A norm functor for quadratic algebras

Abstract

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