Almost Cohen-Macaulayness of Koszul homology
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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 $i$.
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 $i$.
참고문헌 (15)
- W. Bruns, "Cohen-Macaulay Rings" (1993), Cambridge University Press
- L. Chu, "A note on almost Cohen-Macaulay modules", J. Algebra Appl, 14(10), pp.7~ (2015)
- "Macaulay 2, a software system for research in algebraic geometry"[URL]
- C. Huneke, "On the symmetric and Rees algebra of an ideal generated by a d-sequence", J. Algebra, 62(2), pp.268~275 (1980)
- C. Huneke, "The theory of d-sequences and powers of ideals", Adv. in Math, 46(3), pp.249~279 (1982)
- C. Huneke, "Linkage and the Koszul homology of ideals", Amer. J. Math, 104(5), pp.1043~1062 (1982)
- C. Huneke, "The Koszul homology of an ideal", Adv. in Math, 56(3), pp.295~318 (1985)
- C. Ionescu, "More properties of almost Cohen-Macaulay rings", J. Commut. Algebra, 7(3), pp.363~372 (2015)
- M.-C. Kang, "Almost Cohen-Macaulay modules", Comm. Algebra, 29(2), pp.781~787 (2001)
- M.-C. Kang, "Addendum to: “Almost Cohen-Macaulay modules”", Comm. Algebra, 30(2), pp.1049~1052 (2002)
- A. Mafi, "A note on the finiteness property related to derived functors", Acta Math. Vietnam, 34(3), pp.371~374 (2009)
- A. Mafi, "Results on almost Cohen-Macaulay modules", J. Algebr. Syst, 3(2), pp.147~150 (2016)
- S. Tabejamaat, "About a Serre-type condition for modules", J. Algebra Appl, 16(11), pp.6~ (2017)
- S. Tabejamaat, "Amoli, Property of almost Cohen-Macaulay over extension modules", Algebra Colloq, 24(3), pp.509~518 (2017)
- W. Vasconcelos, "Integral Closure, Springer Monographs in Mathematics" (2005), Springer-Verlag