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Sur Le Produit Tensoriel D'algèbres

Mohamed Tabaâ

Abstract


Let $\sigma \colon A\rightarrow B$ and $\rho \colon A\rightarrow C$ 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 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 $\sigma$ is flat, we give a necessary and sufficient condition for $B\otimes_{A}B$ to be a regular ring.

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DOI: http://dx.doi.org/10.7146/math.scand.a-24181

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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