Quasilength, latent regular sequences, and content of local cohomology (Q2654060)

Let \(d\in \mathbb{N}\) be an integer and \(\underline{x}=x_1,\ldots , x_d\) a sequence of elements of a commutative ring \(R\). Let \(I=(\underline{x})R\) and \(M\) a finitely generated \(R\)-module that is annihilated by a power of \(I\). The authors introduced the notion of \(I\)-quasilength of \(M\), \(\mathcal{L}_I^R(M)\), as the length of a shortest filtration of \(M\) with factors that are homomorphic images of \(R/I\). If \(I\) is maximal, then \(\mathcal{L}_I^R(M)\) coincides with the usual length of \(M\). For any \(d\)-tuple \(t=(t_1,\dots ,t_d)\in \mathbb{N}^d\), let \(I_{\underline{t}}:=(x_1^{t_1},\dots , x_d^{t_d})R\) and \[ (I_{\underline{t}}M)^{\text{lim}}:=\bigcup_{\underline{k}=(k_1,\dots ,k_d)\in \mathbb{N}^d}(I_{\underline{t}+\underline{k}}M:_Mx_1^{k_1}\dots x_d^{k_d}). \] Then the authors defined the notions \(\underline{h}_{\underline{x}}^d(M)\) and \(h_{\underline{x}}^d(M)\), respectively, by \[ \underline{h}_{\underline{x}}^d(M):={\text{lim}}_{s\to \infty}\inf \{\frac{\mathcal{L}_I^R(M/(I_{\underline{t}}M)^{\text{lim}})}{t_1\ldots t_d}|t=(t_1,\dots ,t_d)\in \mathbb{N}^d \;\;\text{and every} \;\;t_i\geq s \} \] and \[ h_{\underline{x}}^d(M):={\text{lim}}_{s\to \infty}\inf \{\frac{\mathcal{L}_I^R(M/I_{\underline{t}}M)}{t_1\ldots t_d}|t=(t_1,\ldots ,t_d)\in \mathbb{N}^d \;\;\text{and every} \;\;t_i\geq s \}. \] Let \(\nu(M)\) denote the least number of generators of \(M\). It follows that \[ 0\leq \underline{h}_{\underline{x}}^d(M)\leq h_{\underline{x}}^d(M)\leq \nu(M). \] In particular, this implies that \[ 0\leq \underline{h}_{\underline{x}}^d(R)\leq h_{\underline{x}}^d(R)\leq 1. \] They authors show that the condition \(h_{\underline{x}}^d(R)=1\) depends only on \(d\) and \(I\), and not on specific choice of \(d\) generators for \(I\). Also, they proved that \(h_{\underline{x}}^d(R)=1\) if and only if \(\underline{h}_{\underline{x}}^d(R)=1\). The authors conjectured that: a) \(\underline{h}_{\underline{x}}^d(R)=h_{\underline{x}}^d(R)\), b) \(h_{\underline{x}}^d(R)\) is either 0 or 1; and c) if \(R\) is local and \(\underline{x}\) forms a system of parameters in \(R\), then \(\underline{h}_{\underline{x}}^d(R)=1\). Regarding these conjectures, the authors established the following results. 1) If \(R\) has positive prime characteristic, then \(\underline{h}_{\underline{x}}^d(R)=h_{\underline{x}}^d(R)\) and \(h_{\underline{x}}^d(R)\) is either 0 or 1. 2) Assume that \(R\) is local and \(\underline{x}\) forms a system of parameters in \(R\). If either \(R\) is equicharacteristic or \(R\) is reduced and equidimensional, then \(\underline{h}_{\underline{x}}^d(R)=h_{\underline{x}}^d(R)\). 3) The conjecture \(c)\) implies the direct summand conjecture.