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LHS-spectral sequences for regular extensions of categories

Ergun Yalcin

Abstract

In [F. Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra 212 (2008), 2555-2569], Xu constructs a LHS-spectral sequence for target regular extensions of small categories. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Slominska's spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension.

LHS-spectral sequences for regular extensions of categories

Abstract

In [F. Xu, On the cohomology rings of small categories, J. Pure Appl. Algebra 212 (2008), 2555-2569], Xu constructs a LHS-spectral sequence for target regular extensions of small categories. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Slominska's spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension.
Paper Structure (36 sections, 71 theorems, 169 equations)

This paper contains 36 sections, 71 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Modules over small categories
  3. $R{\mathcal{C}}$-modules
  4. Restriction and induction functors
  5. Tensor products over $R{\mathcal{C}}$
  6. Coinduction functor
  7. Cohomology of small categories
  8. Ext-groups and cohomology of a small category
  9. Homology of a small category
  10. Classifying space of a category
  11. Functoriality and Shapiro's lemma
  12. Functoriality of the cohomology of a category
  13. Functoriality of the homology of a category
  14. Shapiro's isomorphism
  15. Gabriel-Zisman Spectral Sequence
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let ${\mathcal{E}} : {\mathcal{K}} {\,\mathop{\longrightarrow}\limits^{i}\,} {\mathcal{C}} {\,\mathop{\longrightarrow}\limits^{\pi}\,} {\mathcal{D}}$ be a target regular extension. Then for every $R{\mathcal{C}}$-module $M$, there is a spectral sequence where $H^q ({\mathcal{K}}; M)$ denotes the $R{\mathcal{D}}$-module that sends $x\in \mathrm{Ob} ({\mathcal{C}})$ to the cohomology group $H^q (K(

Theorems & Definitions (143)

  • Theorem 1.1: Xu Xu-CohSmall
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Sł omińska Slominska-Homotopy
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 133 more