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Transfer and Norm for Finite Group Schemes

Kostas Karagiannis, Peter Symonds

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.

Transfer and Norm for Finite Group Schemes

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 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.

Paper Structure

This paper contains 22 sections, 23 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline
  4. Preliminaries
  5. Notation & Conventions
  6. Induction & Coinduction
  7. Integrals and Modular functions
  8. Transfer
  9. Definition of the transfer
  10. Properties of the transfer
  11. The Wirthmüller isomorphism
  12. Relative projectivity
  13. $\mathop{\mathrm{Tr}}\nolimits ^G_H \mathop{\mathrm{\mathbf{R}}}\nolimits^G_H$
  14. The double coset formula
  15. Cohomology
  16. ...and 7 more sections

Key Result

Proposition 3.2.1

Let $L,L',M',M,N',N$ be $G$-modules, and let $K,H$ be closed subgroup schemes of $G$ with $K\leq H$.

Theorems & Definitions (53)

  • Definition 2.3.1
  • Definition 3.1.1
  • Proposition 3.2.1
  • proof
  • Corollary 3.2.2
  • Theorem 3.3.1
  • Remark 3.3.2
  • Lemma 3.3.3
  • proof
  • Lemma 3.3.4
  • ...and 43 more