Counterexamples regarding Symmetric Tensors and Divided Powers

Abstract

We investigate the similarities and differences between the module of symmetric tensors TS^n_A(M) and the module of divided powers Γ^n_A(M). There is a canonical map Γ^n_A(M) \to TS^n_A(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 TS_A^n does not in general commute with base change.