Abstract
This paper provides a survey of results on regularity, Betti numbers and reduction numbers, showing their stability in passing from an ideal to its initial ideal if the latter is of Borel type. Similar results happen when one passes from an ideal to its generic initial ideal. We also present analogous results on extremal Betti numbers.
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Aramova, A., Herzog, J., Hibi, T.: Ideals with stable Betti numbers. Adv. Math 152, 72–77 (2000)
Bayer, D., Charalambous, H., Popescu, S.: Extremal Betti numbers and applications to monomial ideals. J. Algebra 221, 497–512 (1999)
Bayer, D., Stillman, M.: A criterion for detecting m-regularity. Invent. Math 87(1), 1–11 (1987)
Bermejo, I., Gimenez, P.: Saturation and Castelmuovo-Mumford regularity. J. Algebra 303, 592–617 (2006)
Bresinsky, H., Hoa, L.T.: On the reduction number of some graded algebras. Proc. Am. Math. Soc. 127, 1257–1263 (1999)
Brienza, F.: Invariants regarding ideals with special properties.Ph.D. thesis,University of L’Aquila (2014)
Bruns, W., Herzog, J.: Cohen Macaulay rings, vol. 59. Cambridge University Press (1993)
Conca, A.: Betti Numbers and Generic Initial Ideals (Lecture, 2),. International School on Computer Algebra:CoCoA 2007 RISC Hangenberg-Linz (Austria) (2007)
Conca, A.: Koszul homology and extremal properties of G i n and L e x. Trans. Am. Math. Soc. 356, 2945–2961 (2004)
Conca, A.: Reduction numbers and initial ideals. Proc. Am. Math. Soc 131, 1015–1020 (2003)
Conca, A., Herzog, J., Hibi, T.: Rigid resolutions and big Betti numbers. Comment. Math. Helv. 79(4), 826–839 (2004)
Hashemi, A., Schweinfurter, M., Seiler, W.M.: Quasi-stability versus Genericity. CASC, Springer 7442, 172–184 (2012)
Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer (2010)
Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Proc. Camb. Philos. Soc. 50, 145–158 (1954)
Seiler, W.: A combinatorial approach to involution and δ−regularity II: Structure analysis of polynomial modules with Pommaret basis. Appl. Alg. Eng. Comm. Comp. 20, 261–338 (2009)
Trung, N.: Gröbner bases, local cohomology and reduction number. Proc. Am. Math. Soc. 129(1), 9–18 (2001)
Trung, N.: Reduction exponent and degree bounds for the defining equations of a graded ring. Proc. Am. Math. Soc. 350, 2813–2832 (1998)
Trung, N.: The Castelnuovo regularity of the Rees algebra and the associated graded ring. Trans. Am. Math. Soc. 350, 2813–2832 (1998)
Trung, N.: The largest non-vanshing degree of graded local cohomology modules. J. Algebra 215, 481–499 (1999)
Trung, N.: Constructive characterization of the reduction numbers. Compos. Math 137, 99–113 (2003)
Vasconcelos, V.W.: The degrees of graded modules. Proceedings of Summer School on Commutative Algebra, Bellaterra 1996, vol. II, CRM Publication, 141–196
Acknowledgments
We thank ProfessorMaria Evelina Rossi for her support and guidance in investigating the subject and the Department of Mathematics of University of Genova for hospitality.
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Dedicated to Professor Ngo Viet Trung on the occasion of his sixtieth birthday
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Brienza, F., Guerrieri, A. Initial Ideals of Borel Type. Acta Math Vietnam 40, 453–462 (2015). https://doi.org/10.1007/s40306-015-0112-5
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DOI: https://doi.org/10.1007/s40306-015-0112-5