Abstract
The first note gives some characterization offlat morphisms by (generalized) Hilbert functions. The second note yields a geometric description of tangential flatness, i.e. the flatness of the induced morphism in the tangent cones. The appendix contains a graded version of the local criterion of flatness as well as some consequences which are used in the main part of the paper.
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Herrmann, M., Orbanz, U. Two notes on flatness. Manuscripta Math 40, 109–133 (1982). https://doi.org/10.1007/BF01168239
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DOI: https://doi.org/10.1007/BF01168239