EVENT DETAILS AND ABSTRACT

Title: Brauer groups of commutative ring spectra
Speaker: David Gepner
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Abstract:

The space of units GL_1(R) of a commutative ring spectrum R is the infinite loop space of a spectrum gl_1(R). Typically, this spectrum is taken to be connective, meaning it has no nonzero negative homotopy groups. However, there are other interesting deloopings of GL_1(R) which carry important algebraic information about R. One in particular has \pi_{-1} gl_1(R) = \pi_0 Pic(R), the Picard group of R, and \pi_{-2} gl_1(R) = \pi_0 Br(R), the Brauer group of R. If R is connective, there is a spectral sequence for computing the homotopy groups of Pic(R) and Br(R) in terms of the homotopy groups of R and the etale cohomology of the multiplicative group on Spec \pi_0 R. If S is a G-Galois extension of R, one obtains a Galois descent spectral sequence which calculates the Picard and Brauer groups of R relative to S.