Sur Le Produit Tensoriel D'algèbres - Tabaâ, Mohamed (MATHEMATICA SCANDINAVICA, 2016)
Let $σ \colon A→ B$ and $ρ \colon A→ C$ be two homomorphisms of noetherian rings such that $B\otimes_{A}C$ is a noetherian ring. We show that if $σ$ is a regular (resp. complete intersection, resp. Gorenstein, resp. Cohen-Macaulay, resp. ($S_{n}$), resp. almost Cohen-Macaulay) homomorphism, so is $σ\otimes I_{C}$ and the converse is true if $ρ$ is faithfully flat. We deduce the transfer of the previous properties of $B$ and $C$ to $B\otimes_{A}C$, and then to the completed tensor product $B\mathbin{\hat\otimes}_{A}C$. If $B\otimes_{A}B$ is noetherian and $σ$ is flat, we give a necessary and sufficient condition for $B\otimes_{A}B$ to be a regular ring.