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.