Polynomial and related algebras as cohomology rings (report on recent progress)

Pub . Ma . UAB Vol . 26 N4 3 Des . 1982 POLYNOMIAL AND RELATEDALGEBRAS AS COHOMOLOGYRINGS (REPORT ON RECENT PROGRESS) Le p be a p ime and La y Smi h § 1 . Rings o In a ian s as Algeb aso e he S een od Algeb a P* = pIx1, . . .,xn! a polynomial algeb a o e 2Z/p . A ques ion o basic impo ance is o decidei P* can occu as he2Z/p cohomology o a opological space . O cou sea necessá y condi ion o P* o be a coho- mology ing is ha i be an uns able algeb a o e he S een od algeb a ( o p í 2 he gene a o s x j all ha e e endeg ee, so he Bocks ein is iden ically ze o and onlyV*, he algeb a o S een od educed powe s is ele an ) . By assuming hisex a s uc u e we can hen y o ei he cons uc a space X wi h P* - H*(X ; a/p), o y o use highe o de cohomology ope a ions, and o , ope a ions in ex ao dina y cohomo- logy o p o e no such space X can exis . Hindsigh now shows ha in ac ano he app oach, using ideas om Galois heo y, and in a ian heo y, p o ides a comple e answe o he ealiza ion p oblem o non- modula polynomial algeb as Ole say ha is non-modula i This hindsigh sugges s a na u al di ision o he ealiza ionp oblem ; namel . i s cons uc a class o examples o uns able polynomial algeb as o e he mod p S een od algeb a, and hen wo y abou which o . hese can occu as cohomology ings . One elegan way o cons uc ing uns able algeb as o e he S een od algeb a is p o ided by in a ian heo y, and was exploi ed o good ad an age by Cla k'and Ewing [6] . We s a wi h G p P*, P[x 1 ,... ,xnl deg xi jí 0 mod 2p : i = 1, . . . .,n .) a ini e g oup G -+ GL(n ; ZZ/p)  a ai h ull ep esen a ion . Le V := Vp e¿ 7 7Z/p  be he ep esen a ion space n o p and o m P k = and O = 0 P(V) = P[V*] he g aded polynomial algeb a on he dual ec j space V* o V, whe e he g ading esul s om he equi emen : deg = 2 . b E V . The ac ion o G can be ex ended o P(V) in he ob ious way and so we can o m he ing o in a ian s H* := P(V) G = { E P(V)Ig = dg E G} . (The s udy o ings o in a ian s wasin ac an im- po an local indus y in U ingen a ound he u n o he cen u y, so i is only na u al ha I should spend a ce ain app en iceship in his a ea .) The S een od algeb a ac s on P(V) in a unique way compa able wi h he Ca an o mula and he uns abili y condi ion, namely, ia he condi ion  k = 0 p : k = 1 0 o he wise :p í 42 o : k = 0 Sq k =  2 : k=2  p=2 0  ,  o he wise (in pa icula O =Sq 1 =0) Mo eo e since G is ac iog by linea ans o ma ions on V and aising o he p h powe is linea in cha ac e is ic p, i ollows ha he ac ion o G commu es wi h he ac ion o he S een od algeb a, and hence H* = P(V) G inhe i s omP(V) he s uc u e o an uns able algeb a o e he S een od algeb a . Example - i-  D#(n) _= P(Y)GL(V) This algeb a was o iginally s udied by Dickson who showed D * (n) =' P[Y1, . . .,yn] deg yi  = 2(p n - p n-i )  1, . . .,n La e onwe will see ha D*(n), which we e e o as he Dickson algeb a, plays a c ucial ole in he classi ica ion o uns able polynomial algeb as o e he S een od algeb a . Fo nowle me jus men ion ha he ' ac ion o he S een od algeb a on D*(n) is comple ely de e mined by he o mulae [17] : p j P yk wi h an analogous o mula o p=2, and he ac ha he Pp J gene a e Example 2 (S een od-Wilke son) : The polynomial algeb a in ques ion is A* := P[x4,x2p+2] whe e j-k = n-1 j = n-1 )  p > 2 o he wise is he c ucial o mula . In [18] S een od e i ied by edious calcula ion ha A* admi s an uns able  *- algeb a s uc u e . (N . B . When p = 3 A*= H*(BSp(2) ; 7Z/3) .) In [20] Wilke sonobse ed ha whe e 2 is he subg oup gene a ed by (N . B .  2l p := p-adic in ege s) A = A* - P(V) G : dim ZZ/p V G " GL(n ; 2Z p ) -1 , 0 B = e+e -1 ,  1 p+1 e + e -1 E 7L p ) and he ac ion o G on V is ia mod p educ ion om 7L p . (The g oup G comes om he Shepa d and Toddlis [11] .) 2 i whe e e = exp{-1 (N . B . one needs o check ha Ou p ima y in e es in in oducing his cons uc ion is howe e o cons uc a la geclass o uns able polynomial algeb as o e he S een od alge i a . Polynomial ings o in a ian s, howe e in cha ac e is ic ze o=we e long known o a ise om he canonical ep esen a ion o he We l g oupo a compac con_nec ed Lie g oup on he uni e sal co e ing space o amaximal o us . These ep esen a ions a egene a ed by eal e lec ions . In [11] Shepa d and Todd in oduced a complex analog o e lec ions, classi ied all he ini e g oups ha admi complex e lec ion ep esen a ions and showed by explici calcula ion ha he esul ing ings o in a ian s we e polynomial algeb as (o e T!) Ewing and Cla k exploi ed he wo ko Shepa d and Todd by ca ying he Shepa d and Todd classi ica ion kicking and sc eaming down o cha ac e is icp . To be mo e 'speci ic one in oduces a cha ac e is ic ee de ini ion o complex e lec ions, namely " De ini ion : An au omo phism is calleda ps_eudó e l F , ec ion i 1-p has ank one . The mo i a ion o his is clea . I you a e going o ha e a e lec ion ac oss a complex hype plane, . hen he o hogonalcomplemen o he hype plane is a complex line = eal 2-plane, so we can alsomake a " unny house mi o " by also o a ing he imagein he o hogonalcomplemen o he mi o . One hen p o es [4] [5] . Theo em (Che alley-Bou baki) : Le p : G-> GL(V)be a ini e dimensional ep esen a ion o he ini e g oup G which is gene a ed by pseudo e lec ions . I IGI Y 0 mod p, whe e p is he cha ac e is ic o he g ound ield, hen whe e : i hen p : V - V p(V) G ='  p[x 19 . . .,x n ] deg x i =  2d i  i  = 1, . . .,n, IGI .= - d 1 . . . dn Thus in he non-modula si ua ion one can cons uc lo s o example o uns able polynomial algeb as o e he S een od algeb a . By u elizing hei mod p educ ion o Shepa d and Todd, Cla k and Ewing can p o ide he ollowing comple e lis o i educiable examples whe e m > 1 and m = q . Numbe 1 1 Rank n I O de (n + 1)! Type [4,6, .. .,2(n + l)] P imes pl(n+1)! 2ak n q , m"_'n! [2m,4m, ... - , 2(n 1)m, 2qn] p;n!,p=1 mod in 2b 2 2m [4,2m] m>2,p-=1modm 31 m [2m) p=_lmodm 4 2 24 [8,12] p-1mod3 5 2 72 [12,24] p-1mod3 6 2 48 [8,24] p-1mod12 7 2 144 [24,24] p-1mod12 8 2 96 [16,24] p-1mod4 9 2 192 116,48] p=_ 1mod9 10 2 288 [24,48] p=_1mod12 11 2 576 [48,481 p-1mod24 12 2 48 112,161 p-1,3mod8, p :P 3 13 2 96 [16,241 p=1mod8 14 2 144 [12,48] p-1,19mod24 15 2 288 (24,481 p-1mod24 16 2 600 [40,60] p-1mod5 17 2 1200 [40,120] p=1mod20 18 2 1800 [60,120] p-1mod15 19 . 2 3600 [120,120] p-1mod60 2 360 [24,601 p =_ 1, 4 mod 15 21 2 720 [24,1201 p- 1, 49 mod 60 22 2 240 124,40] p=_1,9mod20 23 3 120 (4,12,201 p-1,4mod5 24 3 336 [8,12,281 p- 1,2,4mod7 25 648 [12,18,241 p= -- 1mod3 26 3 1296 [12,24,36] p-=lmod3 27 3 2160 [12,24,60] p-1,4mod15 28 4 1152 [4,12,16,241 p 2o 3 29 4 7680 (8,16,24,40) p-lmod4, po5 30 4 14,400 [4,24,40,601 p=1,4mod5 31 4 64 " 6! [16,24,40,481 p_lmod4, p-_5 32 4 2166! [24,36,48,60] p=_1mod3 33 5 72 . 6! [8,12,20,24,36] p-1mod3 34 6 1089! [12,24,36,48,60,84) p-lmod3, p#7 35 6 72 .6! [4,10,12,16,18,24] p#2,3, o 5 36 7 8 .9! [4,12,16,20,24,28,361 p ,-2,3,5, o 7 37 8 19210! 14,16,24,28,36,40,48,601 p ¢ 2,3,5, o 7 I one d ops he non-modula es ic ion, iz . DGI ¢ 0, hen simple examples,e .g . E P[Q1, .. .,an] = p(V) n dim V = n,  E n symme ic g oup show ha he e a e ings o in a ian s ha a e poly- nomial . In ac in he s ic lymodula case, namely when G is a p-g oup, he e is a cha ac e iza ion o he g oups and ep esen a ions [10] o which P(V) G is polynomial . Fo example onehas long known : E xampl e3 :  Le UP(V) = be he subg oup o uppe iangula ma ices . Then P(V) U P( V )  =  P[z 1 ,- ,z n ] deg . . = 2i(p ; .n) Fo example when n = 2 we iad E GL(V) P2 = 2 - 2-1 1 169 In any case i we s a wi h : hen H* := P(V) G is a polynomial algeb a o e he S een odalgeb a o be ound in he lis compiled by Cla k and Ewing and mo eo e he e is a space X such ha H*(X ;7Z/p) = H* . Le me summa ize he p eceeding discussion in he ollowing esul : Theo em 2 (Cla k-Ewing) :  Le p : G --> GL(n ;71/p) be gene a ed by pseudo e lec ions and assume (NMC)  p + IGI Th en p : G y GL(n ; 7Z/p) Algeb aic Pa :  The e is a ep esen a ion G GL(n ;C) o G as a complex pseudo e lec ion g oup, such ha he polynomial algeb as P(O e) G  P(© 2Z/P) G n  n ha e he same ype . Topological Pa :  The e exis a a space X(V ;G) such ha H*(X(V ;G)) = P(V) G ; V =O 7Z/p he ep esen- n a ion spaceo p . Rema ks : (1)  The exis ence o a complex"li ing" o a gi en mod p ep esen a ion can be explained as ollows . We ha e al eady seen ha (NMC) allowsus o cons uc a p-adicli ing G< GL(n ;7l p ) . Bu G being a ini e g oupmeans ha his ep esen a ion is al eady de ined in a ini e ex ension o Q (simplyadjoin enough oo s o uni y), and hence o e 0 . (2)  In addi ion Cla k and Ewing de e mine he cha ac e ields(N .B . Since hey p o e ha he Schu index is always 1 i doesn' ma e which de ini ion o "cha ac e ield" one is using)o he complex hype plane g oups in he Shepa d-Todd lis . Thus s a ing om a complex hype plane g oup one can ead o o e which ini e ields 7Z/p i admi a(pseudo e lec ion) ep esen a ions . Along wi h he ques ion o ealizing P(V) G as a cohomo- logy ing, we should alsolooka he homo opyclassi- ica ion o mapa be ween suchspaces . The cons uc ion o e ed by Cla k and Elaing deli e s a a he explici space X(V ;G) and onecan p o e : (see [14]) (2) he S een odalgeb aac ion on A* li s o an uns able ac ion on P[xi, . . .,xn], and (3) P[y1, . . .,yn] is closed unde his li edac ion . Then he e exis s a opological space A such ha H*(A ; ZZ/p) _ A* The cons uc ion o Cla k and Ewingp o idesmany examples o spaces whose a/p cohomology is a poly- nomial algeb a . Po an odd p ime p ano he e y na u al ques ion o s udy is ha o ealizing symme ic algeb asi .e ., ee commu a i e algeb as as cohomology ings . A med wi h a good ealiza ion heo em o symme ic algeb as, and a co esponding classi ica ion o maps, one could y o mimic wi h hese spaces as building blocks he upside down Pos niko owe (Sulli an's minimal model cons uc ion) o ge mo e comple e in o ma ion abou "Im{H* : Top - UnAl The minu e one s a s o alk abou symme ic algeb as, heBocks einbeha iou becomes impo an . E en in he simpeles case iz and P[x] ® E[Y]  :  B Y ~ 0 P[u] 0 E[ ]  8 : u = 0 P on . 3 :  Wi h he no a ions p eceeding whe e [X(V',G'),X(V11,G")] = Mo p ((V',GT),(V",G")) cP : VI - Vil MO p ((V',G'),(V",G")) :_  (CP i)  1 : Gl -- ," and CP(g' , )= $(g')CP( ') V g'  E G',  '  E V' This classi ica ion o maps comes in handy when one ys . o use he spaces X(X ;G) as "building blocks" o cons uc spaces ealizing o he in e es ing uns able algeb as o e he S een od algeb a as cohomology ings . He e [12] o example is a sample esul in his di ec ion . (This esul has also beenob ained independen ly by Howa dHille .) P op . 4 : Le A* = P[x1, . . .,xn]/(Y11 . . .,yn) be a g aded comple e in e sec ion, ha is an uns able algeb a o e he S een od algeb a . Assume ha : n  n (1)  (_T deg x i )(7F deg Yi) í G(p) ; i=1 i=1 he wo examplesbeha e e y di e en ly, as Aguadé has shown . As a sampleo his esul s one has [3] . P o - p . 5 : (J . Aguadé)  Suppose p en odd p ime, and is an uns able algeb a o e he S een od algeb a . Le 2d = : deg x hen dip-1 and all suchS* occu as cohomology ings . Ske ch o P oo :  To see ha dlp-1 we w i e (3y = x . The e is he Adam ela ion so applying his o y gi es whe e P d y = 0 by uns abili y . Bu hissays P 1 ac s non i ially on S* do dIp-1 . To cons uc he examples whe e Py = x Aguadé p oceeds as ollows . Le CE 7l/p x = 2Z/p-1 be a gene a o , and S* = E[y] 0 P[x] P 1 . =d -1 = ( d - 1)  P d + Pd , p 1 p P d -1 y =  (d - 1)P Pd y  +  Pd Py = 0 + p d (x ) = xp se §  := ep_1/d . Then e induces an ac ion o 7Z/d on K%Z/p ;1) = B 7Z/p . Le Y  := K(E/p ;1) /g be he o bi space . One hen has (whe e deg u = 1) H* (Y ; 2Z/p) = H*(BTl/p) 1 =  (E[u] ® P[Pu])1 -  E(u(p u) d-1 )  0  P(([3u) d ) as equi ed . Recallinghow he cons uc ion o Cla k and Ewing is he many a iable gene aliza ion o he one a iable cons uc ion o Holzsage [9] and Sulli an [19] one is emp ed o y o p oceed analogously s a ing wi hAguadé's cons uc ion o p o e : P op . 6 : Suppose S* -  E(y 1 ,. . . . yn ) 0 P(Py1, . . .,pyn) is an uns able algeb a o en he S een od algeb a whe e n (NMC)  71 - deg Pyi i 0(p) . i=1 Then he e exis s a space Y such ha H*(Y ;7Z/p) 2 S* . The ideao he cons uc ion would be o s a wi h p  : G y GL(n ; 2Z/p) . Le V := ® a/p be he ep esen a ion n space o p . The ac ion o G on V inducesa ee ac ion on BV = K(V ;1) so we can o m he o bi space Y := BV/G B(G x p V) . As in P oposi ion (1), i p + IGI  one ob ains H*(Y ; a/p) _ H*(BV ; a/p) G N [E(V) 0 P(PV) ]G Howe e i is almos ne e he case ha [E(V) ® P(PV N E(Y) 0 P(PY) whe e P(HY) N P(8V) G , (see o example, [4 ; Ex]) e en in he nices uses . Co example, pick an enoi - ous p ime p . Le E l ac on V := ®a/p ia he adjoin n ep esen a ion . Then one sees he ob ious map is-simplyno e en monic . To 1,.... n o V and ecall Bu cp  : E(a1, . . .,an) 0 P(Pa1," .,wn) - .  P(V) G P(a1, .. .,an) 0 E E(V) see his choosea basas 1 + . . . + ñ = 2 + ... + ñ = so p(a19 . . .Pa n ) E ke cp . Thus a p oo o P op_ 6 mus p óceed alóng o he lines . The p oo in [13] uns'mo e o less as ollows : Begin as be o e wí h G < GL(V) . Le *k : X(V ;G) - X(V ;G) be he map induced by he mo phism ( ecall P op . 3) X k  :  (V,G)  _  (V,G)I ak( ) = k , X kg =g . Fo m he ibe squa e (A := diagonal map) Y kX (1, * k) X, X x X X :_ (V,G) de ining Yk . Then o k = p+1 one inds whe e H* (Y p+ 11 a/P) = E(Y) 0 P(PY) P(PY) =POV)G . So o all he examples o ings o in a ian s, and ela ed ings, which we ha eshown o occu as cohomo- logy ings ha esa is ied he non modula i y condi ion . We ha e howe e , a leas as algeb a o e he S een od algeb a, he o he ` ex eme case o P(V) Up (V) , e c, namely P(V) G whe e G is a p-g oup . The ollowing esul se les he ealiza ion ques ion o hese modula ings o in a ian s in he nega i e [15] . P o p .  Le p be an odd p ime and  G < GL(n ; 7L/p) a p-g oup such ha is a polynomial algeb a . Then R* canno a ise as he 7l/p cohomology o a space . Thus a polynomial algeb a al whe e deg x i = 2p , i = 1, . . .,n, and a leas one a l posi i e, ha occu s as a ing o in a ian s can ne e be he cohomology algeb a o a space . The e o e we canno sepa a e he ealiza ion ques ion in o a non- modula heo y, bu mus mo e om a non-modula heo y o a "mixed" heo y . A p o o ype example he e is he Dickson algeb a 18 4 R* := P(V) G ; V = O Tl/p n P*  .= P(x 1 , . .. . x n ) D* (n) := p(V)GL(V),j P[Y1, . . .,yn] deg y i= 2p n - 2p n-i ;  i = 1, . . .,n . One eason o singling ou D*(n) o special s udy is he ollowing esul p o ed join ly wi h Bob Swi ze , in en a emp o cla i y he wo k o Adams andWilke son o be discussed in he nex sec ion . P op . 8 (join wi h R .M . Swi ze ) :  Le H* E UnId/V * ( := Uns able In eg al Domein o e he S een od algeb a .) ANASC ha H* i P(V) G o some G < GL(V), whe e deg =2 d E V, is ha H* be a ini e algeb aic ex ension in UnId/,R* o D*(n) . Finally he ealiza ion ques ion o D*(n)is se led by [17] P o p .  (join wi h R .M . Swi ze ) :  ANASC ha D*(n) occu as a cohomology algeb a is : o n=2 and p<3 . N .B .  Fo n = 1 he example a edue o Holszage and Sulli an . Fo n =2 hey a e all classical, iz ., (CP(W), BSU(3) when p=2 The case P(V) GL(2 ;ZZ/3) has ecen ly been ealizad by A . Zab odsky [23] . We collec some ac s abou V [2 ; § 51 . (1)  I  1 ,... . n is a 7l/p basis o V hen 1 , . . ., n a e algeb aically independen . (2)  The elemen s o V a e uns able, so (3)  he ac ion o IP *  on  P*  commu es wi h he ac ion o GL(n ; IF p ), so P* := P[ 1, . . .p n] E UnId/ > * D*(n)  := P* GL(n ;I p ) < P* < E* a e inclusions in UnId/19* (4)  e e y x E P* is in eg al o e H* . Le A* be he algeb aob ained omH* by adjoining 1, . . . , n . Theh A* E UnId/iq * and we ha e he inclusions H* < A* > P* > D*(n) . Suppose ha we knew ha A* > D*(n) we e an algeb aic ex ension . Then A* > P* is algeb aic . Bu by he algeb aic closu e heo em P* is algeb aically closed and hence A* = P* whence H* < A* = P* is a sepa ablealgeb aic ex ension . Conside he Galois g oup G := Gal(E*> F(H*)) . Clea ly G< GL(n ;IFp ) because he elemen s o G de ine linea ans o ma ions o V and an au omo phism o E* ixing F(H*) is uniquely de e mined by i s ac ion on V . Now we claim H* P* G =IF P[ 1# . . ., niG To see his no e he inclusion H* < P* G is clea . On he o he hand because E* > F(H*) is a Galois ex ension e e y x E P* G lies in F(H*) . Fu he mo e x is in eg ál o e H* by (4) abo e . H* is howe e a polynomial algeb a, hence in eg ally closed, and hus x E H*, i .e . P* G < H* so F ollows . Hence ou p oblem educes o showing A*  >  D* (n) is an algeb aicex ension . In ac we show ha i is an in eg al ex ension by an a gumen li ed om Adams and Wilke son [2] . Fi s o all ecal1 D*(n) _ IFp[y,, . . .lyn] . Le (y1, . . .,yn) deno e he ideal o A* gene a ed by y1' . . .,y n .  I we can show ha A*/(y1, ... . yn ) is ini e dimensional as a ec o space o e IF p , hen in ac an easy a gumen ia induc ion o e he g ading, shows ha e e y elemen o A* is in eg al o e D*(n) . To see ha A*/(y1, . . . . yn) is ini e dimensional o e IF p we p oceed as ollows . Le zE A 2d wi h d q( 0 mod p . Then [2 ; 2 .3] he e is an elemen b E - 9 * such ha Now he de i a ion is P en (bz)  = zp n ynPeo + yn-1P l+ . . .+ y1Pen-1 + pen and hence anishes on E* and he e o e ce ainly on . A* < E* . Thuswe ha e z pn = P en (bz) = -ynPen(bz)-  y1Pen-l (bz) E (y1, . . .,yn) By cons uc ion A* is a ini ely gene a edE p algeb a, wi h gene a o s a l ,. . . y a m  whose deg ees a e ela i ely p ime o p . By he p eceedingcalcula ion he na u al map is su jec i e . Bu IFp[al, . . .,am] y (ap ., . . . .am) IFp[al, ... . a m ]/(ap  , .. .,  am) is isably ini e dimensional . 194 A*/(yl, . . .,yn) REFERENCES 1 . J .F . Adams,On he non-exis en e o elemen so Hop In a ian One, Ann . o Ma h . 72(1960),20-104 2 . J .F . Adams and C . Wilke son, Fini e H-Spaces and Algeb as o e hé S een od Algeb a, Ann . o Ma h .78(1980), 95-143 3 . J . Aguadé, Cohomology Algeb aswi h Two Gene a o s, Ma h . Z . 177(1981), 289-296 4 . N . Bou baki, G oupes e Algeb ás de Lie Ch V He mann Pa is 1968 5 . C . 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