Annihilators of local homology modules

Rezaei, Shahram. "Annihilators of local homology modules." Czechoslovak Mathematical Journal 69.1 (2019): 225-234. <http://eudml.org/doc/294859>.

@article{Rezaei2019,
abstract = {Let $(R,\{\mathfrak \{m\}\})$ be a local ring, $\mathfrak \{a\}$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\{\rm hd\}(\mathfrak \{a\}, M)=n $. We determine the annihilator of the top local homology module $\{\rm H\}_\{n\}^\{\mathfrak \{a\}\}(M)$. In fact, we prove that \[ \{\rm Ann\}\_R(\{\rm H\}\_\{n\}^\{\mathfrak \{a\}\}(M))=\{\rm Ann\}\_R(N(\mathfrak \{a\},M)), \] where $N(\mathfrak \{a\},M)$ denotes the smallest submodule of $M$ such that $\{\rm hd\}(\{\mathfrak \{a\}\},M/N(\mathfrak \{a\},M))<n$. As a consequence, it follows that for a complete local ring $(R,\mathfrak \{m\})$ all associated primes of $\{\rm H\}_\{n\}^\{\mathfrak \{a\}\}(M) $ are minimal.},
author = {Rezaei, Shahram},
journal = {Czechoslovak Mathematical Journal},
keywords = {local homology; Artinian modules; annihilator},
language = {eng},
number = {1},
pages = {225-234},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Annihilators of local homology modules},
url = {http://eudml.org/doc/294859},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Rezaei, Shahram
TI - Annihilators of local homology modules
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 225
EP - 234
AB - Let $(R,{\mathfrak {m}})$ be a local ring, $\mathfrak {a}$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with ${\rm hd}(\mathfrak {a}, M)=n $. We determine the annihilator of the top local homology module ${\rm H}_{n}^{\mathfrak {a}}(M)$. In fact, we prove that \[ {\rm Ann}_R({\rm H}_{n}^{\mathfrak {a}}(M))={\rm Ann}_R(N(\mathfrak {a},M)), \] where $N(\mathfrak {a},M)$ denotes the smallest submodule of $M$ such that ${\rm hd}({\mathfrak {a}},M/N(\mathfrak {a},M))<n$. As a consequence, it follows that for a complete local ring $(R,\mathfrak {m})$ all associated primes of ${\rm H}_{n}^{\mathfrak {a}}(M) $ are minimal.
LA - eng
KW - local homology; Artinian modules; annihilator
UR - http://eudml.org/doc/294859
ER -