The Chern Numbers and Euler Characteristics of Modules

Abstract

The set of the first Hilbert coefficients of parameter ideals relative to a moduleits 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 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.

Sectional genera of parameter ideals

“…In [GhGHOPV,Theorem 7.1], it was proved that, for parameter ideals Q for M, the upper bound χ 1 (Q; M) ≤ hdeg Q (M) − e 0 Q (M) of the first Euler characteristic χ 1 (Q; M) of M relative to Q. In [GO,Theorem 1.3], the authors gave a criterion for the equality χ 1 (Q; M) = hdeg Q (M) − e 0 Q (M).…”

“…In [GO,Theorem 1.3], the authors gave a criterion for the equality χ 1 (Q; M) = hdeg Q (M) − e 0 Q (M). We also have the inequalities 0 ≥ e 1 Q (M) ≥ −T 1 Q (M) for every parameter ideals Q for M ( [MSV,Theorem 3.6], [GhGHOPV,Theorem 6.6 [GhGHOPV] for the characterization of modules which have parameter ideals Q with e 1 Q (M) = 0. Thus the behavior of the first Euler characteristics χ 1 (Q; M) and the first Hilbert coefficients e 1 Q (M) are rather satisfactory understood.…”

Sally Modules and Reduction Numbers Of ideals

“…Since λ(R/Q) = e 0 (I) + χ 1 (Q) ≤ hdeg I (R), by Serre's Theorem ([2, Theorem 4.7.10]) and [9,Theorem 7.2] respectively, the reduction number r(I) can be bounded in terms of I alone. It is worth asking whether a sharper bound holds with replacing λ(R/Q) by e 0 (I) − e 1 (Q) (see [10,Theorem 4.2]).…”

The first Euler characteristics versus the homological degrees

“…The notion of homological degree was introduced by W. V. Vasconcelos and his students [DGV] in 1998, and since then, many authors have been engaged in the development of the theory. Recently, in [GHV,GhGHOPV2,V3], Ghezzi, Hong, Phuong, Vasconcelos, and the authors also made use of homological degrees to obtain bounds for the Hilbert coefficients of parameters. The purpose of our paper is to study the relationship between the first Euler characteristics and the homological degrees of modules.…”

“…In [GhGHOPV2], it was proved that, for parameter ideals Q for M, an upper bound χ 1 (Q; M) ≤ hdeg Q (M) − e 0 Q (M) of χ 1 (Q; M) (Proposition 3.2). It seems natural to ask what happens on the parameters Q for M, when the equality…”