Geodesic
Annihilators of local homology modules
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 225-234 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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))
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 ${\rm hd}({\mathfrak a},M/N(\frak a,M))$. 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.
DOI : 10.21136/CMJ.2018.0263-17
Classification : 13D45, 13E05
Keywords: local homology; Artinian modules; annihilator 
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Rezaei, Shahram. Annihilators of local homology modules. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 225-234. doi: 10.21136/CMJ.2018.0263-17

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