The Chern Numbers and Euler Characteristics of Modules
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The Chern Numbers and Euler Characteristics of Modules L. Ghezzi · S. Goto · J. Hong · K. Ozeki · T. T. Phuong · W. V. Vasconcelos Dedicated to Professors N.V. Trung and G. Valla for their groundbreaking contributions to the theory of Hilbert functions
Received: 1 January 2014 / Revised: 1 April 2014 / Accepted: 28 April 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract The set of the first Hilbert coefficients of parameter ideals relative to a module— its Chern coefficients—over a local Noetherian ring codes for considerable information about its structure–noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another
L. Ghezzi Department of Mathematics, New York City College of Technology-Cuny, 300 Jay Street, Brooklyn, NY 11201, USA e-mail: [email protected] S. Goto () · K. Ozeki Department of Mathematics, School of Science and Technology, Meiji University 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan e-mail: [email protected] K. Ozeki e-mail: [email protected] J. Hong Department of Mathematics, Southern Connecticut State University, 501 Crescent Street, New Haven, CT 06515-1533, USA e-mail: [email protected] T. T. Phuong Department of Information Technology and Applied Mathematics, Ton Duc Thang University, 98 Ngo Tat To Street, Ward 19, Binh Thanh District, Ho Chi Minh City, Vietnam e-mail: [email protected] W. V. Vasconcelos Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA e-mail: [email protected]
L. Ghezzi et al.
series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers. Keywords Hilbert function · Hilbert coefficient · Parameter ideal · Local cohomology · Euler characteristic · Cohen-Macaulay module · Vasconcelos module · Generalized Cohen-Macaulay module · Buchsbaum module Mathematics Subject Classification (2010) 13H10 · 13H15 · 13A30
1 Introduction Let R be a Noetherian local ring with maximal ideal m and let I be an m-primary ideal. There is a great deal of interest on the set of I -good filtrations of R. More concretely, on the set of multiplicative, decreasing filtrations
A = {In | I0 = R, In+1 = I In , n 0} of R ideals which are integral over the I -adic filtration, conveniently coded in the corresponding Rees algebra and its associated graded ring
R(A) =
In t n ,
grA (R) =
n≥0
In /In+1 .
n≥0
Our focus here is on a set of filtrations both broader and more narrowly defined. Let M be a finitely generated R-module. The Hilbert polynomial of the associated
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