P-rings and P-homomorphisms (Q796585)

Let \(\phi:\quad A\to B\) be a local flat homomorphism of local rings. When R is a property of a ring, call \(\phi\) a P-homomorphism if, for each prime ideal p of A, \(B\otimes_ AK\) is locally R for all finite extensions K of the associated field k(p). To answer a question of \textit{A. Grothendieck} [''Éléments de géométrie algébrique'' \(=EGA\), IV. 2 (1965; Zbl 0135.397)], the author seeks results of following type: if \(k\to B\otimes_ Ak\) (where \(k=residue\quad field\quad of\quad A)\) is a P-homomorphism and if, for all prime ideals p of A, \(A_ p\to \hat A_ p\) (completion) is a P-homomorphism then \(\phi\) is a P-homomorphism. Proofs are given: when \(char k=0\) for general nice R; also, in arbitrary characteristic, for \(R=the\) complete intersection property (assuming the residue field of B has finite inseparable multiplicity over k) and for \(R=Cohen\)-Macaulay (assuming there exists a finitely generated Cohen- Macaulay A-module whose Supp is Spec A). The author also discusses a related problem on lifting properties from A/\({\mathfrak I}\) to A where A is a complete Zariski ring with respect to \({\mathfrak I}\).