Abstract
Let \(R=\mathbb {K}[X_1, \ldots , X_n ]\) be a polynomial ring over a field \(\mathbb {K}\). We introduce an endomorphism \(\mathcal {F}^{[m]}: R \rightarrow R \) and denote the image of an ideal I of R via this endomorphism as \(I^{[m]}\) and call it to be the m-th square power of I. In this article, we study some homological invariants of \(I^{[m]}\) such as regularity, projective dimension, associated primes, and depth for some families of ideals, e.g., monomial ideals.
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References
Holger Brenner. A linear bound for Frobenius powers and an inclusion bound for tight closure. Michigan Math. J., 53(3):585–596, 2005.
Holger Brenner and Paul Monsky. Tight closure does not commute with localization. Ann. of Math. (2), 171(1):571–588, 2010.
M. Brodmann. Asymptotic stability of \({\rm Ass}(M/I^{n}M)\). Proc. Amer. Math. Soc., 74(1):16–18, 1979.
Winfried Bruns and Jürgen Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993.
Marc Chardin. Some results and questions on Castelnuovo-Mumford regularity. In Syzygies and Hilbert functions, volume 254 of Lect. Notes Pure Appl. Math., pages 1–40. Chapman & Hall/CRC, Boca Raton, FL, 2007.
S. Dale Cutkosky, Jürgen Herzog, and Ngô Viêt Trung. Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compositio Math., 118(3):243–261, 1999.
Alessandro De Stefani, Luis Núñez Betancourt, and Felipe Pérez. On the existence of \(F\)-thresholds and related limits. Trans. Amer. Math. Soc., 370(9):6629–6650, 2018.
A. V. Geramita, B. Harbourne, J. Migliore, and U. Nagel. Matroid configurations and symbolic powers of their ideals. Trans. Amer. Math. Soc., 369(10):7049–7066, 2017.
Isidoro Gitler, Carlos E. Valencia, and Rafael H. Villarreal. A note on Rees algebras and the MFMC property. Beiträge Algebra Geom., 48(1):141–150, 2007.
Jürgen Herzog and Takayuki Hibi. Monomial ideals, volume 260 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011.
Melvin Hochster and Craig Huneke. Comparison of symbolic and ordinary powers of ideals. Invent. Math., 147(2):349–369, 2002.
Mordechai Katzman. The complexity of Frobenius powers of ideals. J. Algebra, 203(1):211–225, 1998.
Vijay Kodiyalam. Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math. Soc., 128(2):407–411, 2000.
Jorge Neves, Pinto Maria Vaz, and Rafael H. Villarreal. Regularity and algebraic properties of certain lattice ideals. Bull. Braz. Math. Soc. (N.S.), 45(4):777–806, 2014.
Hop Dang Nguyen and Ngo Viet Trung. Depth functions of symbolic powers of homogeneous ideals. Invent. Math., 218(3):779–827, 2019.
Irena Peeva. Graded syzygies, volume 14 of Algebra and Applications. Springer-Verlag London, Ltd., London, 2011.
Acknowledgements
The authors are grateful to Prof. Sarang S. Sane, who initially suggested looking at the behavior of the projective dimension of square powers of monomial ideals from where this article grew up. The authors are thankful to Prof. A. V. Jayanthan for his valuable inputs on several drafts and fruitful conversations regarding making this article. The authors are also thankful to Prof. Rafael Villarreal for many helpful comments and suggestions. The first named author feels privileged and fortunate to get a research environment at the Indian Institute of Technology Madras. The second named author acknowledges support from India’s Sciences and Engineering Research Board (PDF/2020/001436).
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Communicated by Bakshi Gurmeet Kaur.
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Chanda, S., Kumar, A. Properties of analogues of Frobenius powers of ideals. Indian J Pure Appl Math 54, 524–531 (2023). https://doi.org/10.1007/s13226-022-00272-3
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DOI: https://doi.org/10.1007/s13226-022-00272-3