In his announcement of the seminar Hiro mentioned that one can view Frobenius map in positive characteristic as a stable homotopy theory phenomena. In my talk I will explain what this statement really means and why it turns out to be a purely “spectral” phenomena which does not exist on the level of chain complexes. As an application, I will define topological Hochshild homology of commutative rings with the cyclotomic structure on it and hopefully explain its relevance to arithmetic algebraic geometry.