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Tensor-multinomial sums of ideals: Primary decompositions and persistence of associated primes


Authors: Irena Swanson and Robert M. Walker
Journal: Proc. Amer. Math. Soc. 147 (2019), 5071-5082
MSC (2010): Primary 13C05; Secondary 13B22, 14B05
DOI: https://doi.org/10.1090/proc/14630
Published electronically: June 10, 2019
Uncorrected version: Original version posted June 10, 2019
Corrected version: Current version includes an additional grant acknowledgment.
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Abstract: Given a Noetherian tensor product of two Noetherian algebras over a field and proper ideals $ I$ and $ J$ in the two algebras, we determine the associated primes of each power of $ I+J$ in terms of the associated primes of lower powers of $ I$ and of $ J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $ I+J$ from primary decompositions for powers of $ I$ and $ J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $ I$ or $ J$ is a non-zero normal ideal.


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Additional Information

Irena Swanson
Affiliation: Department of Mathematics, Reed College, 3203 SE Woodstock Boulevard, Portland, Oregon 97202
Email: iswanson@reed.edu

Robert M. Walker
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: robmarsw@umich.edu

DOI: https://doi.org/10.1090/proc/14630
Received by editor(s): July 1, 2018
Received by editor(s) in revised form: March 5, 2019
Published electronically: June 10, 2019
Additional Notes: In the course of completing this work, the second author acknowledges support from NSF RTG grant DMS-0943832, NSF DMS-1501625, a 2017–18 Ford Foundation Dissertation Fellowship, and a 2018–19 Rackham Science Award from the Rackham Graduate School at UM-Ann Arbor.
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society