Restriction of divisor classes to hypersurfaces in characteristic p

The injectivity of the restriction homomorphism on divisor class groups to hypersurfaces has been studied by Grothendieck, Danilov, Lipman, and Griffith & Weston, among others. In particular, when A is a Noetherian normal domain of equicharacteristic zero and A/fA satisfies R1, Spiroff established a map Cl(A)→Cl((A/fA)′), where (A/fA)′ represents the integral closure of A/fA, and gave some conditions for injectivity. In this paper, the authors continue in the same vein, but in the case of characteristic p>0. In addition, when the hypersurface A/fA is normal, they provide further enlightenment about the kernel of Cl(A)→Cl(A/fA). Finally, using the second authorʹs previous results, they exhibit a new class of examples for which the kernel is non-trivial.