arXiv subjects
Publications and source records attributed to Mohamed Tabaâ.
In this paper, we study the problem, initiated by Grothendieck, of the transfer of the properties of two preadic noetherian algebras over a field to their completed tensor product.
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.