Abstract
Let \(R=K[x_1,\ldots ,x_n]\) be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog (Eur. J. Combin. 34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) \({\text {height}}(I)=n-1\); (ii) I contains at least \(n-3\) pure powers of the variables \(x_1^d,\ldots ,x_{n-3}^d\); (iii) I is a monomial ideal in at most four variables.
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References
Bandari S and Herzog J, Monomial localizations and polymatroidal ideals, Eur. J. Combin. 34 (2013) 752–763
Conca A, Regularity jumps for powers of ideals, in: Commutative Algebra, Lecture Notes Pure Appl. Math., vol. 244 (2006) (Boca Raton, FL: Chapman & Hall/CRC) pp. 21–32, https://doi.org/10.1201/9781420028324.ch3
Conca A and Herzog J, Castelnuovo–Mumford regularity of products of ideals, Collect. Math. 54 (2003) 137–152
Eisenbud D, Commutative Algebra with a View Towards Algebraic Geometry, GTM., vol. 150 (1995) (Berlin: Springer)
Grayson D R and Stillman M E, Macaulay 2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/
Herzog J and Hibi T, Discrete polymatroids, J. Algebraic Combin. 16 (2002) 239–268
Herzog J and Hibi T, Monomial ideals, GTM., vol. 260 (2011) (Berlin: Springer)
Herzog J, Hibi T and Zheng X, Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004) 23–32
Herzog J, Rauf A and Vladoiu M, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebraic Combin. 37 (2013) 289–312
Herzog J and Takayama Y, Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002) 277–294
Herzog J and Vladoiu M, Monomial ideals with primary components given by powers of monomial prime ideals, Electron. J. Combin. 21 (2014) 69
Karimi Sh. and Mafi A, On stability properties of powers of polymatroidal ideals, Collect. Math. 70 (2019) 357–365
Sturmfels B, Four counterexamples in combinatorial algebraic geometry, J. Algebra 230 (2000) 282–294
Trung T N, Stability of associated primes of integral closures of monomial ideals, J. Comb. Theory Ser. A 116 (2009) 44–54
Acknowledgements
The authors would like to thank the referee for a careful reading of the manuscript and for providing helpful suggestions. The work of the second author has been supported financially by Vice-Chancellorship of Research and Technology, University of Kurdistan under research Project No. 99/11/19299.
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Communicating Editor: Mrinal Kanti Das
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Mafi, A., Naderi, D. Linear resolutions and polymatroidal ideals. Proc Math Sci 131, 25 (2021). https://doi.org/10.1007/s12044-021-00620-z
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DOI: https://doi.org/10.1007/s12044-021-00620-z