v. = r H ( k x) r*(v)rH ( k 2) = r H ( h x) { r H (a) r * ( v ) r H (a ~ v)} r H {h2h ) = r H ( h x) r * ( v ) r H ( h 2) r H (h ) lu s r*(xh) = r * (x )r* (h ) Piich m e a n s t h a t f * ( x , h ) = 0 fo r a ll M 2 - 5 ) , / * € Z 2if (A ,A ). m o d 7 rn+1, b y (3-12). m od77n+1, x e G,he H . S im ila r ly 142 J. A. Green 3*5. T h e tw o la s t le m m a s le a d to th e follow in g th e o re m . Th e o r e m 2. to a li f t i n g L e t A = (A, p) , H r of p if (a) e x t1 (A„, A v) = 0 f o r a ll v e G, v £ H , a n d (b) e x t|f (A, A) = 0. Downloaded from https://royalsocietypublishing.org/ on 27 January 2022 P r o o f .S u p p o se t h a t r .v.., r n h a v e b e e n d efin ed to s a t (1 < m ^ n )and (3*4m) (1 ^ m ^ n — 1). L e t b e a s e t o f d o u b le c o se t repre s e n ta tiv e s o f H in G ,such t h a t 1 e b e ca u se e x t1 (A^, A v)= 0 b y (a). L e m m a 3 th e n sh ow s t h a t th e re is m a tio n r *satisfying (3*13) a n d (3-14). S in ce e x tf j (A, A) = 0, b y (b), / * = d som e c e C ^(A , A ). A p p ly in g le m m a 2, co ro lla ry , to r*, th e re e x ists a n ( w + l ) s j j a p p ro x im a tio n rn+1 w h ic h satisfies r n+l(x ) = r n(x )m o d (x € a n d h en ce b y (3-13), r n+1 satisfies ( 3 - 4 J . T h e re fo re a liftin g seq u e n c e c a n b e defined j a n d th is p ro v e s th e th e o re m . T o p ro v e th e c o ro lla ry a n n o u n c e d a t th e e n d o f § 1-3, a ssu m e n o w t h a t hagij c h a ra c te ris tic p, iGs a fin ite g ro u p su ch t h a t e v e ry p a ir o f d is tin c t S ylow m e e t in {1} a n d t h a t H is th e n o rm a liz e r o f som e fix ed S ylo w p -s u b g ro u p Gp o f G) J A = (A ,p)si a n y ( kC , r)-module. I f A c a n b e lifte d t o o, i t is triv A # = (A, p H). S uppose c o n v erse ly t h a t p H c a n b e lifte d to a re p re s e n ta tio n r H o f on U. (а ) L e t v £v,G e H .Then H , v H v ~ x a re d is tin c t su b g ro u p s o f G (b ec ow n n o rm a liz e r in G), a n d so th e Sylow ^ -s u b g ro u p s Gp , vG p v ~ x w h ic h th e y c ontainil a re d is tin c t, a n d th e re fo re m e e t in {1}. I t follow s t h a t th e o rd e r o f = H rw H v~x I is p rim e to p , th e re fo re (§ 2*3) e v e ry ( k h en ce e x t1 ( A v, A v) = 0. (б ) T h e in d e x ( G : H ) is p rim e to p , b ecau se H ^ Gp . T h e re fo -m odule is ^ - p r o je c tiv e , h e n ce extfy (A, A) = 0. T h u s th e c o n d itio n s of > th e o re m 2 a re satisfied, a n d th e c o ro lla ry is p ro v e d . (k, Re f e r e n c e s Artin, E ., N esbitt, C. J. & Thrall, R. M. 1948 R i n g s w ith m i n i m u m c o n d itio n . Ann Arbor: t U niversity o f Michigan Press. Cartan, H. & Eilenberg, S. 1956 H o m o lo g ic a l a lgeb ra . Princeton: Princeton U niversity P r e s s * Higman, D. G. 1954 D u k e M a th . J . 21, 369-376. Hochschild, G. 1956 T r a n s . A m e r . M a t h . 8 0 c . 82, 246-269.