Smoothing of a Ring Homomorphism Along a Section

  • Michael Artin
  • Jan Denef
Part of the Progress in Mathematics book series (PM, volume 36)


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
  1. (i)

    α is smooth, and

  2. (ii)

    ø is smooth wherever possible — roughly speaking, except above the singular (nonsmooth) locus of π.



Commutative Diagram Local Ring Maximal Rank Ring Homomorphism Residue Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Michael Artin
    • 1
  • Jan Denef
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Departement WiskundeUniversiteit van LeuvenHeverleeBelgium

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