On tensor products of complete intersections

The study of regularity and complete intersection of a tensor product of commutative algebras possessing the same property started with Grothendieck in 1965 and has continued until today. Surprisingly, the homology theory of André and Quillen, developed by these authors in 1967, has never been used for this study. With the help of this theory, we can (slightly) generalize the results known up to now. But even more important, we hope to convince the reader that this homology theory is the adequate tool to handle these problems: the proofs are very short and (assuming some flatness hypothesis) it allows one to see clearly what extra hypotheses we need.