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Noetherian operators, primary submodules and symbolic powers

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Abstract

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides with symbolic powers in many interesting non-smooth settings, and so it could serve as a generalization of the Zariski–Nagata Theorem.

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Acknowledgements

The author is grateful to Bernd Sturmfels for suggesting the initial problem that led to the preparation of this paper, for many helpful discussions, and for his contagious enthusiasm. The author thanks Łukasz Grabowski and Viktor Levandovskyy for useful conversations. The author thanks Greg Brumfiel for a helpful correspondence. The author is grateful to the referee for valuable comments and suggestions for the improvement of this work.

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  1. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Yairon Cid-Ruiz

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Cid-Ruiz, Y. Noetherian operators, primary submodules and symbolic powers. Collect. Math. 72, 175–202 (2021). https://doi.org/10.1007/s13348-020-00285-3

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