Abstract
This paper is an overview of several cohomological extensions of the ordinary multiplicity function of local algebra. It emphasizes the construction of such functions and the development of their main properties. A select set of applications is used to illustrate their usefulness.
MSC 2010: Primary: 13H10, 13H15. Secondary: 13D02, 13D07.
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References
Bayer, D., Mumford, D.: What can be computed in Algebraic Geometry? In: Eisenbud, D., Robbiano, L. (eds.) Computational Algebraic Geometry and Commutative Algebra. Proceedings, Cortona 1991, pp. 1–48. Cambridge University Press, Cambridge (1993)
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1993)
Chardin, M., Ha, D.T., Hoa, L.T.: Castelnuovo–Mumford regularity of Ext modules and homological degree. Trans. Amer. Math. Soc. 363, 3439–3456 (2011). arXiv: math.AC/0903.4535
Dalili, K.: On the number of generators of modules of homomorphisms. J. Algebra 311, 463–491 (2007)
Dalili, K., Vasconcelos, W.V.: Cohomological degrees and the HomAB conjecture. In: Corso, A., Migliore, J., Polini, C. (eds.) Algebra, Geometry and Their Interactions. Contemporary Mathematics, vol. 448, pp. 43–61 (2007)
Doering, L.R., Gunston, T., Vasconcelos, W.V.: Cohomological degrees and Hilbert functions of graded modules. American J. Math. 120, 493–504 (1998)
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)
Gulliksen, T., Levin, G.: Homology of local rings. Queen’s Paper in Pure and Applied Mathematics, vol. 20. Queen’s University, Kingston, Ontario (1969)
Gunston, T.: Cohomological degrees, Dilworth numbers and linear resolution. Ph.D. Thesis, Rutgers University (1998). arxiv:1008.3711[math.AC]
Huneke, C.: Symbolic powers of prime ideals and special graded algebras. Comm. Algebra 9, 339–366 (1981)
Kühl, M.: Über die symmetrische Algebra eines Ideals. Dissertation, Universität Essen, Germany (1981)
Linh, C.H.: Upper bound for the Castelnuovo–Mumford regularity of associated graded modules. Comm. Algebra 33, 1817–1831 (2005)
Mandal, M., Singh, B., Verma, J.K.: On some conjectures about the Chern numbers of filtrations. J. Algebra 325, 147–162 (2011)
Nagata, M.: Local Rings. Interscience, New York (1962)
Nagel, U.: Comparing Castelnuovo–Mumford regularity and extended degree: the borderline cases. Trans. Amer. Math. Soc. 357, 3585–3603 (2005)
Rossi, M.E., Trung, N.V., Valla, G.: Castelnuovo–Mumford regularity and extended degree. Trans. Amer. Math. Soc. 355, 1773–1786 (2003)
Stückrad, J., Vogel, W.: Buchsbaum Rings and Applications. Springer, New York (1986)
Vasconcelos, W.V.: The homological degree of a module. Trans. Amer. Math. Soc. 350, 1167–1179 (1998)
Vasconcelos, W.V.: Integral closure. In: Springer Monographs in Mathematics. Springer, New York (2005)
Yamagishi, K.: Bass number characterization of surjective Buchsbaum modules. Math. Proc. Cambridge Phil. Soc. 110, 261–279 (1991)
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The author was partially supported by the NSF.
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Vasconcelos, W.V. (2013). Cohomological Degrees and Applications. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_22
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DOI: https://doi.org/10.1007/978-1-4614-5292-8_22
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