On tensor products of complete intersections

TL;DR

This paper uses Andre-Quillen homology to analyze tensor products of complete intersections, providing a new, more efficient approach to understanding their regularity and intersection properties.

Contribution

It introduces the application of Andre-Quillen homology to tensor products of complete intersections, generalizing previous results and simplifying proofs.

Findings

Homology theory offers a concise method for studying tensor products.

New conditions for regularity and complete intersection properties are identified.

The approach clarifies the necessary hypotheses for these properties.

Abstract

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 Andre 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 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 to see clearly what extra hypotheses we need.