Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 225-234, 2019
Annihilators of local homology modules
Shahram Rezaei
Received May 28, 2017. Published online August 6, 2018.
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
${\rm Ann}_R({\rm H}_n^{\mathfrak a}(M))={\rm Ann}_R(N(\frak a,M))$,
where $N(\mathfrak a,M)$ denotes the smallest submodule of $M$ such that ${\rm hd}({\mathfrak a},M/N(\frak a,M))<n$. 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.
Keywords: local homology; Artinian modules; annihilator
Affiliations: Shahram Rezaei, Department of Mathematics, Faculty of Science, Payame Noor University (PNU), P.O. Box: 19395-4697, Tehran, Iran, e-mail: Sha.rezaei@gmail.com
