[Paper Review] Hermitian K-theory for stable $\\infty$-categories III: Grothendieck-Witt groups of rings

Baptiste Calmès, E. Dotto+7 more|arXiv (Cornell University)|Sep 15, 2020

Homotopy and Cohomology in Algebraic Topology43 references9 citations

TL;DR

This paper establishes a foundational fibre sequence relating the Grothendieck-Witt spectrum of a ring $ R $ to its $ m C_2 $-homotopy orbits of algebraic $ K $-theory and Ranicki's non-periodic symmetric L-theory, enabling the removal of the $ 2 $-invertibility assumption in classical results. The key contribution is an integral version of Karoubi's splitting, which allows computation of Grothendieck-Witt groups for rings like $ \mathbb{Z} $ and rings of integers in number fields, proving finite generation and equivalence results in high degrees.

ABSTRACT

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-d\\'evissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

Motivation & Objective

• To extend Karoubi's splitting of Grothendieck-Witt groups to the integral setting, without assuming 2 is invertible in the ring.
• To resolve the homotopy limit problem for Dedekind rings with fraction field a number field.
• To compute the higher Grothendieck-Witt groups of $ \mathbb{Z} $ and rings of integers in number fields.
• To prove that the comparison map from quadratic to symmetric Grothendieck-Witt theory is an equivalence in degrees $ \geq d+3 $ for coherent rings of global dimension $ d $.
• To verify a conjecture of Berrick-Karoubi on the structure of $ \mathrm{GW} $-theory via a hermitian localisation-dévissage sequence.

Proposed method

• Construct a fibre sequence of spectra: $ \mathrm{K}(R;M)_{\mathrm{h}C_2} \xrightarrow{\mathrm{hyp}} \mathrm{GW}_{\mathrm{cl}}^s(R;M) \to \mathrm{L}^{\mathrm{short}}(R;M) $, relating $ K $-theory, Grothendieck-Witt theory, and symmetric L-theory.
• Use Ranicki's algebraic surgery and $ \mathrm{C}_2 $-equivariant homotopy theory to define the L-theory spectrum $ \mathrm{L}^{\mathrm{short}}(R;M) $ as a connective spectrum with non-periodic homotopy groups.
• Establish a hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings, enabling computation of $ \mathrm{GW} $-groups via $ \mathrm{K} $- and $ \mathrm{L} $-theoretic data.
• Leverage the fibre sequence to deduce integral results by analyzing the homotopy groups of $ \mathrm{K}(R;M)_{\mathrm{h}C_2} $ and $ \mathrm{L}^{\mathrm{short}}(R;M) $, particularly for $ R = \mathbb{Z} $ and rings of integers.
• Apply the $ \mathrm{C}_2 $-action on $ \mathrm{K}(R;M) $ to recover the $ \mathrm{C}_2 $-invariant part of $ \mathrm{K} $-theory and Witt-theoretic summands after 2-inversion.
• Use the 2-local and $ p $-local structure of $ \mathrm{K} $- and $ \mathrm{L} $-groups to compute $ \mathrm{GW} $-groups of number rings via descent and comparison theorems.

Experimental results

Research questions

• RQ1 Can Karoubi's splitting of Grothendieck-Witt groups be extended to an integral, non-2-inverted setting?
• RQ2 What is the structure of the Grothendieck-Witt groups of $ \mathbb{Z} $ and rings of integers in number fields without inverting 2?
• RQ3 Is the comparison map from quadratic to symmetric Grothendieck-Witt theory an equivalence in high degrees for coherent rings of finite global dimension?
• RQ4 Does the hermitian localisation-dévissage sequence hold for Dedekind rings, and can it resolve the Berrick-Karoubi conjecture?
• RQ5 How do the homotopy groups of $ \mathrm{K}(R;M)_{\mathrm{h}C_2} $ and $ \mathrm{L}^{\mathrm{short}}(R;M) $ combine to determine $ \mathrm{GW} $-groups?

Key findings

• The fibre sequence $ \mathrm{K}(R;M)_{\mathrm{h}C_2} \to \mathrm{GW}_{\mathrm{cl}}^s(R;M) \to \mathrm{L}^{\mathrm{short}}(R;M) $ holds integrally for any ring $ R $, generalizing Karoubi's 2-inverted splitting.
• The Grothendieck-Witt groups of $ \mathbb{Z} $ are computed as $ \mathrm{GW}_1^{-\mathrm{gq}}(\mathbb{Z}) \cong \mathbb{Z}/4 $, $ \mathrm{GW}_3^{-\mathrm{gq}}(\mathbb{Z}) \cong \mathbb{Z}/24 $, and $ \mathrm{GW}_2^{-\mathrm{gq}}(\mathbb{Z}) \to \mathbb{Z} $ has finite cokernel.
• The homotopy limit problem is solved for Dedekind rings with fraction field a number field, confirming the convergence of the spectral sequence computing $ \mathrm{GW} $-groups.
• The comparison map $ \mathrm{GW}^q \to \mathrm{GW}^s $ is an equivalence in degrees $ \geq d+3 $ for coherent rings of global dimension $ d $, removing the need for 2-torsion assumptions.
• The $ \mathrm{C}_2 $-action on $ \mathrm{K}_n(\mathcal{O};\epsilon)[1/2] $ is determined, and the 2-inverted $ \mathrm{L} $-groups of rings of integers are free $ \mathbb{Z}[1/2] $-modules of rank equal to the number of real embeddings.
• The Grothendieck-Witt groups of rings of integers in number fields are finitely generated, as shown via the fibre sequence and finiteness of $ \mathrm{K} $- and $ \mathrm{L} $-groups.