Sur le produit tensoriel d'alg\`ebres

TL;DR

This paper investigates how various algebraic properties such as regularity, Gorenstein, Cohen-Macaulay, and others transfer through tensor products of noetherian rings, providing conditions for these properties to hold in the tensor product and its completion.

Contribution

It establishes the transfer of several homological properties through tensor products of noetherian rings and characterizes when the tensor product is regular under flatness.

Findings

Properties like regularity, Gorenstein, Cohen-Macaulay transfer through tensor products.

Conditions for the tensor product to be regular when the base map is flat.

Transfer of properties to the completed tensor product.

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.