Cohomology of sub-Hopf-algebras of the Steenrod algebra

Abstract

Let A denote the mod 2 Steenrod algebra. The object of this paper is to study certain interesting properties of the cohomology of sub-Hopf-algebras of A which we hope might be helpful for the study of H*(A), the cohomology of A itself. Before stating our main theorems we.recall the work of Margolis [7], AdamsMargolis [4] and Anderson-Davis [S] on the structure of sub-Hopf-algebras of A. For any sequence (nl, n2,. . . ) of non-negative integers (possibly equal to infinity) denote by A(nl, n2,. . . ) the Z2-submodule of A generated by all the Milnor basis elements Sq ( rl, r2, . . . ) with ri C 2”d. (The notation A (nl, n2,. . . ) is adopted after [5].) It is a theorem of Margolis [7] that for any sub-Hopf-algebra B of A there is a sequence (nl, n2,. . . ) such that B = A(nl, n2,. . . ). However not all sequences are realizable as such. It has been shown by Adams and Margolis [4] (also independently by Anderson and Davis [5]) that a sequence (nl, n2,. . . ) is realizable if and only if for i > j a 1, Pzi amin(PZj, a-j j). In particular A (1,2,3, . . . ) is a sub-Hopfalgebra of A; we denote it by Ad. Our first result is to determine which sub-Hopf-algebra of A has nil-free cohomology. (A commutative algebra is nil-free if it has no non-zero nilpotent elements.)