On Lech’s limit formula for modules
Volume 148 / 2017
Colloquium Mathematicum 148 (2017), 27-37
MSC: Primary 13H15; Secondary 13H15, 13B22, 13C14, 13C15, 13C40.
DOI: 10.4064/cm6870-6-2016
Published online: 6 February 2017
Abstract
Let $R=\bigoplus _{n=0}^{\infty } R_n$ be a standard graded algebra and $M=\bigoplus _{n=0}^{\infty } M_n$ a graded Noetherian $R$-module. The main objective of this work is to derive a Lech type formula for a sequence of homogeneous elements $a_1,\dots ,a_m$ of degree one which form a $g$-multiplicity system of $R$. We also extend to this context the well known Serre Theorem, that is, we prove that for $t\gg 0$ the $g$-multiplicity symbol $e_t(a_1,\dots ,a_m;R)$, introduced by Kirby (1987), coincides with the Buchsbaum–Rim multiplicity $e_{\rm BR}(I;R)$ of the $R_0$-module $I$ generated by $a_1,\dots ,a_m.$
