Almost Cohen-Macaulayness of Koszul homology
Bull. Korean Math. Soc. 2019 Vol. 56, No. 2, 471-477
Published online March 1, 2019
Amir Mafi, Samaneh Tabejamaat
University Of Kurdistan; Payame Noor University
Abstract : Let $(R,\mathfrak{m})$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a non-zero finitely generated $R$-module. We show that if $M$ and $H_0(I,M)$ are aCM $R$-modules and $I=(x_1, \ldots ,x_{n+1})$ such that $x_1, \ldots ,x_n$ is an $M$-regular sequence, then $H_i(I,M)$ is an aCM $R$-module for all $i$. Moreover, we prove that if $R$ and $H_i(I,R)$ are aCM for all $i$, then $R/(0:I)$ is aCM. In addition, we prove that if $R$ is aCM and $x_1, \ldots ,x_n$ is an aCM $d$-sequence, then depth $H_i(x_1, \ldots ,x_n;R)\geq i-1$ for all $i$.
Keywords : almost Cohen-Macaulay modules, Koszul homology
MSC numbers : 13C14, 13D07
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