Sur le produit tensoriel d'alg\`ebres

Abstract

Let $\sigma:A\rightarrow B$ and $\rho:A\rightarrow C$\ be two homomorphisms of noetherian rings such that $B\otimes_{A}C$ is a noetherian ring. we show that if $\sigma$ is a regular (resp. complete intersection, resp. Gorenstein, resp. Cohen-Macaulay, resp. $(S_{n}$), resp. almost Cohen-Macaulay) homomorphism, so is $\sigma\otimes I_{C}$ and the converse is true if $\rho$ 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 $\sigma$ is flat, we give a necessary and sufficient condition to $B\otimes_{A}B$ be a regular ring.