On Sweedler's cofree cocommutative coalgebra.
The cofree coalgebra \(C(V)\), together with a linear map \(p\) from \(C(V)\) to \(V\), for a vector space \(V\), satisfies the universal mapping property that for any coalgebra \(D\) and linear map \(f\) from \(D\) to \(V\), there is a unique coalgebra map \(F\) from \(D\) to \(C\) such that \(pF=f\). \(C(V)\) was constructed in \textit{M. E. Sweedler}'s book Hopf algebras [New York: W. A. Benjamin (1969; Zbl 0194.32901)] by first constructing \(C(V^{**})\) using the finite dual functor \(\circ\) on the tensor algebra \(T(V^*)\), where \(C(V^{**})=T(V^*)^\circ\). \(C(V)\) is then constructed using the embedding of \(V\) into \(V^{**}\). The cofree cocommutative coalgebra \(C'(V)\) is then constructed by taking the sum of all cocommutative subcoalgebras of \(C(V)\). In the paper under review, the author gives a proof of the existence of \(C'(V)\) which he calls a direct proof. He has some restrictions on the base field -- algebraically closed and characteristic zero. There are no restrictions in Sweedler's construction. His basic ingredients are local cohomology and residues. In an appendix, he redoes Sweedler's construction using his methods. Here there are no restrictions on the characteristic.
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