On the Chern number of a filtration

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dc.creatorRossi, M. E.
dc.creatorValla, G.
dc.date2008-04-28
dc.date.accessioned2026-07-07T09:35:38Z
dc.date.available2026-07-07T09:35:38Z
dc.descriptionWe study the first Hilbert coefficient (after the multiplicity) $e_1$ 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 $e_1$ 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 $e_1.$ 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.
dc.descriptionwill appear on Rendiconti Seminario Matematico Padova
dc.identifierhttps://arxiv.org/abs/0804.4438
dc.identifierhttp://arxiv.org/abs/0804.4438
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/159896
dc.subjectCommutative Algebra
dc.subjectAlgebraic Geometry
dc.subject13H10; 13H15
dc.titleOn the Chern number of a filtration
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