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Relative Hilbert co-efficients (1512.04315v1)
Published 14 Dec 2015 in math.AC and math.AG
Abstract: Let $(A,\m)$ be a \CM \ local ring of dimension $d$ and let $I \subseteq J$ be two $\m$-primary ideals with $I$ a reduction of $J$. For $i = 0,\ldots,d$ let $e_iJ(A)$ ($e_iI(A)$) be the $i{th}$ Hilbert coefficient of $J$ ($I$) respectively. We call the number $c_i(I,J) = e_iJ(A) - e_iI(A)$ the $i{th}$ relative Hilbert coefficient of $J$ \wrt \ $I$. If $G_I(A)$ is \CM \ then $c_i(I,J)$ satisfy various constraints. We also show that vanishing of some $c_i(I,J)$ has strong implications on $\depth G_{Jn}(A)$ for $n \gg 0$.