Algebra/Topology Seminar

Speaker: Shahar Carmeli

Title: The strict units of the sphere spectrum

Abstract: For a commutative ring spectrum R, there are two candidates for the "multiplicative group" of R. One is the spectrum of units in R, denoted gl_1(R), and the other is the spectrum of "strict units" in R, denoted G_m(R).

The spectrum gl_1(R) is closely related to R itself. For example, the homotopy groups of R and gl_1(R) agree in all degrees above 0. On the other hand, the spectrum G_m(R) is a more subtle object and the subject of active research.

In my talk, I will present a computation of the spectrum G_m(S) of strict units of the sphere spectrum, as well as the connective spectrum of maps from Z to the Picard spectrum of S, which gives a non-trivial delooping of G_m(S). I will then discuss extensions of these computations to other commutative ring spectra such as the spherical Witt vectors associated with a perfect ring of characteristic p.