Algebraic identities to prove that a neat finite free algebra is tracically \'etale

TL;DR

This paper provides an elementary proof that finite free algebras over a commutative ring are tracically étale, using algebraic identities related to matrices of polynomials, confirming a deep connection in commutative algebra.

Contribution

It offers a novel, elementary proof that finite free algebras are tracically étale, based solely on algebraic identities, without relying on existing literature.

Findings

Finite free algebras over a ring are tracically étale.

The trace form of such algebras is nondegenerate.

The proof is based on algebraic identities involving matrices of polynomials.

Abstract

The central objective of this article is to provide an elementary proof of the following theorem, of which we are unaware of any trace in the existing literature. If $B$ is a net finite free algebra over a commutative ring $A$, then it is tracically \'etale (its trace form is nondegenerate) and a fortiori \'etale over A. As indicated in the title, our proof is based on algebraic identities. This confirms the implicit adage that much of the most abstract commutative algebra is concentrated in algebraic identities concerning matrices of polynomials over an arbitrary commutative ring. -- -- -- L'objectif central de cet article est de donner une d\'emonstration \'el\'ementaire du th\'eor\`eme suivant, dont nous ne connaissons pas de trace dans la litt\'erature existante. Si $B$ est une alg\`ebre libre finie nette sur $A$, alors elle est traciquement \'etale (sa forme trace est non d\'eg\'en\'er\'ee) et \`a fortiori \'etale sur $A$. Comme indiqu\'e dans le titre, notre d\'emonstration est bas\'ee sur des identit\'es alg\'ebriques. Cela confirme l'adage implicite selon lequel une grande partie de l'alg\`ebre commutative la plus abstraite se concentre dans des identit\'es alg\'ebriques concernant les matrices de polyn\^omes sur un anneau commutatif arbitraire.