[Paper Review] A Horrocks-Type Theorem for Even Orthogonal $\mathrm{K}_2$

Andrei Lavrenov, Sergey Sinchuk|arXiv (Cornell University)|Sep 5, 2019

Advanced Operator Algebra Research35 references5 citations

TL;DR

This paper establishes a Horrocks-type theorem for the unstable even orthogonal K2-functor, proving that a certain commutative square involving polynomial and Laurent polynomial extensions is a pullback. The proof relies on a local-global principle and a novel decomposition of Steinberg groups over graded rings, extending earlier results on linear K2 to orthogonal groups and confirming the K2-analogue of Serre’s problem for type Dℓ in the presence of 2-torsion invertibility.

ABSTRACT

We prove the Horrocks theorem for unstable even-dimensional orthogonal Steinberg groups. The Horrocks theorem for Steinberg groups is one of the principal ingredients needed for the proof of the $\mathrm{K}_2$-analogue of Serre's problem, whose positive solution is currently known only in the linear case.

Motivation & Objective

• To establish a K2-analogue of Serre’s problem for even-dimensional orthogonal groups, extending the Quillen–Suslin local-global principle to K2-functors.
• To prove that the unstable orthogonal K2-functor KO2(2ℓ, −) satisfies a pullback property over polynomial and Laurent polynomial extensions.
• To generalize the Horrocks theorem to Steinberg groups of type Dℓ, filling a gap in the K2-analogue of Serre’s problem for non-linear Chevalley groups.
• To resolve the injectivity of key homomorphisms in the relative K-theory framework using central extensions and stability theorems.
• To provide a foundational step toward proving the K2-analogue of Serre’s problem for all root systems of rank ≥3, particularly for orthogonal types.

Proposed method

• Reduces the main theorem to proving injectivity of the homomorphism j−: St(Dℓ, A[X−1]) → St(Dℓ, A[X, X−1]) via the local-global principle.
• Applies the local-global principle to reduce the problem to the case where A is a local ring, simplifying the structure of the ring and its ideals.
• Uses relative central extensions from Loday’s theory to analyze the kernel of the map St(Dℓ, B) → St(Dℓ, R), where B = A[X−1] + M[X] and R = A[X, X−1].
• Establishes the surjectivity of a map between relative central extension subgroups C(Dℓ, B, M[X, X−1]) → C(Dℓ, R, M[X, X−1]) using Panin’s stability theorem and the Bass Fundamental Theorem.
• Employs a decomposition of the Steinberg group St(Φ, B) as a quotient of a product of three groups, one of which is St(Φ, A[X−1]), generalizing a method from [38].
• Applies a presentation theorem for Steinberg groups over graded rings, inspired by Rehmann, Soulé, and Tulenbaev, to control the group structure in the decomposition.

Experimental results

Research questions

• RQ1 Does the unstable even orthogonal K2-functor KO2(2ℓ, −) satisfy a Horrocks-type pullback property over polynomial and Laurent polynomial extensions?
• RQ2 Can the injectivity of the homomorphism j−: St(Dℓ, A[X−1]) → St(Dℓ, A[X, X−1]) be established under the assumption that 2 is invertible in A?
• RQ3 Is the local-global principle applicable to the K2-functor of even orthogonal groups, enabling reduction to the local ring case?
• RQ4 Can the relative central extension structure of Steinberg groups be used to prove injectivity of maps between groups over Laurent extensions?
• RQ5 Does the decomposition of St(Φ, B) as a quotient of a product of groups allow for a systematic proof of injectivity in the context of orthogonal K2?

Key findings

• The paper proves that the commutative square involving KO2(2ℓ, A), KO2(2ℓ, A[X]), KO2(2ℓ, A[X−1]), and KO2(2ℓ, A[X, X−1]) is a pullback with all maps injective, for ℓ ≥ 7 and 2 invertible in A.
• The same result holds for the Steinberg group St(Dℓ, −) and the K2-functor K2(Dℓ, −), establishing the K2-analogue of the Horrocks theorem for orthogonal groups.
• The injectivity of j−: St(Dℓ, A[X−1]) → St(Dℓ, A[X, X−1]) is established via a two-step argument: first proving injectivity of jR: St(Dℓ, B) → St(Dℓ, R), then of j−B: St(Dℓ, A[X−1]) → St(Dℓ, B).
• The surjectivity of the map C(Dℓ, B, M[X, X−1]) → C(Dℓ, R, M[X, X−1]) is obtained as a consequence of Panin’s stability theorem and the Bass Fundamental Theorem for higher Grothendieck–Witt groups.
• The decomposition of St(Φ, B) as a quotient of a product of three groups, one being St(Φ, A[X−1]), enables the proof of injectivity of j−B for root systems of type Aℓ (ℓ ≥ 4), Dℓ (ℓ ≥ 5), and E6,7,8.
• The result confirms a key ingredient for the K2-analogue of Serre’s problem in the orthogonal case, particularly for groups of type Dℓ with ℓ ≥ 7, under the assumption that 2 is invertible in the base ring.