Local homology and Gorenstein flat modules

Details Local homology and Gorenstein flat modules Local homology and Gorenstein flat modules

2012-01-15

arxiv.org/abs/1201.3067v1

Mathematics

Commutative Algebra

It will be published in the journal of algebra and its applications

Scientific paper

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $\mathcal{D}(R)$ denote the derived category of $R$-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let $X$ be a homologically bounded to the right complex and $Q$ a bounded to the right complex of Gorenstein flat $R$-modules such that $Q$ and $X$ are isomorphic in $\mathcal{D}(R)$. We establish a natural isomorphism ${\bf L}\Lambda^{\fa}(X)\simeq \Lambda^{\fa}(Q)$ in $\mathcal{D}(R)$ which immediately asserts that $\sup {\bf L}\Lambda^{\fa}(X)\leq \Gfd_RX$. This isomorphism yields several consequences. For instance, in the case $R$ possesses a dualizing complex, we show that $\Gfd_R {\bf L}\Lambda^{\fa}(X)\leq \Gfd_RX$. Also, we establish a criterion for regularity of Gorenstein local rings.

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