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Uniform bounds on symbolic powers in regular rings
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We prove a uniform bound on the growth of symbolic powers of arbitrary (not necessarily radical) ideals in arbitrary (not necessarily excellent) regular rings of all characteristics. This gives a complete answer to a question of Hochster and Huneke. In equal characteristic, this result was proved by Ein, Lazarsfeld, and Smith and by Hochster and Huneke. For radical ideals in excellent regular rings of mixed characteristic, this result was proved by Ma and Schwede. We also prove a generalization of these results involving products of various symbolic powers and a uniform bound for regular local rings related to a conjecture of Eisenbud and Mazur, which are completely new in mixed characteristic. In equal characteristic, these results are due to Johnson and to Hochster-Huneke, Takagi-Yoshida, and Johnson, respectively.
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