Embedding of the derived Brauer group into the secondary \(K\)-theory ring
type: Item
instance of: scholarly article
title: Embedding of the derived Brauer group into the secondary \(K\)-theory ring
zbMATH Open document ID: 1472.16020
DOI: 10.4171/JNCG/379
author: Gonçalo Tabuada
published in: Journal of Noncommutative Geometry
publication date: 2021-06-14T00:00:00Z
review text: Summary: In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree \(\geq 4\) as well as in the case of Mumford's (famous) singular surface.
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 19C30
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 7358356
zbMATH Keywords: derived Brauer group
zbMATH Keywords: secondary \(K\)-theory
zbMATH Keywords: noncommutative motives
zbMATH Keywords: noncommutative algebraic geometry
MaRDI profile type: Publication
cites work: Brauer groups and cohomology
cites work: Q5693333
cites work: Q4106660
cites work: On the Brauer Group of Toric Varieties
cites work: Q4011049
cites work: Some theorems on azumaya algebras
cites work: Q5587673
cites work: Q5489372
cites work: Notes on Motives in Finite Characteristic
cites work: Uniqueness of enhancement for triangulated categories
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: Q3871006
cites work: Ketu and the second invariant of a quadratic space
cites work: Recent developments on noncommutative motives
cites work: A note on secondary \(K\)-theory
cites work: Noncommutative Motives
cites work: Noncommutative motives in positive characteristic and their applications
cites work: Noncommutative motives of separable algebras
cites work: Noncommutative motives of Azumaya algebras
cites work: Derived Azumaya algebras and generators for twisted derived categories
cites work: Chern characters, equivariant traces and derived algebraic geometry
full work available at URL: https://doi.org/10.4171/jncg/379
OpenAlex ID: W3045395916
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description: scientific article; zbMATH DE number 7358356