Sur le produit tensoriel d'algèbres

Abstract

Let $σ:A\rightarrow B$ and $ρ:A\rightarrow C$\ 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 transfert of the previous properties of $B$ and $C$ for $B\otimes_{A}C$, and then for the completed tensor product $B\hat{\otimes}_{A}C$. If $B\otimes_{A}B$ is noetherian and $σ$ is flat, we give a necessary and sufficient condition to $B\otimes_{A}B$ be a regular ring.