Under what conditions on a bimodule $M$ over a noetherian commutative ring $R$ is the tensor algebra $T=T_R(M)$ regular coherent? A theorem of Gersten says this is true for a free bimodule $M$. If $M$ is projective on the right and contains a field so that the enveloping algebra $R\otimes_kR$ is regular (e.g. if $k\to R$ is smooth), then I get that $T$ is left regular. However I don't know what happens with left coherence.