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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. A theorem of Choo, Lam and Luft says that the coproduct of two coherent algebras over a (right) Noetherian ring is coherent. This reduces the question to the case when $M$ is indecomposable as a bimodule. The particular $R$ I'm interested in is a smooth commutative algebra over a field and the $M$'s that show up in my setting are finitely generated and free as a right module, but can be pretty nasty as a left modules.

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 flat on one sideand 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 regular on both sides. However I don't know what happens with coherence. A theorem of Choo, Lam and Luft says that the coproduct of two coherent algebras over a (right) Noetherian ring is coherent. This reduces the question to the case when $M$ is indecomposable as a bimodule. The particular $R$ I'm interested in is a smooth commutative algebra over a field and the $M$'s that show up in my setting are finitely generated and free on the right, but can be pretty nasty as a left modules.

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.

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. A theorem of Choo, Lam and Luft says that the coproduct of two coherent algebras over a (right) Noetherian ring is coherent. This reduces the question to the case when $M$ is indecomposable as a bimodule. The particular $R$ I'm interested in is a smooth commutative algebra over a field and the $M$'s that show up in my setting are finitely generated and free as a right module, but can be pretty nasty as a left modules.