Purely inseparable extensions of complete intersections (Q1311662)

Let \(R\) be a homogeneous complete intersection ring of dimension \(\geq 2\) over an algebraically closed field of characteristic \(p \neq 0\). Assume that \(R\) is a unique factorization domain. Let \(h \in R\) be a product of \(q\) distinct homogeneous irreducible elements of \(R\) with \(\deg (h) \not\equiv 0 \bmod p\). Put \(S = R[z]/(z^ m - h)\). Continuing his previous works, the author proves that if \(m\) is a \(p\)-th power, then \(cl(S) \cong \bigoplus^{q-1}_{i=1} \mathbb{Z}/m \mathbb{Z}\).