The Norm Functor over Schemes

TL;DR

This paper generalizes Ferrand's norm functor from rings to schemes, constructing a global norm morphism for finite locally free morphisms and exploring its properties and applications in algebraic geometry.

Contribution

It introduces a scheme-level construction of the norm functor extending Ferrand's work, and develops a cohomological description and equivalence of algebra stacks.

Findings

Constructed a globalization of Ferrand's norm functor for schemes.

Established a cohomological description of the norm morphism for finite étale covers.

Proved an equivalence of stacks of algebras related to the norm.

Abstract

We construct a globalization of Ferrand's norm functor over rings which generalizes it to the setting of a finite locally free morphism of schemes $T\to S$ of constant rank. It sends quasi-coherent modules over $T$ to quasi-coherent modules over $S$. These functors restrict to the category of quasi-coherent algebras. We also assemble these functors into a norm morphism from the stack of quasi-coherent modules over a finite locally free of constant rank extension of the base scheme into the stack of quasi-coherent modules. This morphism also restricts to the analogous stacks of algebras. Restricting our attention to finite \'etale covers, we give a cohomological description of the norm morphism in terms of the Segre embedding. Using this cohomological description, we show that the norm gives an equivalence of stacks of algebras $A_1^2 \equiv D_2$, akin to the result shown in The Book of Involutions.