The Carlitz shtuka

Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed a finite module analogous to the class group of a number field. In this short note more algebraic constructions of these "unit" and "class" modules are given and they are related to Ext modules in the category of shtukas.