Hilbert coefficients of Hilbert filtrations (Q1379060)

This paper deals with the Hilbert series of a Hilbert filtration \({\mathcal F}\) in a local ring \((A,m)\); its aim is to study the Hilbert function and the Hilbert coefficients of \({\mathcal F}\). Many results already known for the \(I\)-adic case are extended to this more general one: In particular, the Cohen-Macaulay property of \(gr_{\mathcal F} (A)\) is characterized in terms of lower bound of \(e_1 ({\mathcal F})\). The last part of the paper is devoted to the case of a \(I\)-adic filtration, with \(I\) an \(m\)-primary ideal in a local Cohen-Macaulay ring: The behaviour of the Hilbert function is described when some extremal conditions are satisfied by Hilbert coefficients, generalizing results already known in more special cases.