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Computing homological residue fields in algebra and topology

Paul Balmer, James C. Cameron

Abstract

We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.

Computing homological residue fields in algebra and topology

Abstract

We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.

Paper Structure

This paper contains 4 sections, 14 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Background on homological residue fields
  3. The pure-injective $E_{\mathscr{B}}$ in examples
  4. Homological residue fields as comodules

Key Result

Theorem 1

Let $F\colon \mathscr{T} \to \mathscr{F}$ be a monoidal exact functor to a tensor-triangulated field and suppose that $F$ admits a right adjoint $U$. Then the pure-injective object $E_{\mathscr{B}} \in \mathscr{T}$ is a direct summand of $U(\mathbb{1})$.

Theorems & Definitions (33)

  • Theorem : \ref{['thm:EB<U(1)']}
  • Theorem : \ref{['thm:AB-as-comodules']}
  • Lemma 2.2
  • proof
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 23 more