An algebraic introduction to the Steenrod algebra

This paper is a complete presentation of the Steenrod algebra P from a purely algebraic point of view, without referring to cohomology operations. Let Fq be a Galois field of characteristic p, V a finite-dimensional vector space over Fq and Fq[V ] the graded algebra of polynomial functions on V . Fq[−] is a contravariant functor from Fq-vector spaces to graded connected algebras. Then P can be defined as the subalgebra of the endomorphism algebra of Fq[−] generated by the homogeneous components of a perturbation of the Frobenius map [L. Smith, Polynomial invariants of finite groups, A K Peters, Wellesley, MA, 1995; MR1328644 (96f:13008)]. The generators of P obey the Adem-Wu relations; they can be derived by the Bullett-Macdonald identity [S. R. Bullett and I. G. Macdonald, Topology 21 (1982), no. 3, 329–332; MR0649764 (83h:55035)]. The proofs to show that the Adem-Wu relations are a complete set of defining relations for P are rearranged following the strategy of H. Cartan [Comment. Math. Helv. 29 (1955), 40–58; MR0068219 (16,847e)] and J.-P. Serre [Comment. Math. Helv. 27 (1953), 198–232; MR0060234 (15,643c)]. They are extended from the prime field case to the case of an arbitrary Galois field Fq, q = p . The paper also includes the Hopf algebra structure of P [J. Milnor, Ann. of Math. (2) 67 (1958), 150–171; MR0099653 (20 #6092)] with Milnor’s theorems extended from Fp to Fq. {For the entire collection see MR2404071 (2009a:55001)}