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Eventually periodic resolutions with applications to integral group rings

Sean P. Carroll

Abstract

We present a general construction of eventually periodic projective resolutions for modules over quotients of rings of finite left global dimension by a regular central element. Our approach utilizes a construction of Shamash, combined with the iterated mapping cone technique, to systematically 'purge' homology from a complex. The construction is applied specifically to the integral group rings of groups with finite virtual cohomological dimension. We demonstrate the computability of our method through explicit calculations for several families of groups including hyperbolic triangle groups and mapping class groups of the punctured plane.

Eventually periodic resolutions with applications to integral group rings

Abstract

We present a general construction of eventually periodic projective resolutions for modules over quotients of rings of finite left global dimension by a regular central element. Our approach utilizes a construction of Shamash, combined with the iterated mapping cone technique, to systematically 'purge' homology from a complex. The construction is applied specifically to the integral group rings of groups with finite virtual cohomological dimension. We demonstrate the computability of our method through explicit calculations for several families of groups including hyperbolic triangle groups and mapping class groups of the punctured plane.
Paper Structure (18 sections, 32 theorems, 111 equations, 1 figure, 1 table)

This paper contains 18 sections, 32 theorems, 111 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Notation and conventions
  5. Overview and preliminaries
  6. Chain maps to purge homology
  7. General construction
  8. Eventual periodicity
  9. Application to integral group rings
  10. Finiteness conditions and free differential calculus
  11. Eventually 2-periodic resolutions
  12. Explicit calculations
  13. Finite cyclic groups
  14. Amalgamated products of cyclic groups
  15. Central quotients of Heisenberg groups
  16. ...and 3 more sections

Key Result

Lemma 2.3

For any $N\in R{\operatorname{-Mod}}$ and any $i\ge 0$:

Figures (1)

  • Figure 1: Hyperbolic triangle $\tau\subset\mathbb{H}^{2}$

Theorems & Definitions (84)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • ...and 74 more