Expository Notes

• Fun with Traces In this note, we answer a question of Maxime Ramzi about the image of the Hattori-Stallings trace.
• Cyclotomic Idempotents, a brief note proving that the idempotent algebras in (genuine) cyclotomic spectra are in bijection with the idempotent algebras of spectra, giving the smashing spectrum of cyclotomic spectra. We also discuss the smashing spectra of modules over an E_{infty} algebra in cyclotomic spectra.
• Pro-etale stacks, notes to accompany a talk I gave on how to use pro-etale descent to classify certain algebraic groups over an arbitrary field, such as 1-dimensional tori.