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Derived smooth induction with applications

Peter Schneider, Claus Sorensen

Abstract

In natural characteristic, smooth induction from an open subgroup does not always give an exact functor. In this article we initiate a study of the right derived functors, and we give applications to the non-existence of projective representations and duality.

Derived smooth induction with applications

Abstract

In natural characteristic, smooth induction from an open subgroup does not always give an exact functor. In this article we initiate a study of the right derived functors, and we give applications to the non-existence of projective representations and duality.
Paper Structure (14 sections, 29 theorems, 97 equations)

This paper contains 14 sections, 29 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. The derived functors of smooth induction
  3. The connection to higher smooth duality
  4. The top-dimensional derived functor
  5. The case of $p$-adic reductive groups
  6. Notation, conventions, and background
  7. Some uniform subgroups and their cohomology
  8. On the vanishing of certain restriction maps
  9. The proof of Theorem \ref{['introrange']}
  10. The proof of Theorem \ref{['intromain']}
  11. The derived functor $R\underline{\operatorname{Ind}}$
  12. Preliminary remarks
  13. Duality on compact objects
  14. The complex $R\underline{\operatorname{Ind}}(k)$ for reductive groups

Key Result

Theorem 1.1

Let $G={\bf{G}}(\mathfrak{F})$ for a nontrivial connected reductive group $\bf{G}$ defined over a finite extension $\mathfrak{F}/\Bbb{Q}_p$. Then $\operatorname{Mod}_k(G)$ has no nonzero projective objects.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 56 more