Purely inseparable extensions of unique factorization domains (Q1173967)

Let \(k\) be a field of characteristic \(p>0\), \(f_ 1,\ldots,f_{n-1}\) polynomials in the polynomial ring \(A:=k[x_ 1,\ldots,x_ n]\) and \(F\) the variety in \((2n-1)\)-dimensional affine space over \(k\) defined by the equations \(w_ i^ p-f_ i(x_ 1,\ldots,x_ n)\), \(i=1,\ldots,n-1\). The coordinate ring of \(F\) is then isomorphic to \(k[x^ p_ 1,\ldots,x^ p_ n,f_ 1,\ldots,f_{n-1}]\). The author shows under some additional technical assumption that the divisor class group \(\text{Cl}(F)\) is a finite \(p\)-group of type \((p,\ldots,p)\) and order \(p^ N\) with \(N<{M \choose n}\), \(M:=\sum^{n-1}_{i=1}\deg(f_ i)\). Assume furthermore that \(k\) is algebraically closed and that \(f_ 1,\ldots,f_{n-2}\) generate a height \(n-2\) prime ideal \(P\) in \(A\). Let \(W\) be the variety in \((n+1)\)-dimensional affine space defined by \(f_ 1,\ldots,f_{n-2}\), \(w^ p-f_{n-1}\). A similar theorem for the divisor class group \(\text{Cl}(W)\) is then proved under the condition that \(A/P\) is factorial and some more technical assumptions hold.