nLab Brauer ∞-group

Idea

The refinement of the concept of Brauer group from algebra to higher algebra and stable homotopy theory.

Definition

(Szymik 11, def. 3.8, prop. 5.5)

Properties

Relation to Picard $\infty$-group and $\infty$-group of units

Given an E-∞ ring $E$, the looping of the Brauer $\infty$-group is the Picard ∞-group (Szymik 11, theorem 5.7).

$$\Omega Br(E) \simeq Pic(E).$$

The looping of that is the ∞-group of units (Sagave 11, theorem 1.2).

$$\Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E) \,.$$

References

• Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)

• Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)

• Steffen Sagave, Spectra of units for periodic ring spectra (arXiv:1111.6731)