Transfer and Norm for Finite Group Schemes

TL;DR

This paper extends classical transfer and norm concepts to finite group schemes, generalizing results from finite group theory to more complex algebraic structures.

Contribution

It introduces transfer and norm maps for finite group schemes, generalizing Higman's criterion and relating the norm to Mumford's and classical field norms.

Findings

Transfer map for modules and Ext groups characterizes relative projectivity.

The relative norm map coincides with Mumford's norm for finite morphisms.

On fields, the norm reduces to a power of the classical field norm.

Abstract

We develop the theory of transfer and norm maps for finite group schemes, extending classical results from finite group theory to a context where induction and restriction are not necessarily bi-adjoint. In the additive setting, we construct a transfer map for both modules and $\rm Ext $ groups and prove that its surjectivity characterizes relative projectivity, establishing a generalization of Higman's criterion. In the multiplicative setting, we define a relative norm map for algebras with a group scheme action. We compare this norm with other versions in the literature, proving that it coincides with Mumford's norm for finite morphisms and on fields is a power of the classical field norm.