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The sabbatical leave was spent on research on Hibbert coefficients of ideals and integral closures of ideals, resulting in submission for publication, as co-author, of articles, Specialization and integral closure, to the Journal of the London Mathematical Society, and Hilbert polynomials of j-transforms, to the Mathematical Proceedings of the Cambridge Philosophical Society. Two research visits, to Purdue University and to Maiji University in Japan, facilitated work on these publications and laid foundation for new projects, already in progress, on normality of ideals and reduction numbers of ideals.

A degree of a module M is a numerical measure of information carried by M. We highlight some of Vasconcelos’ outstanding contributions to the theory of degrees, bridging commutative algebra and computational algebra. We present several degrees he introduced and developed, including arithmetic degree, jdeg, homological degree, cohomological degrees, canonical degree, and bicanonical degree. For the canonical and bicanonical degrees, we discuss recent developments motivated by our joint works [25, 19, 9]. ©2025 Walter de Gruyter GmbH,Berlin/Boston.

This paper surveys and summarizes Wolmer Vasconcelos’ results surrounding multiplicities, Hilbert coefficients, and their extensions. We particularly focus on Vasconcelos’ results regarding multiplicities and Chern coefficients, and other invariants which they bound. The Sally module is an important instrument introduced by Vasconcelos for this study, which naturally relates Hilbert coefficients to reduction numbers. ©2025 Walter de Gruyter GmbH,Berlin/Boston.

Let R be an analytically unramified local ring with maximal ideal m and d = dimR > 0. If R is unmixed, then e1I (R) = 0 for every m-primary ideal I in R, where e1I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e1I(R) posed by Wolmer V. Vasconcelos is settled affirmatively. © 2010 American Mathematical Society.

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

We study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients. Copyright © 2016 Cambridge Philosophical Society.

Our purpose is to study the cohomological properties of the Rees algebras of a class of ideals generated by quadrics. For all such ideals I⊂. R=. K[. x, y, z] we give the precise value of depth. R[. It] and decide whether the corresponding rational maps are birational. In the case of dimension d≥. 3, when K=R, we give structure theorems for all ideals of codimension d minimally generated by (d+12)-1 quadrics. For arbitrary fields K, we prove a polarized version. © 2014 Elsevier Inc.

In this paper we inject four Hilbert functions in the determination of the defining equations of the Rees algebra of almost complete intersections of finite co-length. Because three of the corresponding modules are Artinian, some of these relationships are very effective, with the novel approach opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones. While not specifically directed towards the extraction of elimination equations, it will show how some of these arise naturally. © 2012 Sociedade Brasileira de Matemática.

For a Noetherian local ring, we analyze conjectural relationships between the first Hilbert coefficient of a parameter ideal and the first partial Euler characteristic of its Koszul complex. Given their similar role as predictors of the Cohen-Macaulay property, we consider a direct comparison between them. For parameter ideals generated by d-sequences these numbers are related in an explicit formula. We then turn to study of families of parameter ideals that have the same Hilbert function. © 2012 Elsevier Inc.

This paper considers the following conjecture: If R is an unmixed, equidimensionallocal ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal J generated by a system of parameters, the Chern coefficient e1(J) < 0 is equivalent to R being non Cohen-Macaulay. The conjecture is established if R is a homomorphic image of a Gorenstein ring, and for all universally catenary integral domains containing fields. Criteria for the detection of Cohen-Macaulayness in equi-generated graded modules are derived. © International Press 2009.

We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method for their calculation in terms of certain Hilbert coefficients. In dimension 2 the structure of the irreducible ideals-always complete intersections by a classical theorem of Serre-leads by a natural approach to the calculation of Sylvester determinants. We introduce a computer-assisted method (with a minimal intervention by the computer) which succeeds, in degree ≤5, in producing the full sets of equations of the ideals. In the process, it answers affirmatively some questions raised by Cox [Cox, D.A., 2006. Four conjectures: Two for the moving curve ideal and two for the Bezoutian. In: Proceedings of Commutative Algebra and its Interactions with Algebraic Geometry, CIRM, Luminy, France, May 2006 (available in CD media)]. © 2007 Elsevier Ltd. All rights reserved.

The purpose of this paper is to introduce new invariants of Cohen–Macaulay local rings. Our focus is the class of Cohen–Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers–the type of R, the reduction number of C–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers–the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C. © 2017 Elsevier Inc.

For a Noetherian local ring (R, m), the first two Hilbert coefficients, e0 and e1, of the I-adic filtration of an m-primary ideal I are known to code for properties of R, of the blowup of Spec(R) along V (I), and even of their normalizations. We give estimations for these coefficients when I is enlarged (in the case of e1 in the same integral closure class) for general Noetherian local rings. © American Mathematical Society.

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen–Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen–Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree. © 2019 Elsevier Inc.

A problem posed by Vasconcelos [33] on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well. © 2011 Elsevier B.V.

The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient e1(Q) is solved affirmatively, where Q is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which e 1(Q) vanishes are derived. The invariance of e1(Q) for parameter ideals Q and its relationship to Buchsbaum rings are studied. © 2010 London Mathematical Society.

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 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. © 2014, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.

Let A be a Noetherian local ring with canonical module KA. We characterize A when KA is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3. If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module KA is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasinormal rings, introduced by W. V. Vasconcelos. We provide several examples as well. ©2025 Walter de Gruyter GmbH,Berlin/Boston.