2008/04/28 by M. E. Rossi, Maria Evelina Rossi, Rossi, M. E. +3 · 1 citation
Mathematics · #13H10 #13H15 #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories #math.AC #math.AG #msc:13H10 #msc:13H15
paper · pdf · doi:10.48550/arxiv.0804.4438
will appear on Rendiconti Seminario Matematico Padova
arxiv created 2008/04/28 · openalex publication_date 2008/04/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the first Hilbert coefficient (after the multiplicity) e1 of a local ring (A,\m). Under various circumstances, it is also called the \bf Chern number of the local ring A. Starting from the work of D.G. Northcott in the 60's, several results have been proved which give some relationships between the Hilbert coefficients, but always assuming the Cohen-Macaulayness of the basic ring. Recent papers of S. Goto, K. Nishida, A. Corso and W. Vasconcelos pushed the interest toward a more general setting. In this paper we extend an upper bound on e1 proved by S. Huckaba and T. Marley. Thus we get the Cohen-Macaulayness of the ring A as a consequence of the extremal behavior of the integer e1. The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness of A. This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements. It is essentially a kind of "Sally machine" for local rings. In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto.