##
**Smoothing of a ring homomorphism along a section.**
*(English)*
Zbl 0555.14002

Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 5-31 (1983).

[For the entire collection see Zbl 0518.00005.]

The paper under review deals with the following problem: given morphisms of affine schemes \(s: \tilde X\to Y,\) \(p: Y\to X\) with p finitely presented, find a morphism \(f: Z\to Y\) such that s factors through f, pf is smooth and f is ”smooth except above the singular locus of p”. This problem is motivated by work on approximation property. The authors solve the problem in two cases: \(\tilde X=X\) and in the isolated singularity case (i.e. \(\tilde X,\) X are spectra of henselian local rings with the same completion and p is smooth at every point of s(X) except the closed point). Meanwhile a complete answer to the above problem was given by M. Cipu and D. Popescu, ”A desingularisation theorem of Néron type” [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984)]; this paper in its turn is based on the papers: D. Popescu, ”General Néron desingularisation” [to appear in Nagoya Math. J. 100 (1985)] and D. Popescu, ”General Néron desingularisation and approximation” (to appear).

The paper under review deals with the following problem: given morphisms of affine schemes \(s: \tilde X\to Y,\) \(p: Y\to X\) with p finitely presented, find a morphism \(f: Z\to Y\) such that s factors through f, pf is smooth and f is ”smooth except above the singular locus of p”. This problem is motivated by work on approximation property. The authors solve the problem in two cases: \(\tilde X=X\) and in the isolated singularity case (i.e. \(\tilde X,\) X are spectra of henselian local rings with the same completion and p is smooth at every point of s(X) except the closed point). Meanwhile a complete answer to the above problem was given by M. Cipu and D. Popescu, ”A desingularisation theorem of Néron type” [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984)]; this paper in its turn is based on the papers: D. Popescu, ”General Néron desingularisation” [to appear in Nagoya Math. J. 100 (1985)] and D. Popescu, ”General Néron desingularisation and approximation” (to appear).

Reviewer: A.Buium

### MSC:

14B25 | Local structure of morphisms in algebraic geometry: étale, flat, etc. |

14B05 | Singularities in algebraic geometry |

14B12 | Local deformation theory, Artin approximation, etc. |