A mod-ℓ vanishing theorem of Beilinson-Soulé type

Let L be a field containing an algebraically closed field and X an equidimensional quasiprojective scheme over L. We prove that CHi(X,n;Z/ℓ)=0 when n>2i and ℓ≠0; this was known previously when i≥dimX and L is itself algebraically closed. This “mod-ℓ” version of the Beilinson-Soulé conjecture implies the equivalence of the rational and integral versions of the conjecture for varieties over fields of this type and can be used to prove the vanishing of the (integral) groups CHi(X,n) (for n>2i) in certain cases.