Truncated Polynomial Algebras over the Steenrod Algebra

MOHAMED ALI It is shown that the classification of polynomial algebras over the mod p Steenrod algebra is an essentially different problem from the classification of polynomial algebras truncated at height greater than p over the Steenrod algebra . 0 . Introduction . Let B = Zp[y2n y2n2, . . . , y2n,] be a polynomial algebra over A(p), the mod p Steenrod algebra, where yen ; has dimension 2ni and p is an odd prime . If each ni is prime to p, the results of [1] and [2] imply that the structure of B is well understood; in particular the set of dimensions {2n1, 2n2 , . . .,2n,} is a union of sets given in the Clark-Ewing list of dimensions in the main Theorem of [3] . Earlier attempts in the 1960's and early 1970's to classify the set of dimensions occurring in B often depended only on the A(p)algebra structure of Bp+1 = Zp[y2n1, y2n2, . . . , yen,]p+1 , the polynomial algebra truncated at height p + 1 . The question of determining the dimensions of the generators of a truncated polynomial algebra A = Zp[x2n1, X2n2, . . . . X2n,]p+1 over A(p), where each ni is prime to p, is not well understood . It appears not to be known if the set of possible dimensions in the two cases coincide as is certainly the case when r = 1 . The purpose of this note is to settle this question, for example, Z11[xs,xlo] 12 supports an ,/4(11)-structure, but Z11[y6,y10] does not . The question has some topological significance . For example, if a product of p-local spheres, IIS(P) , 1 < i < r, supports an A(p) structure in the sense of [5], then each ni E {1, 2, . . p} and there exists a truncated polynomial algebra over A(p), A = Zp [x2n, , x2n 2 , . . . , x2njp+1 [4] . If an addition, 1 < ni < p for each i and there exists B = Zp [y2n y2n2 , . . . , y2n,], then the set of dimensions {2n1, 2n2 , . . . , 2n,.} is given by the Clark-Ewing list . One would then expect that, up to homotopy, IlSn ~ ' supports the structure of a topological group . 1 . Clark's condition . First we apply a well known theorem of A. Clark [2] ; it is clear that the proof holds for a polynomial algebra truncated at height greater than p .