Sequentially Cohen-Macaulay Rees algebras (Q520641)

The authors study the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen-Macaulay. Due to the complexity of the notations and the clearness of the statements of the introduction, we state the main results of this paper and it is mostly paraphrasing from the introduction of the paper. \par Let $R$ be a local ring with maximal ideal $m$ and $t$ be an indeterminate over $R$. Let $\mathcal{F}=\{F_{n}\}_{n\in\mathbb{Z}}$ be a filtration of ideals of $R$ such that $F_{1}\neq R$, $\mathcal{M}=\{M_{n}\}_{n\in\mathbb{Z}}$ a $\mathcal{F}$-filtration of $R$-submodules of $\mathcal{M}$. We write $\mathcal{R}=\sum_{n\geq0}F_{n}t^{n}\subseteq R[t]$, $\mathcal{R}'=\sum_{n\in\mathbb{Z}}F_{n}t^{n}\subseteq R[t,t^{-1}]$, $\mathcal{G}=\mathcal{R}'/t^{-1}\mathcal{R}'$, the Rees algebra, the extended Rees algebra and the associated graded ring of $\mathcal{F}$, respectively. Similarly, we write $\mathcal{R}(\mathcal{M})=\sum_{n\geq0}t^{n}\otimes M_{n}\subseteq R[t]\otimes_{R}M$, $\mathcal{R}'(\mathcal{M})=\sum_{n\in\mathbb{Z}}t^{n}\otimes M_{n}\subseteq R[t,t^{-1}]\otimes_{R}M$, and $\mathcal{G}(\mathcal{M})=\mathcal{R}'(\mathcal{M})/t^{-1}\mathcal{R}'(\mathcal{M})$, the Rees module, the extended Rees module and the associated graded module of $\mathcal{M}$, respectively. Assume that $\mathcal{R}$ is a Noetherian ring and $\mathcal{R}(\mathcal{M})$ is a finitely generated $\mathcal{R}$-module. Set $\mathcal{D}_{i}=\{M_{n}\bigcap D_{i}\}_{n\in\mathbb{Z}}$, $\mathcal{C}_{i}=\{[(M_{n}\cap D_{i})+D_{i-1}]/D_{i-1}\}_{n\in\mathbb{Z}}$ for every $1\le i\le l$. Then $\mathcal{D}_{i}$ (resp. $\mathcal{C}_{i}$) is a $\mathcal{F}$-filtration of $R$-submodules of $D_{i}$ (resp. $C_{i}$). \par Let $\mathfrak{m}$ be a unique graded maximal ideal of $\mathcal{R}$. We write \[ a(N)=\text{max}\{n\text{ in }\mathbb{Z}|[H_{\mathfrak{M}}^{t}(N)]_{n}\neq(0)\}, \] $a$-invariant of $N$ for a finitely generated graded $\mathcal{R}$-module of dimension $t$. Here $\{[H_{\mathfrak{M}}^{t}(N)]_{n}\}_{n\in\mathbb{Z}}$ stands for the homogeneous components of the $t$-th graded local cohomology module $H_{\mathfrak{M}}^{t}(N)$ of $N$ with respect to $\mathfrak{M}.$ \par With notations above, the main results of this paper are the following: \par Theorem 1.1 The following conditions are equivalent. (1) $\mathcal{R}'(M)$ is a sequentially Cohen-Macaulay $\mathcal{R}'$-module. (2) $\mathcal{G}(\mathcal{M})$ is a sequentially Cohen-Macaulay $\mathcal{G}$-module and $\{\mathcal{G}(\mathcal{D}_{i})\}_{0\leq i\leq l}$ is the dimension filtration of $\mathcal{G}(\mathcal{M})$. When this is the case, $M$ is a sequentially Cohen-Macaulay $R$-module. \par Theorem 1.2. Suppose that $M$ is a sequentially Cohen-Macaulay $R$-module and $F_{1}\nsubseteq\mathfrak{p}$ for every $\mathfrak{p}\in\text{Ass}_{R}M$. Then the following conditions are equivalent. (1) $\mathcal{R}(\mathcal{M})$ is a sequentially Cohen-Macaulay $\mathcal{R}$-module. (2) $\mathcal{G}(M)$ is a sequentially Cohen-Macaulay $\mathcal{G}$-module, $\{\mathcal{G}(\mathcal{D}_{i})\}_{0\leq i\leq l}$ is the dimension filtration of $\mathcal{G}(\mathcal{M})$ and $a(\mathcal{G}(\mathcal{C}_{i}))<0$ for every $1\le i\le l$. When this is the case, $\mathcal{R}'(\mathcal{M})$ is a sequentially Cohen-Macaulay $\mathcal{R}'$-module.