Brauer group functor

 algebra
Notation Br⁡:Fld→Ab\Br: \Fld \to \Ab Domain category of fieldsCodomain category of abelian groups External nLab link
The Brauer group Br⁡(K)\Br(K) of a field KK consists of equivalence classes of central simple algebras over KK, where A∼BA \sim B iff A⊗KMn(K)≅B⊗KMm(K)A \otimes_K M_n(K) \cong B \otimes_K M_m(K) for some n,m≥1n,m \geq 1. The group structure is given by [A]⋅[B]≔[A⊗KB][A] \cdot [B] \coloneqq [A \otimes_K B], 1≔[K]1 \coloneqq [K] and [A]−1≔[Aop][A]^{-1} \coloneqq [A^{\op}]. A homomorphism K→LK \to L induces the homomorphism Br⁡(K)→Br⁡(L)\Br(K) \to \Br(L) defined by [A]↦[A⊗KL][A] \mapsto [A \otimes_K L].

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Comments

  • For the image of the functor Br⁡:Fld→Ab\Br : \Fld \to \Ab, see MO/130778.