Department of

February 2016 March 2016 April 2016 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 1 2 3 4 5 1 2 7 8 9 10 11 12 13 6 7 8 9 10 11 12 3 4 5 6 7 8 9 14 15 16 17 18 19 20 13 14 15 16 17 18 19 10 11 12 13 14 15 16 21 22 23 24 25 26 27 20 21 22 23 24 25 26 17 18 19 20 21 22 23 28 29 27 28 29 30 31 24 25 26 27 28 29 30

Tuesday, March 15, 2016

**Abstract:** Let $G$ be a finite $p$-group. The stable module category of $G$ is defined as the quotient of the category of $G$-representations over a field $k$ of characteristic $p$ by those morphisms which factor through a projective. It can also be modeled as the category of module spectra over the Tate construction $k^{tG}$. It is a classical theorem of Dade that the Picard group of the stable module category contains no "exotic" objects when $G$ is abelian. This translates into a statement about the $E_\infty$-ring spectrum $k^{tG}$. We will discuss a general approach to studying the Picard groups of structured ring spectra using descent theory and describe a new proof of Dade's theorem based on Rognes's theory of Galois extensions of ring spectra and Galois descent.