A Norm Functor for Quadratic Algebras

TL;DR

This paper introduces a new norm functor for quadratic algebras over ring extensions, generalizing existing constructions and exploring its relation to discriminant algebras.

Contribution

It constructs a norm functor for quadratic algebras over ring extensions, extending known cases and proposing a conjecture relating it to discriminant algebras.

Findings

Constructed a norm functor for quadratic algebras over ring extensions.

Extended the norm construction to non-étale quadratic algebras.

Proposed a conjectural relationship between discriminant algebras and the new norm functor.

Abstract

Given commutative, unital rings $A$ and $B$ with a ring homomorphism $A\to B$ making $B$ free of finite rank as an $A$-module, we can ask for a "trace" or "norm" homomorphism taking algebraic data over $B$ to algebraic data over $A$. In this paper we we construct a norm functor for the data of a quadratic algebra: given a locally-free rank-$2$ $B$-algebra $D$, we produce a locally-free rank-$2$ $A$-algebra $\mathrm{Nm}_{B/A}(D)$ in a way that is compatible with other norm functors and which extends a known construction for \'etale quadratic algebras. We also conjecture a relationship between discriminant algebras and this new norm functor.