Operations in étale and motivic cohomology

We classify all étale cohomology operations on H et n ( − , μ ℓ ⊗ i ) H_{\operatorname {et}}^n(-,\mu _\ell ^{\otimes i}) , showing that they were all constructed by Epstein. We also construct operations P a P^a on the mod- ℓ \ell motivic cohomology groups H p , q H^{p,q} , differing from Voevodsky’s operations. We use them to classify all motivic cohomology operations on H p , 1 H^{p,1} and H 1 , q H^{1,q} and suggest a general classification.