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Keywords:
local homology; Artinian modules; annihilator
Summary:
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.
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