Research Article Co-Cohen-Macaulay Modules and Local...
4
Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 912643, 3 pages http://dx.doi.org/10.1155/2013/912643 Research Article Co-Cohen-Macaulay Modules and Local Cohomology Hero Saremi 1 and Amir Mafi 2 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 Correspondence should be addressed to Amir Mafi; a mafi@ipm.ir Received 13 August 2012; Revised 6 January 2013; Accepted 19 January 2013 Academic Editor: Feng Feng Copyright © 2013 H. Saremi and A. Mafi. is 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. Let (, m) be a commutative Noetherian local ring and let be a finitely generated -module of dimension . en the following statements hold: (a) if width ( m ()) ≥ −1 for all with 2≤<, then m () is co-Cohen-Macaulay of Noetherian dimension ; (b) if is an unmixed -module and depth ≥−1, then m () is co-Cohen-Macaulay of Noetherian dimension if and only if −1 m () is either zero or co-Cohen-Macaulay of Noetherian dimension −2. As consequence, if m () is co-Cohen- Macaulay of Noetherian dimension for all with 0≤<, then m () is co-Cohen-Macaulay of Noetherian dimension . 1. Introduction roughout this paper, let (, m) be a commutative Noethe- rian local ring and let be a finitely generated -module of dimension . We denote the ith local cohomology module of with respect to m by m (). It is well known that m () is Artinian for all (cf. [1]). e Noetherian dimension of an Artinian -module , denoted by -dim , is defined inductively as follows: when =0, put -dim = −1. en by induction, for any integer ≥0, put -dim = if -dim < is false, and for any ascending chain 0 ⊆ 1 ⊆ 2 ⊆ ⋅⋅⋅ of submodules of there exists an integer 0 such that -dim( +1 / )< for all > 0 . erefore -dim = 0 if and only if is a nonzero Noetherian module. Moreover, if 0 → → → → 0 is an exact sequence of Artinian modules, then -dim = max{-dim ,-dim }. Let 1 ,..., ∈ m. 1 ,..., is an -coregular sequence if 0: ( 1 ,..., −1 ) →0: ( 1 ,..., −1 ) is surjective for = 1,..., and 0: ( 1 ,..., ) ̸ =0. e width of , denoted by width (), is the length of any maximal -coregular sequence in m. For any -coregular element ∈ m, we have that -dim(0: ) = -dim −1 and width(0: ) = width −1. Details about -dim and width can be found in Roberts [2], Kirby [3], and Ooishi [4]; there is a general fact: for any Artinian -module width ≤-dim <∞ holds and is co-Cohen-Macaulay if and only if width = -dim holds (cf. [5–7]). Tang [8] has shown that if either ≤2 or is Cohen-Macaulay, then m () is co-Cohen- Macaulay (see also [9]). Following Nagata [10], is unmixed if dim( /p)= for all p ∈ Ass . e main aim of this paper is to prove the following theorem. eorem 1. e following statements are true. (a) If width ( m ()) ≥ −1 for all with 2 ≤ < , then m () is co-Cohen-Macaulay of Noetherian dimension . (b) If is an unmixed -module and depth ≥− 1, then m () is co-Cohen-Macaulay of Noetherian dimension if and only if −1 m () is either zero or co-Cohen-Macaulay of Noetherian dimension −2. 2. The Results Following Macdonald [11], every Artinian -module has minimal secondary representation = 1 +⋅⋅⋅+ , where is p secondary. e set { 1 ,..., } is independent of the choice of the minimal secondary representation of . is set is called the set of attached prime ideals of and denoted
Transcript of Research Article Co-Cohen-Macaulay Modules and Local...
Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 912643 3 pageshttpdxdoiorg1011552013912643
Research ArticleCo-Cohen-Macaulay Modules and Local Cohomology
Hero Saremi1 and Amir Mafi2
1 Department of Mathematics Sanandaj Branch Islamic Azad University Sanandaj Iran2Department of Mathematics University of Kurdistan Pasdaran Street PO Box 416 Sanandaj Iran
Correspondence should be addressed to Amir Mafi a mafiipmir
Received 13 August 2012 Revised 6 January 2013 Accepted 19 January 2013
Academic Editor Feng Feng
Copyright copy 2013 H Saremi and A Mafi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Let (119877m) be a commutative Noetherian local ring and let119872 be a finitely generated 119877-module of dimension 119889 Then the followingstatements hold (a) if width (119867119894m (119872)) ge 119894minus1 for all 119894with 2 le 119894 lt 119889 then119867119889m (119872) is co-Cohen-Macaulay of Noetherian dimension119889 (b) if119872 is an unmixed 119877-module and depth119872 ge 119889 minus 1 then119867119889m (119872) is co-Cohen-Macaulay of Noetherian dimension 119889 if andonly if 119867119889minus1m (119872) is either zero or co-Cohen-Macaulay of Noetherian dimension 119889 minus 2 As consequence if 119867119894m (119872) is co-Cohen-Macaulay of Noetherian dimension 119894 for all 119894 with 0 le 119894 lt 119889 then119867119889m (119872) is co-Cohen-Macaulay of Noetherian dimension 119889
1 Introduction
Throughout this paper let (119877m) be a commutative Noethe-rian local ring and let119872 be a finitely generated 119877-module ofdimension 119889 We denote the ith local cohomology module of119872with respect tom by119867119894m(119872) It is well known that119867119894m(119872)
is Artinian for all 119894 (cf [1])The Noetherian dimension of an Artinian 119877-module 119860
denoted by119873-dim119860 is defined inductively as follows when119860 = 0 put119873-dim119860 = minus1 Then by induction for any integer119903 ge 0 put 119873-dim119860 = 119903 if 119873-dim119860 lt 119903 is false and forany ascending chain 119860
0sube 1198601sube 1198602sube sdot sdot sdot of submodules
of 119860 there exists an integer 1198990such that 119873-dim(119860
119899+1119860119899) lt
119903 for all 119899 gt 1198990 Therefore 119873-dim119860 = 0 if and only
if 119860 is a nonzero Noetherian module Moreover if 0 rarr
1198601015840 rarr 119860 rarr 11986010158401015840 rarr 0 is an exact sequence of Artinianmodules then 119873-dim119860 = max119873-dim1198601015840 119873-dim11986010158401015840 Let1199091 119909
119899isin m 119909
1 119909
119899is an 119860-coregular sequence if
0119860(1199091 119909
119894minus1)119877119909119894
997888rarr 0119860(1199091 119909
119894minus1)119877 is surjective for 119894 =
1 119899 and 0119860(1199091 119909
119899)119877 = 0 The width of 119860 denoted by
width (119860) is the length of anymaximal119860-coregular sequencein m For any 119860-coregular element 119909 isin m we have that119873-dim(0
119860119909) = 119873-dim119860minus1 and width(0
119860119909) = width119860minus1
Details about119873-dim119860 and width119860 can be found in Roberts[2] Kirby [3] and Ooishi [4] there is a general fact forany Artinian 119877-module 119860width119860 le 119873-dim119860 lt infin holds
and 119860 is co-Cohen-Macaulay if and only if width119860 =
119873-dim119860 holds (cf [5ndash7]) Tang [8] has shown that if either119889 le 2 or 119872 is Cohen-Macaulay then 119867
119889
m(119872) is co-Cohen-Macaulay (see also [9]) FollowingNagata [10]119872 is unmixedif dim(p) = 119889 for all p isin Ass
The main aim of this paper is to prove the followingtheorem
Theorem 1 The following statements are true
(a) If width (119867119894m(119872)) ge 119894 minus 1 for all 119894 with 2 le 119894 lt
119889 then 119867119889m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889
(b) If 119872 is an unmixed 119877-module and depth 119872 ge 119889 minus
1 then 119867119889m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889 if and only if 119867119889minus1m (119872) is either zero orco-Cohen-Macaulay of Noetherian dimension 119889 minus 2
2 The Results
Following Macdonald [11] every Artinian 119877-module 119860 hasminimal secondary representation 119860 = 119860
1+ sdot sdot sdot + 119860
119899 where
119860119894is p119894secondary The set 119901
1 119901
119899 is independent of the
choice of the minimal secondary representation of 119860 Thisset is called the set of attached prime ideals of 119860 and denoted
2 Journal of Mathematics
by Att119860 The set of all minimal elements of Att 119860 is exactlythe set of all minimal elements of Var(Ann119860) A sequence(1199091 119909
119899) of elements inm is called a strict f-sequence of119872
if 119909119895+1
notin p for all p isin cup119889minus119895
119894=1Att(119867119894m(119872(119909
1 119909
119895)119872) m
for all 119895 = 0 1 119899 minus 1 This notion was introduced in [12]
Lemma 2 (see [9]) For all integer 0 le 119894 lt 119889 one has119873-dim (119867119894m(119872)) le 119894 and119873-dim (119867119889m(119872)) = 119889
Lemma 3 Let 119909 isin m be a strict 119891-sequence of 119872 Then thefollowing statements are true
(i) Suppose that 119889 ge 3 and119867119889m(119872) is co-Cohen-Macaulayof Noetherian dimension 119889 Then119867119889minus1m (119872119909119872) is alsoco-Cohen-Macaulay of Noetherian dimension 119889 minus 1
(ii) Suppose that 119867119889minus1m (119872119909119872) is co-Cohen-Macaulay ofNoetherian dimension 119889 minus 1 and width (119867119889m(119872)) ge
3 Then 119867119889m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889
Proof (i) By our hypothesis and using [1 Exercise 1139] wehave ℓ(0
119872119909) lt infin Hence from the exact sequences
0 997888rarr (0119872119909) 997888rarr 119872 997888rarr
119872
0119872119909997888rarr 0
0 997888rarr119872
(0119872119909)
119909
997888rarr 119872 997888rarr119872
119909119872997888rarr 0
(1)
we get the exact sequence
0 997888rarr119867119894m (119872)
119909119867119894m (119872)997888rarr 119867
119894
m (119872
119909119872) 997888rarr (0
119867119894+1
m (119872)119909) 997888rarr 0 (dagger)
(2)
for all 119894 = 1 2 119889 minus 1 Thus in case 119894 = 119889 minus 1 wehave that (0
119867119889
m(119872)119909) is co-Cohen-Macaulay of Noetherian
dimension 119889 minus 1 By the choice of 119909 isin m the module119867119889minus1m (119872)119909119867119889minus1m (119872) is an 119877-module of finite length More-over since width (119867119889minus1m (119872119909119872)) ge min2 119889 minus 1 gt 0 by[8 Proposition 24] we have119867m
0(119867119889minus1
m (119872119909119872)) = 0 by [13Theorem 411] Thus by [13 Corollary 37] the long exactlocal homology sequence with respect to 119894 = 119889 minus 1 overthe exact sequence (dagger) provides 119867119889minus1m (119872)119909119867119889minus1m (119872) = 0Hence there is an isomorphism 119867119889minus1m (119872119909119872) cong (0
119867119889
m(119872)119909)
and so 119867119889minus1m (119872119909119872) is a co-Cohen-Macaulay module ofNoetherian dimension 119889 minus 1
The proof of (ii) follows by the same arguments as in theproof of (i)
Brodmann and Sharp [14] for all integer 119894 ge 0 defined theset p isin Spec(119877) 119867119894minusdim119877pp119877p
(119872p) = 0 the ith pseudo supportof119872 and denoted by Psupp119894(119872) Note that if 119877 is completewith respect to m-adic topology then by [15 Theorem 31]Var(Ann (119867119894m(119872))) = Psupp119894(119872) The module 119872 satisfiesSerrersquos condition 119878
119899 where 119899 is nonnegative integer provided
depth119872p ge min119899 dim119872p for all p isin Supp(119872) Note that
119872 satisfies the condition Serre 1198781if and only if 119872 has no
imbedded primes that is Ass119872 = min Ass119872 is unmixed
Lemma 4 Let 119872 be unmixed and 119889 le 2 Then 119867119889minus1m (119872) iseither zero or co-Cohen-Macaulay of Noetherian dimension 119889minus2
Proof When 119889 = 0 it is trivial We assume that 119889 = 1Since 119872 is unmixed then depth119872 ge 1 and so we have1198670
m(119872) = 0 Thus the result has been proved in this caseNowassume that119889 = 2 and1198671m(119872) = 0 By using [6Theorem14] we can assume that 119877 is complete with respect to m-adic topology Let p isin Psupp1(119872) Then1198671minusdim119877pp119877p
(119872p) = 0Therefore 1 le depth119872p le 1 minus dim119877p and so dim119877p =
0 Hence Psupp1(119872) = m and so by [15 Theorem 31]Att(1198671m(119872)) sube m This implies that 1198671m(119872) is of finitelength (see [1 Corollary 7212]) Hence1198671m(119872) is co-Cohen-Macaulay of Noetherian dimension zero
Theorem 5 Let 119872 be an unmixed 119877-module If depth119872 ge
119889 minus 1 then the following statements are equivalent
(i) the module119867119889m(119872) is co-Cohen-Macaulay of Noethe-rian dimension 119889
(ii) the module 119867119889minus1
m (119872) is either zero or co-Cohen-Macaulay of Noetherian dimension 119889 minus 2
Proof (i) rArr (ii) We use induction on 119889 The case 119889 le 2
follows by Lemma 4 Let 119889 ge 3 Let 119909 isin m be a strict 119891-sequence on 119872 By [1 Exercise 1139] 119909 notin p for all p isin
Ass119872 m Note that m notin Ass119872 since depth119872 gt 0Therefore 119909 is 119872 regular Thus by Lemma 3 119867119889minus1m (119872119909119872)
is co-Cohen-Macaulay of Noetherian dimension 119889 minus 1 Since119872119909119872 unmixed and depth119872119909119872 ge 119889 minus 2 it follows fromthe inductive hypothesis that 119867119889minus2m (119872119909119872) is either zero orco-Cohen-Macaulay of Noetherian dimension 119889 minus 3 Hencefrom the exact sequence
0 997888rarr119867119889minus2
m (119872)
119909119867119889minus2m (119872)997888rarr 119867
119889minus2
m (119872
119909119872) 997888rarr (0
119867119889minus1
m (119872)119909) 997888rarr 0
(3)
and our assumption we get 119867119889minus2m (119872119909119872) cong (0119867119889minus1
m (119872)119909)
and so (0119867119889minus1
m (119872)119909) is either zero or co-Cohen-Macaulay of
Noetherian dimension 119889 minus 3 Since 119909 is a coregular sequenceon 119867119889minus1m (119872) we have 119867119889minus1m (119872) being either zero or co-Cohen-Macaulay of Noetherian dimension 119889minus2 as required
(ii) rArr (i) We prove by induction on 119889 By [8 Corollary25] we can assume that 119889 ge 3 By our hypothesis there exists119909 isin m (cuppisinAss(119872)p) cup (cupqisinAtt(119867119889minus1m (119872))
q) Hence from theexact sequence
0 997888rarr119867119894m (119872)
119909119867119894m (119872)997888rarr 119867
119894
m (119872
119909119872) 997888rarr (0
119867119894+1
m (119872)119909) 997888rarr 0
(4)
we have the isomorphism 119867119894m(119872119909119872) cong (0119867119894+1
m (119872)119909)(Dagger)
for 119894 isin 119889 minus 2 119889 minus 1 Since 119909 is a coregular sequence on
Journal of Mathematics 3
119867119889minus1m (119872) (0119867119889minus1
m (119872)119909) is either zero or co-Cohen-Macaulay
of Noetherian dimension 119889 minus 3 and so is 119867119889minus2m (119872119909119872)Hence by induction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulay of Noetherian dimension 119889 minus 1 Since 119909 is acoregular sequence on119867119889m(119872) it follows by (Dagger) that119867119889m(119872)
is co-Cohen-Macaulay of Noetherian dimension 119889 Thiscomplete the proof
The following consequence follows byTheorem 5
Corollary 6 Let 119872 be a Cohen-Macaulay module Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension 119889
The following theorem extends [16 Corollary 36]
Theorem7 Let width (119867119894m(119872)) ge 119894minus1 for all 119894with 2 le 119894 lt 119889Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension119889
Proof We use induction on 119889 Let 119889 = 3 Then by [1Corollary 217] and our assumption there exists 119909 isin
m (cuppisinAss(119872)p) cup (cupqisinAtt(1198672m(119872))q) Hence from the exactsequence
0 997888rarr 119872119909
997888rarr 119872 997888rarr119872
119909119872997888rarr 0 (lowast) (5)
we get the exact sequence
0 997888rarr 1198672
m (119872
119909119872) 997888rarr 119867
3
m (119872)119909
997888rarr 1198673
m (119872) 997888rarr 0 (6)
Therefore1198672m(119872119909119872) cong (01198673
m(119872)119909) and so (0
1198673
m(119872)119909) is co-
Cohen-Macaulay of Noetherian dimension 2Thus1198673m(119872) isco-Cohen-Macaulay of Noetherian dimension 3 The resulthas been proved in this case Now suppose that 119889 gt 3 andassume that our assertion is true for 119889 minus 1 There exists 119909 isin
m (cuppisinAss(119872)p)cup (cup119889minus1
119894=2cupqisinAtt(119867119894m(119872))q) and so from the exact
sequence (lowast) we have the following long exact sequence
sdot sdot sdot 997888rarr 119867119894
m (119872
119909119872) 997888rarr 119867
119894
m (119872)119909
997888rarr 119867119894
m (119872) 997888rarr sdot sdot sdot
(7)
Thus there is an isomorphism119867119894m(119872119909119872) cong (0119867119894+1
m (119872)119909)(⋆)
for all 119894 with 2 le 119894 lt 119889 Hence width (119867119894m(119872119909119872)) ge
119894 minus 1 for all 119894 with 2 le 119894 lt 119889 minus 1 and so by theinduction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulayof Noetherian dimension 119889 minus 1 Therefore in view of (⋆)the module 119867
119889
m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889 as required
The following corollary immediately follows by Theo-rem 7 and [8 Corollary 25]
Corollary 8 Let 119867119894m(119872) be co-Cohen-Macaulay of Noethe-rian dimension 119894 for all 119894 with 0 le 119894 lt 119889 Then 119867119889m(119872) isco-Cohen-Macaulay of Noetherian dimension 119889
Acknowledgments
Theauthors are deeply grateful to the referees for their carefulreading of the paper and the helpful suggestions
References
[1] M P Brodmann and R Y Sharp Local Cohomology AnAlgebraic Introduction with Geometric Applications CambridgeUniversity Press 1998
[2] R N Roberts ldquoKrull dimension for Artinian modules overquasi local commutative ringsrdquoQuarterly Journal of Mathemat-ics vol 26 no 1 pp 269ndash273 1975
[3] D Kirby ldquoDimension and length for Artinian modulesrdquo Quar-terly Journal of Mathematics vol 41 no 4 pp 419ndash429 1990
[4] A Ooishi ldquoMatlis duality and width of a modulerdquo HiroshimaMathematical Journal vol 6 no 3 pp 573ndash587 1976
[5] Z Tang and H Zakeri ldquoCo-Cohen-Macaulay modules andmodules of generalized fractionsrdquo Communications in Algebravol 22 no 6 pp 2173ndash2204 1994
[6] I H Denizler and R Y Sharp ldquoCo-cohen-macaulay artinianmodules over commutative ringsrdquo Glasgow Mathematical Jour-nal vol 38 no 3 pp 359ndash366 1996
[7] M R Pournaki and M Tousi ldquoA note on the countable unionof prime submodulesrdquo International Journal ofMathematics andMathematical Sciences vol 27 no 10 pp 641ndash643 2001
[8] Z Tang ldquoLocal homology and local cohomologyrdquo AlgebraColloquium vol 11 no 4 pp 467ndash476 2004
[9] A Mafi ldquoCo-Cohen-Macaulay modules and generalized localcohomologyrdquo Algebra Colloquium vol 18 no Spec01 pp 807ndash813 2011
[10] M Nagata Local Rings Interscience New York NY USA 1962[11] I G Macdonald ldquoSecondary representation of modules over a
commutative ringrdquo Symposia Mathematica vol 11 pp 23ndash431973
[12] N T Cuong M Morales and L T Nhan ldquoThe finitenessof certain sets of attached prime ideals and the length ofgeneralized fractionsrdquo Journal of Pure and Applied Algebra vol189 no 1ndash3 pp 109ndash121 2004
[13] N T Cuong and T T Nam ldquoA local homology theory forlinearly compact modulesrdquo Journal of Algebra vol 319 no 11pp 4712ndash4737 2008
[14] M P Brodmann and R Y Sharp ldquoOn the dimension and mul-tiplicity of local cohomology modulesrdquo Nagoya MathematicalJournal vol 167 pp 217ndash233 2002
[15] T N An and L T Nhan ldquoOn the unmixedness and universalcatenaricity of local rings and local cohomology modulesrdquoJournal of Algebra vol 321 no 1 pp 303ndash311 2009
[16] A Mafi ldquoSome criteria for the Cohen-Macaulay property andlocal cohomologyrdquo Acta Mathematica Sinica (English Series)vol 25 no 6 pp 917ndash922 2009
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2 Journal of Mathematics
by Att119860 The set of all minimal elements of Att 119860 is exactlythe set of all minimal elements of Var(Ann119860) A sequence(1199091 119909
119899) of elements inm is called a strict f-sequence of119872
if 119909119895+1
notin p for all p isin cup119889minus119895
119894=1Att(119867119894m(119872(119909
1 119909
119895)119872) m
for all 119895 = 0 1 119899 minus 1 This notion was introduced in [12]
Lemma 2 (see [9]) For all integer 0 le 119894 lt 119889 one has119873-dim (119867119894m(119872)) le 119894 and119873-dim (119867119889m(119872)) = 119889
Lemma 3 Let 119909 isin m be a strict 119891-sequence of 119872 Then thefollowing statements are true
(i) Suppose that 119889 ge 3 and119867119889m(119872) is co-Cohen-Macaulayof Noetherian dimension 119889 Then119867119889minus1m (119872119909119872) is alsoco-Cohen-Macaulay of Noetherian dimension 119889 minus 1
(ii) Suppose that 119867119889minus1m (119872119909119872) is co-Cohen-Macaulay ofNoetherian dimension 119889 minus 1 and width (119867119889m(119872)) ge
3 Then 119867119889m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889
Proof (i) By our hypothesis and using [1 Exercise 1139] wehave ℓ(0
119872119909) lt infin Hence from the exact sequences
0 997888rarr (0119872119909) 997888rarr 119872 997888rarr
119872
0119872119909997888rarr 0
0 997888rarr119872
(0119872119909)
119909
997888rarr 119872 997888rarr119872
119909119872997888rarr 0
(1)
we get the exact sequence
0 997888rarr119867119894m (119872)
119909119867119894m (119872)997888rarr 119867
119894
m (119872
119909119872) 997888rarr (0
119867119894+1
m (119872)119909) 997888rarr 0 (dagger)
(2)
for all 119894 = 1 2 119889 minus 1 Thus in case 119894 = 119889 minus 1 wehave that (0
119867119889
m(119872)119909) is co-Cohen-Macaulay of Noetherian
dimension 119889 minus 1 By the choice of 119909 isin m the module119867119889minus1m (119872)119909119867119889minus1m (119872) is an 119877-module of finite length More-over since width (119867119889minus1m (119872119909119872)) ge min2 119889 minus 1 gt 0 by[8 Proposition 24] we have119867m
0(119867119889minus1
m (119872119909119872)) = 0 by [13Theorem 411] Thus by [13 Corollary 37] the long exactlocal homology sequence with respect to 119894 = 119889 minus 1 overthe exact sequence (dagger) provides 119867119889minus1m (119872)119909119867119889minus1m (119872) = 0Hence there is an isomorphism 119867119889minus1m (119872119909119872) cong (0
119867119889
m(119872)119909)
and so 119867119889minus1m (119872119909119872) is a co-Cohen-Macaulay module ofNoetherian dimension 119889 minus 1
The proof of (ii) follows by the same arguments as in theproof of (i)
Brodmann and Sharp [14] for all integer 119894 ge 0 defined theset p isin Spec(119877) 119867119894minusdim119877pp119877p
(119872p) = 0 the ith pseudo supportof119872 and denoted by Psupp119894(119872) Note that if 119877 is completewith respect to m-adic topology then by [15 Theorem 31]Var(Ann (119867119894m(119872))) = Psupp119894(119872) The module 119872 satisfiesSerrersquos condition 119878
119899 where 119899 is nonnegative integer provided
depth119872p ge min119899 dim119872p for all p isin Supp(119872) Note that
119872 satisfies the condition Serre 1198781if and only if 119872 has no
imbedded primes that is Ass119872 = min Ass119872 is unmixed
Lemma 4 Let 119872 be unmixed and 119889 le 2 Then 119867119889minus1m (119872) iseither zero or co-Cohen-Macaulay of Noetherian dimension 119889minus2
Proof When 119889 = 0 it is trivial We assume that 119889 = 1Since 119872 is unmixed then depth119872 ge 1 and so we have1198670
m(119872) = 0 Thus the result has been proved in this caseNowassume that119889 = 2 and1198671m(119872) = 0 By using [6Theorem14] we can assume that 119877 is complete with respect to m-adic topology Let p isin Psupp1(119872) Then1198671minusdim119877pp119877p
(119872p) = 0Therefore 1 le depth119872p le 1 minus dim119877p and so dim119877p =
0 Hence Psupp1(119872) = m and so by [15 Theorem 31]Att(1198671m(119872)) sube m This implies that 1198671m(119872) is of finitelength (see [1 Corollary 7212]) Hence1198671m(119872) is co-Cohen-Macaulay of Noetherian dimension zero
Theorem 5 Let 119872 be an unmixed 119877-module If depth119872 ge
119889 minus 1 then the following statements are equivalent
(i) the module119867119889m(119872) is co-Cohen-Macaulay of Noethe-rian dimension 119889
(ii) the module 119867119889minus1
m (119872) is either zero or co-Cohen-Macaulay of Noetherian dimension 119889 minus 2
Proof (i) rArr (ii) We use induction on 119889 The case 119889 le 2
follows by Lemma 4 Let 119889 ge 3 Let 119909 isin m be a strict 119891-sequence on 119872 By [1 Exercise 1139] 119909 notin p for all p isin
Ass119872 m Note that m notin Ass119872 since depth119872 gt 0Therefore 119909 is 119872 regular Thus by Lemma 3 119867119889minus1m (119872119909119872)
is co-Cohen-Macaulay of Noetherian dimension 119889 minus 1 Since119872119909119872 unmixed and depth119872119909119872 ge 119889 minus 2 it follows fromthe inductive hypothesis that 119867119889minus2m (119872119909119872) is either zero orco-Cohen-Macaulay of Noetherian dimension 119889 minus 3 Hencefrom the exact sequence
0 997888rarr119867119889minus2
m (119872)
119909119867119889minus2m (119872)997888rarr 119867
119889minus2
m (119872
119909119872) 997888rarr (0
119867119889minus1
m (119872)119909) 997888rarr 0
(3)
and our assumption we get 119867119889minus2m (119872119909119872) cong (0119867119889minus1
m (119872)119909)
and so (0119867119889minus1
m (119872)119909) is either zero or co-Cohen-Macaulay of
Noetherian dimension 119889 minus 3 Since 119909 is a coregular sequenceon 119867119889minus1m (119872) we have 119867119889minus1m (119872) being either zero or co-Cohen-Macaulay of Noetherian dimension 119889minus2 as required
(ii) rArr (i) We prove by induction on 119889 By [8 Corollary25] we can assume that 119889 ge 3 By our hypothesis there exists119909 isin m (cuppisinAss(119872)p) cup (cupqisinAtt(119867119889minus1m (119872))
q) Hence from theexact sequence
0 997888rarr119867119894m (119872)
119909119867119894m (119872)997888rarr 119867
119894
m (119872
119909119872) 997888rarr (0
119867119894+1
m (119872)119909) 997888rarr 0
(4)
we have the isomorphism 119867119894m(119872119909119872) cong (0119867119894+1
m (119872)119909)(Dagger)
for 119894 isin 119889 minus 2 119889 minus 1 Since 119909 is a coregular sequence on
Journal of Mathematics 3
119867119889minus1m (119872) (0119867119889minus1
m (119872)119909) is either zero or co-Cohen-Macaulay
of Noetherian dimension 119889 minus 3 and so is 119867119889minus2m (119872119909119872)Hence by induction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulay of Noetherian dimension 119889 minus 1 Since 119909 is acoregular sequence on119867119889m(119872) it follows by (Dagger) that119867119889m(119872)
is co-Cohen-Macaulay of Noetherian dimension 119889 Thiscomplete the proof
The following consequence follows byTheorem 5
Corollary 6 Let 119872 be a Cohen-Macaulay module Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension 119889
The following theorem extends [16 Corollary 36]
Theorem7 Let width (119867119894m(119872)) ge 119894minus1 for all 119894with 2 le 119894 lt 119889Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension119889
Proof We use induction on 119889 Let 119889 = 3 Then by [1Corollary 217] and our assumption there exists 119909 isin
m (cuppisinAss(119872)p) cup (cupqisinAtt(1198672m(119872))q) Hence from the exactsequence
0 997888rarr 119872119909
997888rarr 119872 997888rarr119872
119909119872997888rarr 0 (lowast) (5)
we get the exact sequence
0 997888rarr 1198672
m (119872
119909119872) 997888rarr 119867
3
m (119872)119909
997888rarr 1198673
m (119872) 997888rarr 0 (6)
Therefore1198672m(119872119909119872) cong (01198673
m(119872)119909) and so (0
1198673
m(119872)119909) is co-
Cohen-Macaulay of Noetherian dimension 2Thus1198673m(119872) isco-Cohen-Macaulay of Noetherian dimension 3 The resulthas been proved in this case Now suppose that 119889 gt 3 andassume that our assertion is true for 119889 minus 1 There exists 119909 isin
m (cuppisinAss(119872)p)cup (cup119889minus1
119894=2cupqisinAtt(119867119894m(119872))q) and so from the exact
sequence (lowast) we have the following long exact sequence
sdot sdot sdot 997888rarr 119867119894
m (119872
119909119872) 997888rarr 119867
119894
m (119872)119909
997888rarr 119867119894
m (119872) 997888rarr sdot sdot sdot
(7)
Thus there is an isomorphism119867119894m(119872119909119872) cong (0119867119894+1
m (119872)119909)(⋆)
for all 119894 with 2 le 119894 lt 119889 Hence width (119867119894m(119872119909119872)) ge
119894 minus 1 for all 119894 with 2 le 119894 lt 119889 minus 1 and so by theinduction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulayof Noetherian dimension 119889 minus 1 Therefore in view of (⋆)the module 119867
119889
m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889 as required
The following corollary immediately follows by Theo-rem 7 and [8 Corollary 25]
Corollary 8 Let 119867119894m(119872) be co-Cohen-Macaulay of Noethe-rian dimension 119894 for all 119894 with 0 le 119894 lt 119889 Then 119867119889m(119872) isco-Cohen-Macaulay of Noetherian dimension 119889
Acknowledgments
Theauthors are deeply grateful to the referees for their carefulreading of the paper and the helpful suggestions
References
[1] M P Brodmann and R Y Sharp Local Cohomology AnAlgebraic Introduction with Geometric Applications CambridgeUniversity Press 1998
[2] R N Roberts ldquoKrull dimension for Artinian modules overquasi local commutative ringsrdquoQuarterly Journal of Mathemat-ics vol 26 no 1 pp 269ndash273 1975
[3] D Kirby ldquoDimension and length for Artinian modulesrdquo Quar-terly Journal of Mathematics vol 41 no 4 pp 419ndash429 1990
[4] A Ooishi ldquoMatlis duality and width of a modulerdquo HiroshimaMathematical Journal vol 6 no 3 pp 573ndash587 1976
[5] Z Tang and H Zakeri ldquoCo-Cohen-Macaulay modules andmodules of generalized fractionsrdquo Communications in Algebravol 22 no 6 pp 2173ndash2204 1994
[6] I H Denizler and R Y Sharp ldquoCo-cohen-macaulay artinianmodules over commutative ringsrdquo Glasgow Mathematical Jour-nal vol 38 no 3 pp 359ndash366 1996
[7] M R Pournaki and M Tousi ldquoA note on the countable unionof prime submodulesrdquo International Journal ofMathematics andMathematical Sciences vol 27 no 10 pp 641ndash643 2001
[8] Z Tang ldquoLocal homology and local cohomologyrdquo AlgebraColloquium vol 11 no 4 pp 467ndash476 2004
[9] A Mafi ldquoCo-Cohen-Macaulay modules and generalized localcohomologyrdquo Algebra Colloquium vol 18 no Spec01 pp 807ndash813 2011
[10] M Nagata Local Rings Interscience New York NY USA 1962[11] I G Macdonald ldquoSecondary representation of modules over a
commutative ringrdquo Symposia Mathematica vol 11 pp 23ndash431973
[12] N T Cuong M Morales and L T Nhan ldquoThe finitenessof certain sets of attached prime ideals and the length ofgeneralized fractionsrdquo Journal of Pure and Applied Algebra vol189 no 1ndash3 pp 109ndash121 2004
[13] N T Cuong and T T Nam ldquoA local homology theory forlinearly compact modulesrdquo Journal of Algebra vol 319 no 11pp 4712ndash4737 2008
[14] M P Brodmann and R Y Sharp ldquoOn the dimension and mul-tiplicity of local cohomology modulesrdquo Nagoya MathematicalJournal vol 167 pp 217ndash233 2002
[15] T N An and L T Nhan ldquoOn the unmixedness and universalcatenaricity of local rings and local cohomology modulesrdquoJournal of Algebra vol 321 no 1 pp 303ndash311 2009
[16] A Mafi ldquoSome criteria for the Cohen-Macaulay property andlocal cohomologyrdquo Acta Mathematica Sinica (English Series)vol 25 no 6 pp 917ndash922 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
119867119889minus1m (119872) (0119867119889minus1
m (119872)119909) is either zero or co-Cohen-Macaulay
of Noetherian dimension 119889 minus 3 and so is 119867119889minus2m (119872119909119872)Hence by induction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulay of Noetherian dimension 119889 minus 1 Since 119909 is acoregular sequence on119867119889m(119872) it follows by (Dagger) that119867119889m(119872)
is co-Cohen-Macaulay of Noetherian dimension 119889 Thiscomplete the proof
The following consequence follows byTheorem 5
Corollary 6 Let 119872 be a Cohen-Macaulay module Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension 119889
The following theorem extends [16 Corollary 36]
Theorem7 Let width (119867119894m(119872)) ge 119894minus1 for all 119894with 2 le 119894 lt 119889Then119867119889m(119872) is co-Cohen-Macaulay of Noetherian dimension119889
Proof We use induction on 119889 Let 119889 = 3 Then by [1Corollary 217] and our assumption there exists 119909 isin
m (cuppisinAss(119872)p) cup (cupqisinAtt(1198672m(119872))q) Hence from the exactsequence
0 997888rarr 119872119909
997888rarr 119872 997888rarr119872
119909119872997888rarr 0 (lowast) (5)
we get the exact sequence
0 997888rarr 1198672
m (119872
119909119872) 997888rarr 119867
3
m (119872)119909
997888rarr 1198673
m (119872) 997888rarr 0 (6)
Therefore1198672m(119872119909119872) cong (01198673
m(119872)119909) and so (0
1198673
m(119872)119909) is co-
Cohen-Macaulay of Noetherian dimension 2Thus1198673m(119872) isco-Cohen-Macaulay of Noetherian dimension 3 The resulthas been proved in this case Now suppose that 119889 gt 3 andassume that our assertion is true for 119889 minus 1 There exists 119909 isin
m (cuppisinAss(119872)p)cup (cup119889minus1
119894=2cupqisinAtt(119867119894m(119872))q) and so from the exact
sequence (lowast) we have the following long exact sequence
sdot sdot sdot 997888rarr 119867119894
m (119872
119909119872) 997888rarr 119867
119894
m (119872)119909
997888rarr 119867119894
m (119872) 997888rarr sdot sdot sdot
(7)
Thus there is an isomorphism119867119894m(119872119909119872) cong (0119867119894+1
m (119872)119909)(⋆)
for all 119894 with 2 le 119894 lt 119889 Hence width (119867119894m(119872119909119872)) ge
119894 minus 1 for all 119894 with 2 le 119894 lt 119889 minus 1 and so by theinduction hypothesis 119867119889minus1m (119872119909119872) is co-Cohen-Macaulayof Noetherian dimension 119889 minus 1 Therefore in view of (⋆)the module 119867
119889
m(119872) is co-Cohen-Macaulay of Noetheriandimension 119889 as required
The following corollary immediately follows by Theo-rem 7 and [8 Corollary 25]
Corollary 8 Let 119867119894m(119872) be co-Cohen-Macaulay of Noethe-rian dimension 119894 for all 119894 with 0 le 119894 lt 119889 Then 119867119889m(119872) isco-Cohen-Macaulay of Noetherian dimension 119889
Acknowledgments
Theauthors are deeply grateful to the referees for their carefulreading of the paper and the helpful suggestions
References
[1] M P Brodmann and R Y Sharp Local Cohomology AnAlgebraic Introduction with Geometric Applications CambridgeUniversity Press 1998
[2] R N Roberts ldquoKrull dimension for Artinian modules overquasi local commutative ringsrdquoQuarterly Journal of Mathemat-ics vol 26 no 1 pp 269ndash273 1975
[3] D Kirby ldquoDimension and length for Artinian modulesrdquo Quar-terly Journal of Mathematics vol 41 no 4 pp 419ndash429 1990
[4] A Ooishi ldquoMatlis duality and width of a modulerdquo HiroshimaMathematical Journal vol 6 no 3 pp 573ndash587 1976
[5] Z Tang and H Zakeri ldquoCo-Cohen-Macaulay modules andmodules of generalized fractionsrdquo Communications in Algebravol 22 no 6 pp 2173ndash2204 1994
[6] I H Denizler and R Y Sharp ldquoCo-cohen-macaulay artinianmodules over commutative ringsrdquo Glasgow Mathematical Jour-nal vol 38 no 3 pp 359ndash366 1996
[7] M R Pournaki and M Tousi ldquoA note on the countable unionof prime submodulesrdquo International Journal ofMathematics andMathematical Sciences vol 27 no 10 pp 641ndash643 2001
[8] Z Tang ldquoLocal homology and local cohomologyrdquo AlgebraColloquium vol 11 no 4 pp 467ndash476 2004
[9] A Mafi ldquoCo-Cohen-Macaulay modules and generalized localcohomologyrdquo Algebra Colloquium vol 18 no Spec01 pp 807ndash813 2011
[10] M Nagata Local Rings Interscience New York NY USA 1962[11] I G Macdonald ldquoSecondary representation of modules over a
commutative ringrdquo Symposia Mathematica vol 11 pp 23ndash431973
[12] N T Cuong M Morales and L T Nhan ldquoThe finitenessof certain sets of attached prime ideals and the length ofgeneralized fractionsrdquo Journal of Pure and Applied Algebra vol189 no 1ndash3 pp 109ndash121 2004
[13] N T Cuong and T T Nam ldquoA local homology theory forlinearly compact modulesrdquo Journal of Algebra vol 319 no 11pp 4712ndash4737 2008
[14] M P Brodmann and R Y Sharp ldquoOn the dimension and mul-tiplicity of local cohomology modulesrdquo Nagoya MathematicalJournal vol 167 pp 217ndash233 2002
[15] T N An and L T Nhan ldquoOn the unmixedness and universalcatenaricity of local rings and local cohomology modulesrdquoJournal of Algebra vol 321 no 1 pp 303ndash311 2009
[16] A Mafi ldquoSome criteria for the Cohen-Macaulay property andlocal cohomologyrdquo Acta Mathematica Sinica (English Series)vol 25 no 6 pp 917ndash922 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
