Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE OHM-RUSH CONTENT FUNCTION III: COMPLETION, GLOBALIZATION, AND POWER-CONTENT ALGEBRAS

  • Epstein, Neil (Department of Mathematical Sciences George Mason University) ;
  • Shapiro, Jay (Department of Mathematical Sciences George Mason University)
  • Received : 2020.08.26
  • Accepted : 2021.03.10
  • Published : 2021.11.01
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Abstract

One says that a ring homomorphism R → S is Ohm-Rush if extension commutes with arbitrary intersection of ideals, or equivalently if for any element f ∈ S, there is a unique smallest ideal of R whose extension to S contains f, called the content of f. For Noetherian local rings, we analyze whether the completion map is Ohm-Rush. We show that the answer is typically 'yes' in dimension one, but 'no' in higher dimension, and in any case it coincides with the content map having good algebraic properties. We then analyze the question of when the Ohm-Rush property globalizes in faithfully flat modules and algebras over a 1-dimensional Noetherian domain, culminating both in a positive result and a counterexample. Finally, we introduce a notion that we show is strictly between the Ohm-Rush property and the weak content algebra property.

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Acknowledgement

We offer our thanks to the anonymous referee, whose careful reading and numerous comments improved the presentation of the paper.

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