process

Axioms for the norm residue isomorphism 431cohomology operations generated by the P i and ˇ is isomorphic to the topologists’Steenrod algebra A , developed in [5].In addition, the motivic operations satisfy the usual Cartan formula P n .xy/ DP P i .x/P n i .y/, P i .x/ D x` if x 2 H 2i;i .Y I Z=`/, and P i D 0 on H p;q .Y I Z=`/when .p; q/ is in the region q i, p 2, the dual to the usual Steenrod algebra A is a gradedcommutativealgebra on generators i in (even) degrees .2`i 2; `i 1/ and i in(odd) degrees .2`i 1; `i 1/; see [9, 12.6]. The dual to i is the motivic cohomologyoperation Q i , which has bidegree .2`i 1; `i 1/. Because it is true in A , the Q iare derivations which form an exterior subalgebra of all (stable) motivic cohomologyoperations. They may be inductively defined by Q 0 D ˇ and Q iC1 D ŒP `i;Q i .We now quickly establish those portions of [8] that we need, concerning theseoperations on the cohomology of X and the unreduced simplicial suspension †X ofX (the cofiber of X ! cone.X/). The proofs we give are due to Voevodsky, and onlydepend upon [10] and [9]. In particular, they do not depend upon the missing lemmasin op. cit., or upon the Axioms 0.3.Fix q