Abstract
This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.
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Acknowledgements
The research leading to this paper was initiated during the week-long workshop “Women in Commutative Algebra” (19w5104) which took place at the Banff International Research Station (BIRS). The authors would like to thank the organizers and acknowledge the hospitality of BIRS and the additional support provided by the National Science Foundation, DMS-1934391. For this work Liana Şega was supported in part by a grant from the Simons Foundation (#354594), and Susan Cooper and Sara Faridi were supported by Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are grateful to the two anonymous referees, Bernd Ulrich, Claudia Polini and Alexandra Seceleanu for comments that helped improved this paper. The computations for this project were done using the computer algebra program Macaulay2 [12]. For the last author, this material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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This paper is dedicated to Jürgen Herzog, whose interest in powers of ideals has been an inspiration to many, in honor of his \(80^{th}\) birthday.
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Cooper, S.M., El Khoury, S., Faridi, S. et al. Powers of graphs & applications to resolutions of powers of monomial ideals. Res Math Sci 9, 31 (2022). https://doi.org/10.1007/s40687-022-00324-4
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DOI: https://doi.org/10.1007/s40687-022-00324-4