Centrality of ${\mathrm{K}}_2$-functor revisited

Abstract

We prove that $\mathrm{St}(n,A)$ is a crossed module over $\mathrm{GL}(n,A)$ under a local stable rank condition on an algebra $A$ over a commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and stability results for ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$. We also prove similar result for the Steinberg group associated with any sufficiently isotropic general linear group constructed by a quasi-finite algebra.

arXiv commentI corrected the statement of proposition 3 and some minor misprints

References in corpus (2)

• On centrality of $K_2$ for Chevalley groups of type $E_l$
• On centrality of even orthogonal ${\mathrm{K}}_2$

Cited by in corpus (7)

• Explicit presentation of relative Steinberg groups
• Centrality of ${\mathrm{K}}_2$ for Chevalley groups: a pro-group approach
• Centrality of odd unitary $K_2$-functor
• Groups with ${\mathsf{A}}_{\ell }$-commutator relations
• Locally isotropic elementary groups
• Another presentation of orthogonal Steinberg groups
• A presentation of relative unitary Steinberg groups