Journal of Mathematics

Volume 2013 (2013), Article ID 912643, 3 pages

http://dx.doi.org/10.1155/2013/912643

## Co-Cohen-Macaulay Modules and Local Cohomology

^{1}Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran^{2}Department of Mathematics, University of Kurdistan, Pasdaran Street, P.O. Box 416, Sanandaj, Iran

Received 13 August 2012; Revised 6 January 2013; Accepted 19 January 2013

Academic Editor: Feng Feng

Copyright © 2013 Hero Saremi and Amir Mafi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a commutative Noetherian local ring and let be a finitely generated -module of dimension . Then the following statements hold: (a) if width for all with , then is co-Cohen-Macaulay of Noetherian dimension ; (b) if is an unmixed -module and depth , then is co-Cohen-Macaulay of Noetherian dimension if and only if is either zero or co-Cohen-Macaulay of Noetherian dimension . As consequence, if is co-Cohen-Macaulay of Noetherian dimension for all with , then is co-Cohen-Macaulay of Noetherian dimension .

#### 1. Introduction

Throughout this paper, let be a commutative Noetherian local ring and let be a finitely generated -module of dimension . We denote the th local cohomology module of with respect to by . It is well known that is Artinian for all (cf. [1]).

The *Noetherian dimension* of an Artinian -module , denoted by , is defined inductively as follows: when , put . Then by induction, for any integer , put if is false, and for any ascending chain of submodules of there exists an integer such that for all . Therefore if and only if is a nonzero Noetherian module. Moreover, if is an exact sequence of Artinian modules, then . Let . is an -coregular sequence if is surjective for and . The width of , denoted by , is the length of any maximal -coregular sequence in . For any -coregular element , we have that and . Details about and can be found in Roberts [2], Kirby [3], and Ooishi [4]; there is a general fact: for any Artinian -module holds and is co-Cohen-Macaulay if and only if holds (cf. [5–7]). Tang [8] has shown that if either or is Cohen-Macaulay, then is co-Cohen-Macaulay (see also [9]). Following Nagata [10], is unmixed if for all .

The main aim of this paper is to prove the following theorem.

Theorem 1. *The following statements are true.*(a)*If for all with , then is co-Cohen-Macaulay of Noetherian dimension . *(b)*If is an unmixed -module and , then is co-Cohen-Macaulay of Noetherian dimension if and only if is either zero or co-Cohen-Macaulay of Noetherian dimension .*

#### 2. The Results

Following Macdonald [11], every Artinian -module has minimal secondary representation , where is secondary. The set is independent of the choice of the minimal secondary representation of . This set is called *the set of attached prime ideals* of and denoted by . The set of all minimal elements of is exactly the set of all minimal elements of . A sequence of elements in is called a *strict f-sequence* of if for all for all . This notion was introduced in [12].

Lemma 2 (see [9]). *For all integer , one has and . *

Lemma 3. *Let be a strict -sequence of . Then the following statements are true.*(i)*Suppose that and is co-Cohen-Macaulay of Noetherian dimension . Then is also co-Cohen-Macaulay of Noetherian dimension . *(ii)*Suppose that is co-Cohen-Macaulay of Noetherian dimension and . Then is co-Cohen-Macaulay of Noetherian dimension . *

*Proof. *(i) By our hypothesis and using [1, Exercise 11.3.9], we have . Hence from the exact sequences
we get the exact sequence
for all . Thus in case we have that is co-Cohen-Macaulay of Noetherian dimension . By the choice of the module is an -module of finite length. Moreover, since by [8, Proposition 2.4], we have by [13, Theorem 4.11]. Thus, by [13, Corollary 3.7], the long exact local homology sequence with respect to over the exact sequence provides . Hence there is an isomorphism and so is a co-Cohen-Macaulay module of Noetherian dimension .

The proof of (ii) follows by the same arguments as in the proof of (i).

Brodmann and Sharp [14], for all integer , defined the set , the th *pseudo support* of , and denoted by . Note that if is complete with respect to -adic topology, then by [15, Theorem 3.1] . The module satisfies Serre's condition , where is nonnegative integer, provided for all . Note that satisfies the condition Serre if and only if has no imbedded primes, that is, is unmixed.

Lemma 4. *Let be unmixed and . Then is either zero or co-Cohen-Macaulay of Noetherian dimension . *

*Proof. *When , it is trivial. We assume that . Since is unmixed then and so we have . Thus the result has been proved in this case. Now assume that and . By using [6, Theorem 1.4], we can assume that is complete with respect to -adic topology. Let . Then . Therefore and so . Hence and so by [15, Theorem 3.1] . This implies that is of finite length (see [1, Corollary 7.2.12]). Hence is co-Cohen-Macaulay of Noetherian dimension zero.

Theorem 5. *Let be an unmixed -module. If , then the following statements are equivalent: *(i)*the module is co-Cohen-Macaulay of Noetherian dimension ; *(ii)*the module is either zero or co-Cohen-Macaulay of Noetherian dimension . *

*Proof. *(i) (ii). We use induction on . The case follows by Lemma 4. Let . Let be a strict -sequence on . By [1, Exercise 11.3.9], for all . Note that , since . Therefore is regular. Thus, by Lemma 3, is co-Cohen-Macaulay of Noetherian dimension . Since unmixed and , it follows from the inductive hypothesis that is either zero or co-Cohen-Macaulay of Noetherian dimension . Hence, from the exact sequence
and our assumption we get and so is either zero or co-Cohen-Macaulay of Noetherian dimension . Since is a coregular sequence on , we have being either zero or co-Cohen-Macaulay of Noetherian dimension , as required.

(ii) (i). We prove by induction on . By [8, Corollary 2.5], we can assume that . By our hypothesis there exists . Hence, from the exact sequence
we have the isomorphism for . Since is a coregular sequence on , is either zero or co-Cohen-Macaulay of Noetherian dimension and so is . Hence by induction hypothesis is co-Cohen-Macaulay of Noetherian dimension . Since is a coregular sequence on , it follows, by , that is co-Cohen-Macaulay of Noetherian dimension . This complete the proof.

The following consequence follows by Theorem 5.

Corollary 6. *Let be a Cohen-Macaulay module. Then is co-Cohen-Macaulay of Noetherian dimension . *

The following theorem extends [16, Corollary 3.6].

Theorem 7. *Let for all with . Then is co-Cohen-Macaulay of Noetherian dimension . *

*Proof. *We use induction on . Let . Then, by [1, Corollary 2.1.7] and our assumption, there exists . Hence from the exact sequence
we get the exact sequence
Therefore and so is co-Cohen-Macaulay of Noetherian dimension 2. Thus is co-Cohen-Macaulay of Noetherian dimension 3. The result has been proved in this case. Now suppose that and assume that our assertion is true for . There exists and so from the exact sequence we have the following long exact sequence
Thus there is an isomorphism for all with . Hence for all with and so by the induction hypothesis is co-Cohen-Macaulay of Noetherian dimension . Therefore, in view of , the module is co-Cohen-Macaulay of Noetherian dimension , as required.

The following corollary immediately follows by Theorem 7 and [8, Corollary 2.5].

Corollary 8. *Let be co-Cohen-Macaulay of Noetherian dimension for all with . Then is co-Cohen-Macaulay of Noetherian dimension . *

#### Acknowledgments

The authors are deeply grateful to the referees for their careful reading of the paper and the helpful suggestions.

#### References

- M. P. Brodmann and R. Y. Sharp,
*Local Cohomology: An Algebraic Introduction with Geometric Applications*, Cambridge University Press, 1998. - R. N. Roberts, “Krull dimension for Artinian modules over quasi local commutative rings,”
*Quarterly Journal of Mathematics*, vol. 26, no. 1, pp. 269–273, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Kirby, “Dimension and length for Artinian modules,”
*Quarterly Journal of Mathematics*, vol. 41, no. 4, pp. 419–429, 1990. View at Publisher · View at Google Scholar · View at Scopus - A. Ooishi, “Matlis duality and width of a module,”
*Hiroshima Mathematical Journal*, vol. 6, no. 3, pp. 573–587, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Tang and H. Zakeri, “Co-Cohen-Macaulay modules and modules of generalized fractions,”
*Communications in Algebra*, vol. 22, no. 6, pp. 2173–2204, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. H. Denizler and R. Y. Sharp, “Co-cohen-macaulay artinian modules over commutative rings,”
*Glasgow Mathematical Journal*, vol. 38, no. 3, pp. 359–366, 1996. View at Google Scholar · View at Scopus - M. R. Pournaki and M. Tousi, “A note on the countable union of prime submodules,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 27, no. 10, pp. 641–643, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Tang, “Local homology and local cohomology,”
*Algebra Colloquium*, vol. 11, no. 4, pp. 467–476, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Mafi, “Co-Cohen-Macaulay modules and generalized local cohomology,”
*Algebra Colloquium*, vol. 18, no. Spec01, pp. 807–813, 2011. View at Google Scholar - M. Nagata,
*Local Rings*, Interscience, New York, NY, USA, 1962. - I. G. Macdonald, “Secondary representation of modules over a commutative ring,”
*Symposia Mathematica*, vol. 11, pp. 23–43, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. T. Cuong, M. Morales, and L. T. Nhan, “The finiteness of certain sets of attached prime ideals and the length of generalized fractions,”
*Journal of Pure and Applied Algebra*, vol. 189, no. 1–3, pp. 109–121, 2004. View at Publisher · View at Google Scholar · View at Scopus - N. T. Cuong and T. T. Nam, “A local homology theory for linearly compact modules,”
*Journal of Algebra*, vol. 319, no. 11, pp. 4712–4737, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. P. Brodmann and R. Y. Sharp, “On the dimension and multiplicity of local cohomology modules,”
*Nagoya Mathematical Journal*, vol. 167, pp. 217–233, 2002. View at Google Scholar · View at Scopus - T. N. An and L. T. Nhan, “On the unmixedness and universal catenaricity of local rings and local cohomology modules,”
*Journal of Algebra*, vol. 321, no. 1, pp. 303–311, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Mafi, “Some criteria for the Cohen-Macaulay property and local cohomology,”
*Acta Mathematica Sinica (English Series)*, vol. 25, no. 6, pp. 917–922, 2009. View at Publisher · View at Google Scholar · View at Scopus