Note on length and multiplicity of modules over two-dimensional regular local rings

TL;DR

This work bounds the Buchsbaum-Rim multiplicity $e(M)$ of finitely generated torsion-free modules over a two-dimensional regular local ring in terms of the ideal of maximal minors $I(M)$ and its adjoint $adj(I)$. It establishes precise equalities and criteria for when $M$ is integrally closed, yielding a formula $e(M)=λ(F/M)+λ(R/adj(I))$ for integrally closed $M$ and a length-multiplicity identity $e(I)-e(M)=λ(R/I)-λ(F/M)$ in that case. The authors also prove dual bounds involving $λ(R/adj(I))$ and give a one-to-one correspondence between integrally closed modules of rank $r$ with $I(M)=I$ and contracted modules with $I(K)=I$ and $I_{r-1}(K)=adj(I)$, extending classical results for integrally closed ideals. The results integrate reductions, Fitting ideals, and adjoint theory to classify and relate modules, with concrete examples and remarks on mixed multiplicities. Overall, the paper generalizes known ideal-theoretic formulas to modules and provides tools for systematically analyzing integrally closed modules in this setting.

Abstract

We give lower and upper bounds on the Buchsbaum-Rim multiplicity of finitely generated torsion-free modules over two-dimensional regular local rings, and conditions for them to attain the bounds. As consequences, we have formulae on the multiplicity of integrally closed modules.

Theorems & Definitions (26)

• Theorem 1
• Proposition 2
• Theorem 3
• Theorem 4
• Lemma 5
• Proposition 6
• proof
• Proposition 7
• Proposition 8
• proof : Proof of Proposition $\ref{['lower2']}$