Truncated polynomial algebras over The Steenrod algebra

Publicacions Ma emá iques, Vol 34 (1990), 335-339 . TRUNCATED POLYNOMIAL ALGEBRAS OVER THE STEENROD ALGEBRA Abs ac MOHAMED ALI I is shown ha he classi ica ion o polynomial algeb as o e he mod p S een od algeb a is an essen ially di e en p oblem om he classi ica- ion o polynomial algeb as unca ed a heigh g ea e han p o e he S een od algeb a . 0 . In oduc ion . Le B = Zp[y2n  y2n2, . . . , y2n,] be a polynomial algeb a o e A(p), he mod p S een od algeb a, whe e y en ; has dimension 2ni and p is an odd p ime . I each ni is p ime o p, he esul s o [1] and [2] imply ha he s uc u e o B is well unde s ood ; in pa icula he se o dimensions {2n1, 2n 2 , .. .,2n,} is a union o se s gi en in he Cla k-Ewing lis o dimensions in he main Theo em o [3] . Ea lie a emp s in he 1960's and ea ly 1970's o classi y he se o dimensions occu ing in B o en depended only on he A(p)- algeb a s uc u e o Bp+ 1 = Zp[y2n1, y2n2, . . . , yen,] p+1 , he polynomial algeb a unca ed a heigh p + 1 . The ques ion o de e mining he dimensions o he gene a o s o a unca ed polynomial algeb a A = Zp[x2n1, X2n2, . . . . X2n,]p+1 o e A(p), whe e each ni is p ime o p, is no well unde s ood . I appea s no o be known i he se o possible dimensions in he wo cases coincide as is ce ainly he case when = 1 . The pu pose o his no e is o se le his ques ion, o example, Z11[xs,xlo] 12 suppo s an ,/4(11)-s uc u e, bu Z11[y6,y10] does no . The ques ion has some opological signi icance . Fo example, i a p oduc o p-local sphe es, IIS(P) , 1 < i < , suppo s an A(p) s uc u e in he sense o [5], hen each ni E {1, 2,  . . p} and he e exis s a unca ed polynomial algeb a o e A(p), A = Z p [x2n, , x2n 2, . . . , x2nj p+1 [4] .  I an addi ion, 1 < ni < p o each i and he e exis s B =Z p [y2n  y2n 2 ,.. ., y2n,], hen he se o dimensions {2n1, 2n 2 ,.. . , 2n, .} is gi en by he Cla k-Ewing lis .  One would hen expec ha , up o homo opy, IlS n ~ ' suppo s he s uc u e o a opological g oup . 1 . Cla k's condi ion . Fi s we apply a well known heo em o A . Cla k [2] ; i is clea ha he p oo holds o a polynomial algeb a unca ed a heigh g ea e han p . 336  M . ALi Theo em 1 .1 . Le A be a polynomial algeb a unca ed a heigh p -1 -1 o e ¡he mod p S een od algeb a . I 2m is he deg ee o a gene a o o A, hen ei he m - 0 mod po hese exis s a gene a o o dimension 2n wi h n - 1 - p mod I we es ic a en ion o A= Zp[s2n,x2,n]P+1 as abo e, ou ine calcula ions imply ha , o small p imes, he pai s o in ege s {2n, 2m} mus lie in he able below . p = 3  {4,4} p = 5  {4,4},{4,6},{4,8} p = 7  {4,4}, {4,6}, {4,8} , {4,12}, {6,6}, {6,12}, {8,12}, {12,12} p = 11  {4,4},{4,6},{4,8},{4,10},{4,12},{4,20},{6,10}, {8,20},{10,10},{10,20},{12,16},{20,20} p = 13  {4,4},{4,6},{4,8},{4,12},{4,14}, {4,24},{6,6},{6,10}, {6,12},{6,24},{8,8},{8,12},{8,16},{8,24},{12,12}, {12,18},{12,24},{16,24},{24,24} Compa ing his lis wi h he lis o possible dimensions o he gene a o s o a polynomial algeb a B = Zp[y2n, y 2 , n ] o e he S een od algeb a [3], we see ha he only pai s which do no appea in he la e a e {6,10} and {8, 20} when p = 11 and {6,10}, {8,16} and {12,18} when p = 13 . We mus he e o e conside he possibili y o de ining he ac ion o he S een od algeb a on A= Z p [x 2n , x 2 , n ]P+ 1 in hese i e cases . 2 . S een od heo em . In his las published pape , S een od conside ed he ques ion o de ining he cyclic educed powe s on g aded polynomial algeb a . We belie e ha his esul s ha e ne e been applied . We summa ize hem in ou con ex . Le A be a g aded algeb a o e he (uns able) mod p S een od algeb a con- cen a ed in e en dimensions . Recall ha he cyclic educed powe s a e homo- mo phisms pq : A 2n -s A2n+2q(p - 1) sa is ying : (1) p ° = Iden i y, (2) pqx = xP i 2q = dimx, (3) pqx = 0 i 2q > dim x, ( 4 ) p9(xy) =El-0 pixp4-'y, (5)I a<pb, TRUNCATED POLYNOMIAL ALGEBRAS  33 7 1 l  a+  (p - 1)(b - ) - 1  a+b- 0 a-p  )p  p Now le A = Z p [x2n, , x2np ... . ,x2n, ] a+l .  Suppose ha o each i, he e a e de ined homomo phisms Pq by se ing P g x2ni = q,i, 0 < q < 2ni, whe e ,i E A has dimension 2ni + 2q(p -1), Pni x2n ; = XZni and P g x2n i = 0 o q > ni and ex ending Pq o e A by linea i y and he Ca an o mula (4) . Theo em 6 .2 and Lemma 4 .1 o [7] imply ha hese Pq de ine an ac ion o he S een od algeb a p o ided ha R(a, b)x2 ni = 0 o all i whe e (a, b) = (p , b), >_ 0 . We now es ic a en ion o he i e examples A= Zp[x2n,x2m] n+1 lis ed abo e whe e n < m . Suppose ha P I x2n = 91,n,P l x2m = h 1 , m . We de ine Yx2n induc i ely o 1 < i < n by equi ing R(1, i - 1)x2n = 0, o equi a- len ly, P I x2n = ( i) -1 P 1 P i-l x2n, P n x2n =x2n wi h simila de ini ions o x 2 , n . Dimensional easons imply ha p- = 0 in all cases as 2mp - 2n < 2p(p -- 1) and he e o e R(p , b) = 0 o > 0 . Thus ó show ha he de ini ión o he P' de ines an ac ion o he S een od algeb a i is su icien o check ha wi h he chosen 91,n and hl,,n, R(I,n - 1)x2n = 0and R(1, m - 1)x 2 ,, = 0 . Theo em 2 .1 . (a) p = 11, ZI1 [x s , x 1o ] 12 and Zll [x8, X211 12 suppo an ac ion o he S een- od algeb a which is unique up o algeb a isomo phism o e he S een od algeb a . (b) p = 13, Z13[xg,x10]14, Z13[x12, x18] 14 and Z13[x8,x1B] 14 do no suppo an ac ion o ¡he S een od algeb a . We conside he examples in u n . (a) (1) p = 11, {6,10} . We i s p o e uniqueness . Assume ha ZI I [x s , xlo] 12 suppo s an A(p)-ac ion . Fo dimensional easons p l x s = ax s xi o1 P xio = %x lo + - yx i . Rou ine compu a ions show ha p 3 xó = (3!) -1 [2ay 2 x6 l + a(a + 20)(a + 4P)xsxi o + 5a (a + 2p)xsxi o ] = xsl The e o e ay 2 = 3 and a+20 = 0 . We can choose y o ha e any non-ze o alue . So we se y = 5 . Then a = 1 and ,Q = 5 . We deduce ha i A suppo s an ac ion o he S een od algeb a, he ac ion is unique up o algeb a homomo phism o e he S een od algeb a and we can se P I xs = xsx 2 o, p 1 XIO = 5(x3 1  lo + xs) . The e o e le 91,3 = xsxio, hi,s = 5(xó + . xs) and de ine P g xs, PgXIO as de- sc ibed abo e .  The calcula ion abo e implies ha R(1, 2)X6 = 0 and so we need only check ha R(1,4)xl0 = 0 . Rou ine calcula ion e i ies ha his is ue . (a) (2) p = 11, {8, 20} . Again assume ha Z11 [x8, x12] 12 suppo s an A(p)- ac ion . Fo dimensional easons, p1 x8 = "8x20, p 1 x20 = 0x20 +yx8 . Rou ine compu a ions show ha p 4x8 = (4!) -1 [a(a + 0)(a + 2 0)(a + 30 )x8 x2o + a7(a - p)(a + 2p)x6x6o `  + ay 2(a + 30)x8 1 ] = x81 . The e o e by sui able choice o x2o, we can assume ha a = 1 . I ollows ha p= 5, 7 = 4 . Replacing x8 by x8 i necessa y, we can assume ha y= 4, and he e o e i A suppo s an ac ion o he S een od algeb a, he ac ion is unique up o algeb a homomo phism o e he S een od algeb a and we can se p1 x8 = x8xlo, p1 x2o = 5x2 0 + 4x8 . The e o e le 91,4 = XSx20, hijo = 5x30 + 4x8 and de ine 0 9 x6, pgx2o as desc ibed ábo e . . Clea ly R(1, 3)x8 = 0 and so we need jus o check ha R(1, 9)X20 = 0 . Rou ine calcula ion e i ies ha his is ue . (b) (3) p = 13, {6,10} . Again suppose ha Z13[x6,x1o] 14 suppo s an A(p) ac ion . Fo dimensional easons p1 x 6 = ax 55 + px 3 o, p 1 x10 = yxsxlo . Rou ine calcula ions show ha 03x6 = (3!) -1 [6a 3 x6 3 + p{6a 2 + (4a + 3y)(5a + 3y)}x610 + 0 2 (5a + 3y)x6 1  = x63 The e o e a 3 = 1, p{6a 2 + (4a + 3y)(5a + 3y)} and p2 (5a + 3y) = 0 in Z13 . This is possible only i p = 0, bu his implies ha 0 9 x10 E (x 6 ) which is no possible as p 5 xlo = X13 . Thus Z 13 [x 6 , x10] 14 will no suppo an A(13) ac ion . 10 (b) (4) p = 13, {8,16} . Again suppose ha Z13 [X8, x16] 14 suppo s an A(p) ac ion . Fo dimensional easons p1 x8 = ax8 -1- pxi6+yxáxls, 0'x16 = 6x8X16+ /Zx8x16 + 0x8 . Rou ine calcula ions show ha p4 x8 = (4!) [{7a 4 + 12a0B 2 + 8ayic0 + 6a 2 8y + 11y202 + 608 2 / + y8p 2 + 2BZyb}x83+ {9a 2 yN, + 3app + 6ayp 2 + PBp2 + yp 3 + 80 , yó + 3a3y+ 4cey 2 0 + 9a 2 pB + 8aO-y6 + py0 2 + 12pO 2 6}x8 1 x16+ {5x 3 0 + 110 2 0 2 + 60Byp + 11apO-y + 4a 2 y 2 + 6ay 2 p + 8a 2 -yb + 6«060+ a 2 pp + 8ayóp + 30Bóp + 7yóp 2 +8& 3 + 9B-y 2 6 + 80y0 + 2By 3 }x8 16 + {a 2 py + 11poy 2 + 6ay 3 + 7y 3 [1 + 11a-Y26 + 12PB6y + 5,Y 2 6FU + 120-yp2+ 6a 2p6 + 100ó 2 p + 12apbp + 12-í 62,U + 6p c 2 ó + 11ap 2 B + 3p2B c}xáxi6+ {90, 2 0 2 + 6p 2 0y + 5apy 2 + 3py 2 ~ + apy6 + 110 2 06 + ap 2 F¿ + 9py6,¿+ 90 2 0 2 + lly 4 +7^í '3 6 + 3y262 + 6_Y63 +3 apb 2 + 9ppb2}x8xi6+ {5ap 2 y+0 3 B+4yp 2 p+0y 3 +4py 2 ó+12pyó 2 +10«0 2 6+0 2 Pb+1106 3 }xáxi6+ {llapa + 120 3 0 + 302y2 + 602y6 + 90262 }x8x66] =x63 TRUNCATED POLYNOMIAL ALGEBRAS  33 9 One can show ha his implies ha 0 = 0, and so p g xls E (x8) which is alse as p 8 xls =xis . Thus Z13[x8,x16] 14 will no suppo an A(13)-ac ion . (b) (5) p = 13, {12,18} . Simila ly as in he las examples, we can see ha , o dimensional easons, p 1 x 1 2 = axil +Oxi81 il l xl8 = yxi2x18 and hen p s x12 = (6!) -1 [{ 8 a i xiz + ( 3a ' y + 9a l y 2 +4a 2 y 3 + 4ay 4 + 6,y5 }x 10 x12 i8 + i 2 {5a 4 +7a 2 y+5a 2 y 2 + 3 ce - y 3 }xi2x i8 +,~ 3 {7a 3 +9a 2 y-~ lla-y 2 +y 3 }xi2x 6 8 + q4 {7a 2 + 12a-y+ 6y 2 }X12X881 = X13 12 . This is ue only i ,Q = 0, bu his implies ha p e x 1 8 E (x12) which is impossible as p9 x18 = X18 and so Z13[x12,x18] 14 will no suppo an .4(13)-ac ion . Re e ences 1 .  J .F . ADAMS AND C.W . WILBERSON, Fini e H-spaces and algeb as o e he S een od algeb a, Ann . Ma h . 111 (1980), 95-143 . 2 .  A . CLARK, On I 3 o ini e dimensional H-spaces, Ann . Ma h . 78 (1963), 193-195 . 3 .  A . CLARK AND J . EwING,The ealiza ion o polynomial algeb as as co- homology ings, Paci ic J . Ma h . 50 (1974), 425-434 . 4 .  J .R . HUBBUCK AND M . MIMURA, Ce ain p- egula H-spaces, A ch . Ma h . 49 (1987), 79-82 . 5 .  N . IWASE, H-spaces wi h gene a ing subspaces, P oc . A . Roy . Soc . Edin . ( o appea ) . 6 .  J . STASHEFF, Homo opy associa i i y o H-spaces, I and II, T ans . Ame . Ma h . Soc . 10 8 (1963), 275-292 and 293-312 . 7 .  N .E . STEENROD,"Polynomial algeb as o e he algeb a o cohomology op- e a ions," H-spaces, Neucha el (Suisse) 1970, Lec u e no es in Ma h . 196, 1971, pp . 85-99 . Depa men o Ma hema ical Sciences Abe deen Uni e si y Abe deen AB9 2TY SCOTLAND Rebu el 18 de Desemb e de 1989