Invariant hypersurfaces for derivations in positive characteristic (Q886223)

Let \(k\) be an algebraically closed field of characteristic \(p>0,\) \(A\) an integral \(k\)-algebra of finite type, \(Q(A)\) the field of fractions of \(A\), \(\text{ Der}_k(A)\) the set of the \(k\)-derivations of \(A.\) If \({\mathcal D}\) is a subset of \(\text{ Der}_k(A)\), an element \(f\in Q(A)^*\) is an algebraic solution of \({\mathcal D}\), if \(D(f)/f\in A, \forall D\in{\mathcal D}.\) The algebraic solutions form a multiplicative group and the first integrals of \({\mathcal D}\) a subgroup. Denote by \(\Pi(A,{\mathcal D})\) the quotient group. The author proves that \(\dim_{\mathbb{F}_p}(\Pi(A,{\mathcal D}))<\infty\) and he also gives an estimation for this dimension. As application, he shows that if \(B\) is a \(k-\)algebra between \(A\) and \(A^p\), the kernel of the pull-back morphism \(\text{ Pic}(B)\rightarrow\text{ Pic}(A)\) has finite dimension as an \(\mathbb{F}_p\)-vector space. As another application, he computes the Picard group in a special case.