On Lech’s limit formula for modules

R. Callejas-Bedregal¹ ; V. H. Jorge Pérez²

¹ Universidade Federal da Paraíba–DM 58.051-900, João Pessoa, PB, Brazil
² Departament of Mathematics Universidade de São Paulo–ICMC Caixa Postal 668 13560-970, São Carlos, SP, Brazil

Colloquium Mathematicum, Tome 148 (2017) no. 1, pp. 27-37

Résumé

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 $a_1,\dots ,a_m.$

Détail

DOI : 10.4064/cm6870-6-2016

Mots-clés : bigoplus infty standard graded algebra bigoplus infty graded noetherian r module main objective work derive lech type formula sequence homogeneous elements dots degree which form g multiplicity system extend context known serre theorem prove g multiplicity symbol dots introduced kirby coincides buchsbaum rim multiplicity module nbsp generated dots