Counterexamples regarding symmetric tensors and divided powers

We investigate the similarities and differences between the module of symmetric tensors TSAn(M) and the module of divided powers ΓAn(M). There is a canonical map ΓAn(M)→TSAn(M) which is an isomorphism in many important cases. We give examples showing that this map need neither be surjective nor injective in general. These examples also show that the functor TSAn does not in general commute with base change.