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On Lech’s limit formula for modules

Volume 148 / 2017

R. Callejas-Bedregal, V. H. Jorge Pérez 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


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 $a_1,\dots ,a_m.$


  • R. Callejas-BedregalUniversidade Federal da Paraíba–DM
    58.051-900, João Pessoa, PB, Brazil
  • V. H. Jorge PérezDepartament of Mathematics
    Universidade de São Paulo–ICMC
    Caixa Postal 668
    13560-970, São Carlos, SP, Brazil

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