Type Construction and Cycle Detection

We introduce $F$-gauges over a prism, construct syntomic cycle classes, and prove the prismatic Poincaré duality for proper smooth schemes.

We show that almost all permutations have some power that is a cycle of prime length. The proof includes a theorem giving a strong upper bound on the proportion of elements of the symmetric group having no cycles with length in a given set.

This is the second part of two parts, titled " cone construction". In this part we prove the Lefschetz cohomologicity of the cone operator $Con$.