  We 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.

Commutative Algebra;Algebraic Geometry