# Picard groups and the algebraic K-theory of cuspidal singularities.

## Seminar/Forum

107
Peter Hall
Hesselholt has a conjectural calculation of the algebraic K-theory of $$k[x,y]/(x^b-y^a)$$. It has remained a conjecture until now because nobody has been able to prove that a certain $$S^1$$-equivariant space that comes up in the calculation is built from representation spheres in a specified way. I will explain how to sidestep this issue by computing the Picard group of the category of p-complete $$C{p^n}$$-spectra. This lets us use homological data to recognize, up to p-completion, when a $$C{p^n}$$-spectrum looks like a virtual representation sphere.