Matlis dual of local cohomology modules

Naal, Batoul, and Khashyarmanesh, Kazem. "Matlis dual of local cohomology modules." Czechoslovak Mathematical Journal 70.1 (2020): 1-7. <http://eudml.org/doc/297342>.

@article{Naal2020,
abstract = {Let $(R,\mathfrak \{m\})$ be a commutative Noetherian local ring, $\mathfrak \{a\}$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $\mathfrak \{a\} M\ne M$ and $\{\rm cd\}(\mathfrak \{a\},M) - \{\rm grade\}(\mathfrak \{a\},M)\le 1$, where $\{\rm cd\}(\mathfrak \{a\},M)$ is the cohomological dimension of $M$ with respect to $\mathfrak \{a\}$ and $\{\rm grade\}(\mathfrak \{a\},M)$ is the $M$-grade of $\mathfrak \{a\}$. Let $D(-) := \{\rm Hom\}_R(-,E)$ be the Matlis dual functor, where $E := E(R/\mathfrak \{m\})$ is the injective hull of the residue field $R/\mathfrak \{m\}$. We show that there exists the following long exact sequence \begin\{eqnarray*\} 0 \longrightarrow & H^\{n-2\}\_\{\mathfrak \{a\}\}(D(H^\{n-1\}\_\{\mathfrak \{a\}\}(M))) \longrightarrow H^\{n\}\_\{\mathfrak \{a\}\}(D(H^\{n\}\_\{\mathfrak \{a\}\}(M))) \longrightarrow D(M) \\ \longrightarrow & H^\{n-1\}\_\{\mathfrak \{a\}\}(D(H^\{n-1\}\_\{\mathfrak \{a\}\}(M))) \longrightarrow H^\{n+1\}\_\{\mathfrak \{a\}\}(D(H^\{n\}\_\{\mathfrak \{a\}\}(M))) \\ \longrightarrow & H^\{n\}\_\{\mathfrak \{a\}\}(D(H^\{n-1\}\_\{(x\_1, \ldots ,x\_\{n-1\})\}(M))) \longrightarrow H^\{n\}\_\{\mathfrak \{a\}\}(D(H^\{n-1\}\_\mathfrak \{(\}M))) \longrightarrow \ldots , \end\{eqnarray*\} where $n:=\{\rm cd\}(\mathfrak \{a\},M)$ is a non-negative integer, $x_1, \ldots ,x_\{n-1\}$ is a regular sequence in $\mathfrak \{a\}$ on $M$ and, for an $R$-module $L$, $H^i_\{\mathfrak \{a\}\}(L)$ is the $i$th local cohomology module of $L$ with respect to $\mathfrak \{a\}$.},
author = {Naal, Batoul, Khashyarmanesh, Kazem},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology module; Matlis dual functor; filter regular sequence},
language = {eng},
number = {1},
pages = {1-7},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Matlis dual of local cohomology modules},
url = {http://eudml.org/doc/297342},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Naal, Batoul
AU - Khashyarmanesh, Kazem
TI - Matlis dual of local cohomology modules
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 1
SP - 1
EP - 7
AB - Let $(R,\mathfrak {m})$ be a commutative Noetherian local ring, $\mathfrak {a}$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $\mathfrak {a} M\ne M$ and ${\rm cd}(\mathfrak {a},M) - {\rm grade}(\mathfrak {a},M)\le 1$, where ${\rm cd}(\mathfrak {a},M)$ is the cohomological dimension of $M$ with respect to $\mathfrak {a}$ and ${\rm grade}(\mathfrak {a},M)$ is the $M$-grade of $\mathfrak {a}$. Let $D(-) := {\rm Hom}_R(-,E)$ be the Matlis dual functor, where $E := E(R/\mathfrak {m})$ is the injective hull of the residue field $R/\mathfrak {m}$. We show that there exists the following long exact sequence \begin{eqnarray*} 0 \longrightarrow & H^{n-2}_{\mathfrak {a}}(D(H^{n-1}_{\mathfrak {a}}(M))) \longrightarrow H^{n}_{\mathfrak {a}}(D(H^{n}_{\mathfrak {a}}(M))) \longrightarrow D(M) \\ \longrightarrow & H^{n-1}_{\mathfrak {a}}(D(H^{n-1}_{\mathfrak {a}}(M))) \longrightarrow H^{n+1}_{\mathfrak {a}}(D(H^{n}_{\mathfrak {a}}(M))) \\ \longrightarrow & H^{n}_{\mathfrak {a}}(D(H^{n-1}_{(x_1, \ldots ,x_{n-1})}(M))) \longrightarrow H^{n}_{\mathfrak {a}}(D(H^{n-1}_\mathfrak {(}M))) \longrightarrow \ldots , \end{eqnarray*} where $n:={\rm cd}(\mathfrak {a},M)$ is a non-negative integer, $x_1, \ldots ,x_{n-1}$ is a regular sequence in $\mathfrak {a}$ on $M$ and, for an $R$-module $L$, $H^i_{\mathfrak {a}}(L)$ is the $i$th local cohomology module of $L$ with respect to $\mathfrak {a}$.
LA - eng
KW - local cohomology module; Matlis dual functor; filter regular sequence
UR - http://eudml.org/doc/297342
ER -