A norm functor for quadratic algebras

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.

0 citations (Crossref) [2023-10-31]