Lech's inequality, the Stückrad-Vogel conjecture, and uniform behavior of Koszul homology (Q1735478)

Let \((R,\mathfrak{m})\) be a local noetherian ring and \(M\) a finitely generated \(R\)-module. Consider the set \[ \Big\{\frac{\ell(M/IM)}{\operatorname{e}(I,M)}\mid I\text{ ideal of }R\text{ with }\sqrt{I + \operatorname{Ann}(M)}=\mathfrak{m}\Big\} \] whose infimum and supremum are denoted by \(m(M)\) and \(n(M)\), respectively. (Here the Hilbert-Samuel multiplicity of \(M\) with respect to \(I\) is defined by \(\operatorname{e}(I,M)=\lim_{n \to \infty}d!\frac{\ell(M/I^nM)}{n^d}\) where \(d=\dim M\).) The authors prove that \(n(M)< \infty\) whenever the completion \(\widehat{M}\) is equidimensional, positively answering a question of \textit{J. Stückrad} and \textit{W. Vogel} [``On composition series and new invariants of local algebra'', Preprint, 96--99]. This is first proved by using the degree functions introduced by \textit{W. V. Vasconcelos} [Trans. Am. Math. Soc. 350, No. 3, 1167--1179 (1998; Zbl 0903.13007)], but an alternative proof that avoids the use of these functions is also provided. Several results regarding the behavior of \(n(M)\) under some base changes (localization, local flat extensions, and the killing of a parameter) are proved. The authors leave open the question whether the supremum \(n(M)\) is actually attained for some ideal \(I\). Another important result in the paper generalizes an inequality first proved by \textit{C. Lech} [Ark. Mat. 4, 63--86 (1960; Zbl 0192.13901)]. The authors show that \(m(M) \geq \frac{1}{d!\operatorname{e}({R/\operatorname{Ann}(M)})}\) for every finitely generated \(R\)-module \(M\), an inequality which recovers Lech's result for \(M=R\). As an application, if \(H_i\) denotes the \(i\)-th Koszul homology, the authors obtain several results regarding the uniform behavior of the sequences \[ \Big\{\frac{\ell(H_i(x_1^t, \ldots, x_d^t; M))}{\ell(M/(x_1^t, \ldots, x_d^t)M)}\Big\}_{t\geq 1} \] with respect to all systems of parameters \(\underline{x}=x_1,\ldots, x_d\) of \(M\).