The Carlitz algebras

Abstract

The Carlitz F q -algebra C = C ν , ν ∈ N , is generated over an algebraically closed field K (which contains a non-discrete locally compact field of positive characteristic p > 0 , i.e. K ≃ F q [ [ x , x − 1 ] ] , q = p ν ), by the (power of the) Frobenius map X = X ν : f ↦ f q , and by the Carlitz derivative Y = Y ν . It is proved that the Krull and global dimensions of C are 2, classifications of simple C -modules and ideals are given, there are only countably many ideals, they commute ( I J = J I ) , and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple C -module is a sum of eigenspaces of the element Y X (the set of eigenvalues for Y X is given explicitly for each simple C -module). This fact is crucial in finding the group Aut F q ( C ) of F q -algebra automorphisms of C and in proving that any two distinct Carlitz rings are not isomorphic ( C ν ≄ C μ if ν ≠ μ ). The centre of C is found explicitly, it is a UFD that contains countably many elements.