@JoeBerner I don't know, but don't purely transcendental extensions increase étale cohomological dimension? In other words, might the spectrum of k(X_1, X_2, X_3, ...) be a counterexample?
Does the Hopf ring star-indecomposables functor have a (right) adjoint? I thought I knew, but I don't. I think the ring-ring construction has the property that there's a natural isomorphism R –> Q^* Z[R], so that's a candidate, but the existence of a counit Z[Q^* H] –> H seems unlikely
