## 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).
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.

Zbl 0518.00005