Part of the Progress in Mathematics book series (PM, volume 36)
Smoothing of a Ring Homomorphism Along a Section
This paper studies the problem of smoothing a homomorphis.n of commutative rings along a section. The data needed to pose the problem make up a commutative diagram of al:fine schemes, such that Y is finitely presented over X. Our standard notation is that X, X, Y are the spectra of A, Ā, B respectively, and that B is a finitely presented A-algebra. (In the body of the text, we work primarily with the rings rather than with their spectra. This reverses the arrows.) The problem is to embed the commutative diagram (0.1) into a larger one, such that
α is smooth, and
ø is smooth wherever possible — roughly speaking, except above the singular (nonsmooth) locus of π.
KeywordsCommutative Diagram Local Ring Maximal Rank Ring Homomorphism Residue Field
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