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On the Homological Stability of Orthogonal and Spin Groups

Marco Schlichting, Sunny Sood

TL;DR

This work establishes improved homological stability ranges for orthogonal and related groups over commutative local rings with infinite residue fields (and invertible 2). By constructing a chain complex of totally isotropic unimodular sequences and employing localisation of hyperhomology spectral sequences, the authors extend stability ranges for $O_{n,n}$, $SO_{n,n}$, $EO_{n,n}$ and $Spin_{n,n}$. Key ideas include transitivity of group actions on unimodular sequences, explicit stabiliser computations, local $R^{*}$-actions, and the gluing of local actions into global spectral sequence actions, followed by a precise analysis of the localized $d^{1}$ differentials. The results yield isomorphisms $H_k(G_{n})\to H_k(G_{n+1})$ for $k\le n-1$ and surjectivity for $k\le n$ across all four families, with Spin stability obtained via Hochschild-Serre arguments. These findings generalise and sharpen previous stability ranges, connecting to Hermitian K-Theory and providing comprehensive stability results for Spin and EO groups over local rings.

Abstract

We improve homological stability ranges for the orthogonal group, special orthogonal group, elementary orthogonal group and the spin group over a commutative local ring $R$ with infinite residue field such that $2 \in R^{*}$.

On the Homological Stability of Orthogonal and Spin Groups

TL;DR

This work establishes improved homological stability ranges for orthogonal and related groups over commutative local rings with infinite residue fields (and invertible 2). By constructing a chain complex of totally isotropic unimodular sequences and employing localisation of hyperhomology spectral sequences, the authors extend stability ranges for , , and . Key ideas include transitivity of group actions on unimodular sequences, explicit stabiliser computations, local -actions, and the gluing of local actions into global spectral sequence actions, followed by a precise analysis of the localized differentials. The results yield isomorphisms for and surjectivity for across all four families, with Spin stability obtained via Hochschild-Serre arguments. These findings generalise and sharpen previous stability ranges, connecting to Hermitian K-Theory and providing comprehensive stability results for Spin and EO groups over local rings.

Abstract

We improve homological stability ranges for the orthogonal group, special orthogonal group, elementary orthogonal group and the spin group over a commutative local ring with infinite residue field such that .
Paper Structure (28 sections, 65 theorems, 224 equations)

This paper contains 28 sections, 65 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The complex of totally isotropic unimodular sequences
  4. The chain complex
  5. Proving acyclicity
  6. Homological stability for $O_{n,n}$
  7. Transitivity of the group action
  8. Analysis of stabilisers
  9. Computation of stabilisers
  10. The local $R^*$-action
  11. A global action on the spectral sequence
  12. Computation of the localised $d^{1}$ differentials, and proof of homological stability
  13. Homological stability for $SO_{n,n}$
  14. Local $R^*$-actions and transitivity
  15. Computation of $\overline{T}_{n}$ and a local $R^{*}$-action
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $R$ be a commutative local ring with infinite residue field such that $2 \in R^{*}$. Then, the natural homomorphism is an isomorphism for $k \leq n-1$ and surjective for $k \leq n$.

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 141 more