Abstract
Let \({\varphi : A \rightarrow B}\) be a flat morphism of Artin local rings with the same embedding dimension. Denote by \({\mathfrak{m}_A}\) the maximal ideal of A. Bart de Smit asked whether any finite B-module that is A-flat is B-flat. We prove the conjecture in embedding dimension one or two. In embedding dimension n, we prove the conjecture under an additional assumption on \({B/\mathfrak{m}_{A}B}\).
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Acknowledgments
We would like to thank Bart de Smit for raising his interesting conjecture to our attention. We also thank the referee for reading very carefully our paper and for numerous valuable suggestions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Brochard, S., Mézard, A. About de Smit’s question on flatness. Math. Z. 267, 385–401 (2011). https://doi.org/10.1007/s00209-009-0624-6
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DOI: https://doi.org/10.1007/s00209-009-0624-6