A characterization of rings over which Abelian Hopf algebras form an Abelian category (Q1117024)

Let K be a commutative ring and let \(H_ K\) denote the category of all abelian Hopf algebras over K, i.e. all strictly commutative, cocommutative, connected, graded Hopf K-algebras. It is well-known that \(H_ K\) is a Grothendieck category, whenever K is a field, and although \(H_ K\) is always additive, with colimits and generators, it is in general not abelian. In the present paper, the author proves that \(H_ K\) is abelian if and only if for any prime number p, the ring K/pK is a von Neumann regular ring and \(pK=p^ 2K\). In particular, \(H_ K\) is abelian if K is either a von Neumann regular ring or if K is a Q-algebra. Actually, in the latter case, \(H_ K\) is equivalent to the category of reduced graded K-modules. Note also that examples may be (and are) given of rings which are not of the previous two types but such that \(H_ K\) is still abelian.