K(1)-local K-theory of Azumaya algebras

2025/09/01 by Ramzi, Maxime
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2509.01516

Abstract

We study the decategorification process that takes an Azumaya algebra to its K(1)-local K-theory. We prove various injectivity statements and relate precisely certain Brauer groups (more precisely, spectra) to certain K-theoretic strict Picard spectra or strict unit spectra. For example, we prove that for fields F of characteristic ≠ p, Br(F)[p^∞] ≃ \mathbbG_pic(L_K(1)K(F)⊗ \mathbbS_W(\mathbb F_p))[p^∞] where \mathbbG_pic is Carmeli's strict Picard spectrum, and Br(F) denotes the Brauer space of the field F.

Citations

• Localizing motives of Azumaya algebras
• The spectrum of units of algebraic K-theory
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• A note on secondary K-theory
• A note on secondary K-theory II
• Noncommutative motives of separable algebras
• Additive invariants of toric and twisted projective homogeneous\n varieties via noncommutative motives
• Motives of Azumaya algebras
• The Merkuriev-Suslin theorem for any semi-local ring

Related

• Localizing motives of Azumaya algebras
• The derived Picard group of an affine Azumaya algebra
• Brauer-Wall Groups and Truncated Picard Spectra of K-theory
• Derived Azumaya algebras and twisted K-theory
• Azumaya algebras over unramified extensions of function fields