Cousin complexes with applications to local cohomology and
Transcript of Cousin complexes with applications to local cohomology and
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Cousin complexes with applicationsto
local cohomology and commutative rings
Raheleh JafariPhD Thesis under supervison of:
Mohammad T. Dibaei
Faculty of Mathematical Sciences and ComputerTarbiat Moallem University
19 June 201129 Khordad 1390
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Cousin complexes
Uniform local cohomological
annihilators
Attached primes of local
cohomologies
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Cousin complexes
Uniform local cohomological
annihilators
Attached primes of local
cohomologies
Cohen-Macaulay locus
Cohen-Macaulay formal fibres
Generalized Cohen-Mcaulay
modules
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History
- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes
- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology
- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations
- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes
- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules
- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus
- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay
- Some comments
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Contents
I Finite Cousin complexes
- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal
I Attached primes of local cohomology modules
- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules- Application to generalized Cohen-Macaulay modules
I Cohen-Macaulay loci of modules
- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay- Some comments
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Def.A filtration of Spec R is a descending sequence F = (Fi )i≥0 ofsubsets of Spec (R), F0 ⊇ F1 ⊇ F2 ⊇ · · · ⊇ Fi ⊇ · · · , with theproperty that, for each i ∈ N0, every member of ∂Fi = Fi \ Fi+1 isa minimal member of Fi with respect to inclusion. We say thefiltration F admits M if Supp M ⊆ F0.
Notation
For each i ≥ 0, set
Hi = {p ∈ Supp M | htMp ≥ i}.
The sequence (Hi )i≥0 is a filtration of Spec R which admits M andis called the height filtration of M and is denoted by H(M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Def.A filtration of Spec R is a descending sequence F = (Fi )i≥0 ofsubsets of Spec (R), F0 ⊇ F1 ⊇ F2 ⊇ · · · ⊇ Fi ⊇ · · · , with theproperty that, for each i ∈ N0, every member of ∂Fi = Fi \ Fi+1 isa minimal member of Fi with respect to inclusion. We say thefiltration F admits M if Supp M ⊆ F0.
Notation
For each i ≥ 0, set
Hi = {p ∈ Supp M | htMp ≥ i}.
The sequence (Hi )i≥0 is a filtration of Spec R which admits M andis called the height filtration of M and is denoted by H(M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cousin Complexes
Let F = (Fi )i≥0 be a filtration of Spec R which admits anR–module M. The Cousin complex of M with respect to F , is thecomplex
C(F ,M) : 0d−2
−→ M−1d−1
−→ M0 d0
−→ M1 d1
−→ · · · dn−1
−→ Mn dn
−→ Mn+1 −→ · · ·,
where M−1 = M and for n > −1,Mn = ⊕
p∈Fn
(Coker dn−2)p.
I We denote the Cousin complex of M with respect toM–height filtration, H(M), by CR(M).
I We denote the nth cohomology of CR(M) byHn
M := Ker dn/Im dn−1.
I We call the Cousin complex CR(M) finite whenever each HnM
is finite as R–module.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cousin Complexes
Let F = (Fi )i≥0 be a filtration of Spec R which admits anR–module M. The Cousin complex of M with respect to F , is thecomplex
C(F ,M) : 0d−2
−→ M−1d−1
−→ M0 d0
−→ M1 d1
−→ · · · dn−1
−→ Mn dn
−→ Mn+1 −→ · · ·,
where M−1 = M and for n > −1,Mn = ⊕
p∈Fn
(Coker dn−2)p.
I We denote the Cousin complex of M with respect toM–height filtration, H(M), by CR(M).
I We denote the nth cohomology of CR(M) byHn
M := Ker dn/Im dn−1.
I We call the Cousin complex CR(M) finite whenever each HnM
is finite as R–module.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cousin Complexes
Let F = (Fi )i≥0 be a filtration of Spec R which admits anR–module M. The Cousin complex of M with respect to F , is thecomplex
C(F ,M) : 0d−2
−→ M−1d−1
−→ M0 d0
−→ M1 d1
−→ · · · dn−1
−→ Mn dn
−→ Mn+1 −→ · · ·,
where M−1 = M and for n > −1,Mn = ⊕
p∈Fn
(Coker dn−2)p.
I We denote the Cousin complex of M with respect toM–height filtration, H(M), by CR(M).
I We denote the nth cohomology of CR(M) byHn
M := Ker dn/Im dn−1.
I We call the Cousin complex CR(M) finite whenever each HnM
is finite as R–module.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cousin Complexes
Let F = (Fi )i≥0 be a filtration of Spec R which admits anR–module M. The Cousin complex of M with respect to F , is thecomplex
C(F ,M) : 0d−2
−→ M−1d−1
−→ M0 d0
−→ M1 d1
−→ · · · dn−1
−→ Mn dn
−→ Mn+1 −→ · · ·,
where M−1 = M and for n > −1,Mn = ⊕
p∈Fn
(Coker dn−2)p.
I We denote the Cousin complex of M with respect toM–height filtration, H(M), by CR(M).
I We denote the nth cohomology of CR(M) byHn
M := Ker dn/Im dn−1.
I We call the Cousin complex CR(M) finite whenever each HnM
is finite as R–module.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cousin Complexes
Throughout R is a commutative, noetherian ring with non–zeroidentity and M is a finitely generated R–module.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Sharp (1970) shows that M is Cohen-Macaulay if and only if theCousin complex of M, C(M), is exact.
Sharp and Schenzel (1994) show that M satisfies the condition(Sn) if and only if CR(M) is exact at ith term for i ≤ n − 2.
Question
What rings or modules admit finite Cousin complexes and whatproperties these rings or modules have?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Sharp (1970) shows that M is Cohen-Macaulay if and only if theCousin complex of M, C(M), is exact.
Sharp and Schenzel (1994) show that M satisfies the condition(Sn) if and only if CR(M) is exact at ith term for i ≤ n − 2.
Question
What rings or modules admit finite Cousin complexes and whatproperties these rings or modules have?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Sharp (1970) shows that M is Cohen-Macaulay if and only if theCousin complex of M, C(M), is exact.
Sharp and Schenzel (1994) show that M satisfies the condition(Sn) if and only if CR(M) is exact at ith term for i ≤ n − 2.
Question
What rings or modules admit finite Cousin complexes and whatproperties these rings or modules have?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
CM
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
CM
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Dibaei and Tousi (1998) show that if a local ring R which satisfiesthe condition (S2), possesses a dualizing complex, then the Cousincomplex of the canonical module of R is finite.
Dibaei and Tousi (1998) show that if a local ring R satisfies thecondition (S2) and has a canonical module K , then finiteness ofcohomologies of the Cousin complex of K with respect to a certainfiltration is necessary and sufficient condition for R to possess adualizing complex.
Dibaei and Tousi (2001) show that CR(M) is finite, when R is alocal ring which has a dualizing complex and M is equidimensionalwhich satisfies the condition (S2).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Dibaei and Tousi (1998) show that if a local ring R which satisfiesthe condition (S2), possesses a dualizing complex, then the Cousincomplex of the canonical module of R is finite.
Dibaei and Tousi (1998) show that if a local ring R satisfies thecondition (S2) and has a canonical module K , then finiteness ofcohomologies of the Cousin complex of K with respect to a certainfiltration is necessary and sufficient condition for R to possess adualizing complex.
Dibaei and Tousi (2001) show that CR(M) is finite, when R is alocal ring which has a dualizing complex and M is equidimensionalwhich satisfies the condition (S2).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
Dibaei and Tousi (1998) show that if a local ring R which satisfiesthe condition (S2), possesses a dualizing complex, then the Cousincomplex of the canonical module of R is finite.
Dibaei and Tousi (1998) show that if a local ring R satisfies thecondition (S2) and has a canonical module K , then finiteness ofcohomologies of the Cousin complex of K with respect to a certainfiltration is necessary and sufficient condition for R to possess adualizing complex.
Dibaei and Tousi (2001) show that CR(M) is finite, when R is alocal ring which has a dualizing complex and M is equidimensionalwhich satisfies the condition (S2).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
History
”Dibaei (2005)” proves that when all formal fibres of R areCohen-Macaulay and M satisfies (S2), if M is equidimensional,then CR(M) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
![Page 40: Cousin complexes with applications to local cohomology and](https://reader036.fdocuments.in/reader036/viewer/2022071602/613d5fa0736caf36b75c907e/html5/thumbnails/40.jpg)
Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if M is equidimensional and
(i) R is universally catenary,
(ii) all the formal fibers of all the localizations of R areCohen-Macaulay,
(iii) the Cohen-Macaulay locus of each finitely generatedR–algebra is open,
Then CR(M) is finite and finitely many of its cohomologies arenon–zero.
Kawasaki (2008) shows that if R is a catenary ring. Then thefollowing statements are equivalent.
R satisfies the conditions (i), (ii) and (iii).
for any finitely generated equidimensional R–module M,CR(M) is finite and finitely many of its cohomologies arenon–zero.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if R is a universally catenary local ringwith Cohen-Macaulay formal fibres and M is an equidimensionalR–module, then CR(M) is finite.
In the proof of the above result, the assumption that R isuniversally catenary, is used to show that M is equidimensional.So one may consider this theorem as a generalization of the resultof Dibaei (2005).
•••Assume that (R,m) is a local ring such that all of its formal fibersare Cohen-Macaulay and M is an equidimensional R–module.Then CR(M) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if R is a universally catenary local ringwith Cohen-Macaulay formal fibres and M is an equidimensionalR–module, then CR(M) is finite.
In the proof of the above result, the assumption that R isuniversally catenary, is used to show that M is equidimensional.So one may consider this theorem as a generalization of the resultof Dibaei (2005).
•••Assume that (R,m) is a local ring such that all of its formal fibersare Cohen-Macaulay and M is an equidimensional R–module.Then CR(M) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Kawasaki (2008) shows that if R is a universally catenary local ringwith Cohen-Macaulay formal fibres and M is an equidimensionalR–module, then CR(M) is finite.
In the proof of the above result, the assumption that R isuniversally catenary, is used to show that M is equidimensional.So one may consider this theorem as a generalization of the resultof Dibaei (2005).
•••Assume that (R,m) is a local ring such that all of its formal fibersare Cohen-Macaulay and M is an equidimensional R–module.Then CR(M) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
•••Assume that (R,m) is a local ring.
I If 0 −→ Lf−→ M
g−→ N is an exact sequence of R–moduleswith the property that htMp ≥ 2 for all p ∈ Supp N, then
CR(L)′ ∼= CR(M)′;
in particular, CR(L) is finite if and only if CR(M) is finite.
I If Lf−→ M
g−→ N −→ 0 is an exact sequence of R–moduleswith the property that htMp ≥ 1 for all p ∈ Supp L, then
CR(M)′ ∼= CR(N)′;
in particular, CR(M) is finite if and only if CR(N) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
•••Assume that (R,m) is a local ring.
I If 0 −→ Lf−→ M
g−→ N is an exact sequence of R–moduleswith the property that htMp ≥ 2 for all p ∈ Supp N, then
CR(L)′ ∼= CR(M)′;
in particular, CR(L) is finite if and only if CR(M) is finite.
I If Lf−→ M
g−→ N −→ 0 is an exact sequence of R–moduleswith the property that htMp ≥ 1 for all p ∈ Supp L, then
CR(M)′ ∼= CR(N)′;
in particular, CR(M) is finite if and only if CR(N) is finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
Cor .
If (R,m) is local, then there is a finitely generated R–module Nwhich satisfies the condition (S1) with Supp N = Supp M andHi
N∼= Hi
M for all i ≥ 0.
Cor .
If (R,m) is a homomorphic image of a Gorenstein local ring, thenthere is a finitely generated R–module N which satisfies thecondition (S2) with Supp N = Supp M and Hi
N∼= Hi
M for all i ≥ 0.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
Cor .
If (R,m) is local, then there is a finitely generated R–module Nwhich satisfies the condition (S1) with Supp N = Supp M andHi
N∼= Hi
M for all i ≥ 0.
Cor .
If (R,m) is a homomorphic image of a Gorenstein local ring, thenthere is a finitely generated R–module N which satisfies thecondition (S2) with Supp N = Supp M and Hi
N∼= Hi
M for all i ≥ 0.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
Rem.
If M is of finite dimension and that CR(M) is finite, then⋂i≥−1
(0 :R Hi ) 6⊆⋃
p∈MinM
p.
•••Assume that a is an ideal of R such that aM 6= M. Then, for eachinteger r with 0 ≤ r < htMa,
r−1∏i=−1
(0 :R Hi ) ⊆r⋂
i=0
(0 :R Ext iR(R/a,M)) ⊆
r⋂i=0
(0 :R Hia(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohomology modules of Cousin complexes
Rem.
If M is of finite dimension and that CR(M) is finite, then⋂i≥−1
(0 :R Hi ) 6⊆⋃
p∈MinM
p.
•••Assume that a is an ideal of R such that aM 6= M. Then, for eachinteger r with 0 ≤ r < htMa,
r−1∏i=−1
(0 :R Hi ) ⊆r⋂
i=0
(0 :R Ext iR(R/a,M)) ⊆
r⋂i=0
(0 :R Hia(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Def.x ∈ R is called a uniform local cohomological annihilator ofM, if x ∈ R \ ∪p∈MinMp and for each maximal ideal m of R,
xHim(M) = 0 for all i < dim Mm.
x is called a strong uniform local cohomological annihilator ofM if x is a uniform local cohomological annihilator of Mp forevery prime ideal p in Supp M.
We use the notation u.l.c.a instead of uniform local cohomologicalannihilator, for short.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Def.x ∈ R is called a uniform local cohomological annihilator ofM, if x ∈ R \ ∪p∈MinMp and for each maximal ideal m of R,
xHim(M) = 0 for all i < dim Mm.
x is called a strong uniform local cohomological annihilator ofM if x is a uniform local cohomological annihilator of Mp forevery prime ideal p in Supp M.
We use the notation u.l.c.a instead of uniform local cohomologicalannihilator, for short.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Def.x ∈ R is called a uniform local cohomological annihilator ofM, if x ∈ R \ ∪p∈MinMp and for each maximal ideal m of R,
xHim(M) = 0 for all i < dim Mm.
x is called a strong uniform local cohomological annihilator ofM if x is a uniform local cohomological annihilator of Mp forevery prime ideal p in Supp M.
We use the notation u.l.c.a instead of uniform local cohomologicalannihilator, for short.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Def.x ∈ R is called a uniform local cohomological annihilator ofM, if x ∈ R \ ∪p∈MinMp and for each maximal ideal m of R,
xHim(M) = 0 for all i < dim Mm.
x is called a strong uniform local cohomological annihilator ofM if x is a uniform local cohomological annihilator of Mp forevery prime ideal p in Supp M.
We use the notation u.l.c.a instead of uniform local cohomologicalannihilator, for short.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
•••Assume that M is of finite dimension and that CR(M) is finite.Then M has a u.l.c.a
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Zhou (2006) proves that over a ring R of finite dimension d , thefollowing conditions are equivalent.
R has a u.l.c.a.
R is locally equidimensional, and R/p has a u.l.c.a for allminimal prime ideal p of R.
•••The following conditions are equivalent.
M has a u.l.c.a.
M is locally equidimensional and R/p has a u.l.c.a for allp ∈ Min M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Cor .
Let M and N be finitely generated R–modules of finite dimensionssuch that Supp M = Supp N. Then the following conditions areequivalent.
M has a u.l.c.a.
N has a u.l.c.a.
Cor .
If M has a u.l.c.a, then M is equidimensional and R/0 :R M isuniversally catenary.
•••If CR(M) is finite, M is equidimensional and R/0 :R M isuniversally catenary.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Cor .
Let M and N be finitely generated R–modules of finite dimensionssuch that Supp M = Supp N. Then the following conditions areequivalent.
M has a u.l.c.a.
N has a u.l.c.a.
Cor .
If M has a u.l.c.a, then M is equidimensional and R/0 :R M isuniversally catenary.
•••If CR(M) is finite, M is equidimensional and R/0 :R M isuniversally catenary.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Cor .
Let M and N be finitely generated R–modules of finite dimensionssuch that Supp M = Supp N. Then the following conditions areequivalent.
M has a u.l.c.a.
N has a u.l.c.a.
Cor .
If M has a u.l.c.a, then M is equidimensional and R/0 :R M isuniversally catenary.
•••If CR(M) is finite, M is equidimensional and R/0 :R M isuniversally catenary.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Cor .
Let M and N be finitely generated R–modules of finite dimensionssuch that Supp M = Supp N. Then the following conditions areequivalent.
M has a u.l.c.a.
N has a u.l.c.a.
Cor .
If M has a u.l.c.a, then M is equidimensional and R/0 :R M isuniversally catenary.
•••If CR(M) is finite, M is equidimensional and R/0 :R M isuniversally catenary.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
Cor .
Let M and N be finitely generated R–modules of finite dimensionssuch that Supp M = Supp N. Then the following conditions areequivalent.
M has a u.l.c.a.
N has a u.l.c.a.
Cor .
If M has a u.l.c.a, then M is equidimensional and R/0 :R M isuniversally catenary.
•••If CR(M) is finite, M is equidimensional and R/0 :R M isuniversally catenary.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Example
Consider a noetherian local ring R of dimension d > 2. Chooseany pair of prime ideals p and q of R with conditions dim R/p = 2,dim R/q = 1, and p 6⊆ q. Then Min R/pq = {p, q} and so R/pq isnot an equidimensional R–module and thus its Cousin complex isnot finite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
•••Let (R,m) be local and M be a finitely generated R–module withdimension d = dim M > 1. Then the following statements areequivalent.
M has a u.l.c.a.
R/0 :R M is catenary and equidimensional, there exists aparameter element x of M such thatMin (M/xM) ∩ Ass M = ∅ and all modules M/x tM, t ∈ N,have a common u.l.c.a.
•••Let (R,m) be a local ring. If CR(M) is finite, then M/xM has au.l.c.a for any parameter element x of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
•••Let (R,m) be local and M be a finitely generated R–module withdimension d = dim M > 1. Then the following statements areequivalent.
M has a u.l.c.a.
R/0 :R M is catenary and equidimensional, there exists aparameter element x of M such thatMin (M/xM) ∩ Ass M = ∅ and all modules M/x tM, t ∈ N,have a common u.l.c.a.
•••Let (R,m) be a local ring. If CR(M) is finite, then M/xM has au.l.c.a for any parameter element x of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
•••Let (R,m) be local and M be a finitely generated R–module withdimension d = dim M > 1. Then the following statements areequivalent.
M has a u.l.c.a.
R/0 :R M is catenary and equidimensional, there exists aparameter element x of M such thatMin (M/xM) ∩ Ass M = ∅ and all modules M/x tM, t ∈ N,have a common u.l.c.a.
•••Let (R,m) be a local ring. If CR(M) is finite, then M/xM has au.l.c.a for any parameter element x of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Uniform annihilators of local cohomology
•••Let (R,m) be local and M be a finitely generated R–module withdimension d = dim M > 1. Then the following statements areequivalent.
M has a u.l.c.a.
R/0 :R M is catenary and equidimensional, there exists aparameter element x of M such thatMin (M/xM) ∩ Ass M = ∅ and all modules M/x tM, t ∈ N,have a common u.l.c.a.
•••Let (R,m) be a local ring. If CR(M) is finite, then M/xM has au.l.c.a for any parameter element x of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some partial characterizations
•••Assume that R is universally catenary and all formal fibres of R areCohen-Macaulay. If M is equidimensional, then CR(M) is finite; inparticular M has a uniform local cohomological annihilator.
•••Assume that (R,m) is local and all formal fibers of R areCohen-Macaulay. Then the following statements are equivalent fora finitely generated R–module M.
M ia an equidimensional R–module.
The Cousin complex of M is finite.
M has a u.l.c.a.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some partial characterizations
•••Assume that R is universally catenary and all formal fibres of R areCohen-Macaulay. If M is equidimensional, then CR(M) is finite; inparticular M has a uniform local cohomological annihilator.
•••Assume that (R,m) is local and all formal fibers of R areCohen-Macaulay. Then the following statements are equivalent fora finitely generated R–module M.
M ia an equidimensional R–module.
The Cousin complex of M is finite.
M has a u.l.c.a.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some partial characterizations
•••Assume that R is universally catenary and all formal fibres of R areCohen-Macaulay. If M is equidimensional, then CR(M) is finite; inparticular M has a uniform local cohomological annihilator.
•••Assume that (R,m) is local and all formal fibers of R areCohen-Macaulay. Then the following statements are equivalent fora finitely generated R–module M.
M ia an equidimensional R–module.
The Cousin complex of M is finite.
M has a u.l.c.a.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some partial characterizations
•••Assume that R is universally catenary and all formal fibres of R areCohen-Macaulay. If M is equidimensional, then CR(M) is finite; inparticular M has a uniform local cohomological annihilator.
•••Assume that (R,m) is local and all formal fibers of R areCohen-Macaulay. Then the following statements are equivalent fora finitely generated R–module M.
M ia an equidimensional R–module.
The Cousin complex of M is finite.
M has a u.l.c.a.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Using our approach, we may have the following result which Zhou(2006) has also proved it for M = R.
•••Assume that x is a u.l.c.a of M. Then Mx is a Cohen-MacaulayRx–module.
We may recover, partially, another result of Zhou (2006) over localrings.
•••Assume that (R,m) is a local ring and x is a u.l.c.a of M, then apower of x is a strong u.l.c.a of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Using our approach, we may have the following result which Zhou(2006) has also proved it for M = R.
•••Assume that x is a u.l.c.a of M. Then Mx is a Cohen-MacaulayRx–module.
We may recover, partially, another result of Zhou (2006) over localrings.
•••Assume that (R,m) is a local ring and x is a u.l.c.a of M, then apower of x is a strong u.l.c.a of M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Height of an ideal
•••For any finitely generated R–module M and any ideal a of R withaM 6= M, ∏
−1≤i(0 :R H i ) ⊆ 0 :R HhtMa−1
a (M).
Question
Does the inequality∏−1≤i
(0 :R H i ) ⊆ 0 :R HhtMaa (M)
hold?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Height of an ideal
•••For any finitely generated R–module M and any ideal a of R withaM 6= M, ∏
−1≤i(0 :R H i ) ⊆ 0 :R HhtMa−1
a (M).
Question
Does the inequality∏−1≤i
(0 :R H i ) ⊆ 0 :R HhtMaa (M)
hold?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Height of an ideal
•••Assume that M has finite dimension and CR(M) is finite. Then
htMa = inf{r :∏−1≤i
(0 :R HiM) 6⊆ 0 :R Hr
a(M)},
for all ideals a with aM 6= M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes of local cohomology
Throughout this section (R,m) is a local ring and M is a finitelygenerated R–module of dimension d .
a(M) =⋂
i<dimM
(0 :R Him(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes related to cohomologies of Cousincomplexes
•••Assume that CR(M) is finite. Let t, 0 ≤ t < d , be an integer suchthat dimHi
M ≤ t − i − 1, for all i ≥ −1. Then
Att Htm(M) =
t−1⋃i=−1{p ∈ AssHi
M : dim R/p = t − i − 1}.
•••Assume that CR(M) is finite. Let l < d be an integer. Thefollowing statements are equivalent.
Hjm(M) = 0 for all j , l < j < d .
dimHiM ≤ l − i − 1 for all i ≥ −1.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes related to cohomologies of Cousincomplexes
•••Assume that CR(M) is finite. Let t, 0 ≤ t < d , be an integer suchthat dimHi
M ≤ t − i − 1, for all i ≥ −1. Then
Att Htm(M) =
t−1⋃i=−1{p ∈ AssHi
M : dim R/p = t − i − 1}.
•••Assume that CR(M) is finite. Let l < d be an integer. Thefollowing statements are equivalent.
Hjm(M) = 0 for all j , l < j < d .
dimHiM ≤ l − i − 1 for all i ≥ −1.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes related to cohomologies of Cousincomplexes
•••Assume that CR(M) is finite. Let t, 0 ≤ t < d , be an integer suchthat dimHi
M ≤ t − i − 1, for all i ≥ −1. Then
Att Htm(M) =
t−1⋃i=−1{p ∈ AssHi
M : dim R/p = t − i − 1}.
•••Assume that CR(M) is finite. Let l < d be an integer. Thefollowing statements are equivalent.
Hjm(M) = 0 for all j , l < j < d .
dimHiM ≤ l − i − 1 for all i ≥ −1.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes related to cohomologies of Cousincomplexes
•••Assume that CR(M) is finite. Then
I Att Hd−1m (M) =
⋃d−2i=−1{p ∈ AssHi
M : dim R/p = d − i − 2}.
I Hd−1m (M) 6= 0 if and only if dimHi
M = d − i − 2 for some i ,−1 ≤ i ≤ d − 2.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Attached primes related to cohomologies of Cousincomplexes
•••Assume that CR(M) is finite. Then
I Att Hd−1m (M) =
⋃d−2i=−1{p ∈ AssHi
M : dim R/p = d − i − 2}.I Hd−1
m (M) 6= 0 if and only if dimHiM = d − i − 2 for some i ,
−1 ≤ i ≤ d − 2.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2005) show that for any ideal a of R,
Att Hda (M) = {p ∈ Supp M : cd (a,R/p) = d},
where cd (a,K ) is the cohomological dimension of an R–module Kwith respect to a, that is cd (a,K ) = sup{i ∈ Z : Hi
a(K ) 6= 0}.
Rem.
Att Hda (M) ⊆ Assh M.
Att Hdm(M) = Assh M (known before)
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2005) show that for any ideal a of R,
Att Hda (M) = {p ∈ Supp M : cd (a,R/p) = d},
where cd (a,K ) is the cohomological dimension of an R–module Kwith respect to a, that is cd (a,K ) = sup{i ∈ Z : Hi
a(K ) 6= 0}.
Rem.
Att Hda (M) ⊆ Assh M.
Att Hdm(M) = Assh M (known before)
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2005) show that for any ideal a of R,
Att Hda (M) = {p ∈ Supp M : cd (a,R/p) = d},
where cd (a,K ) is the cohomological dimension of an R–module Kwith respect to a, that is cd (a,K ) = sup{i ∈ Z : Hi
a(K ) 6= 0}.
Rem.
Att Hda (M) ⊆ Assh M.
Att Hdm(M) = Assh M (known before)
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2005) show that for any ideal a of R,
Att Hda (M) = {p ∈ Supp M : cd (a,R/p) = d},
where cd (a,K ) is the cohomological dimension of an R–module Kwith respect to a, that is cd (a,K ) = sup{i ∈ Z : Hi
a(K ) 6= 0}.
Rem.
Att Hda (M) ⊆ Assh M.
Att Hdm(M) = Assh M (known before)
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2005) show that for any ideal a of R,
Att Hda (M) = {p ∈ Supp M : cd (a,R/p) = d},
where cd (a,K ) is the cohomological dimension of an R–module Kwith respect to a, that is cd (a,K ) = sup{i ∈ Z : Hi
a(K ) 6= 0}.
Rem.
Att Hda (M) ⊆ Assh M.
Att Hdm(M) = Assh M (known before)
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Dibaei and Yassemi (2007) prove that for any pair of ideals a andb of a complete ring R, if Att Hd
a (M) = Att Hdb (M), then
Hda (M) ∼= Hd
b (M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Question
For any subset T of Assh M, is there an ideal a of R such thatAtt Hd
a (M) = T ?
Rem.
If dim M = 1, then any subset T of Assh M is equal to the setAtt H1
a(M) for some ideal a of R.
•••Assume that R is complete and T ⊆ Assh M, then there exists anideal a of R such that T = Att Hd
a (M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Question
For any subset T of Assh M, is there an ideal a of R such thatAtt Hd
a (M) = T ?
Rem.
If dim M = 1, then any subset T of Assh M is equal to the setAtt H1
a(M) for some ideal a of R.
•••Assume that R is complete and T ⊆ Assh M, then there exists anideal a of R such that T = Att Hd
a (M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
Question
For any subset T of Assh M, is there an ideal a of R such thatAtt Hd
a (M) = T ?
Rem.
If dim M = 1, then any subset T of Assh M is equal to the setAtt H1
a(M) for some ideal a of R.
•••Assume that R is complete and T ⊆ Assh M, then there exists anideal a of R such that T = Att Hd
a (M).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
•••Assume that R is complete. Then the number of non–isomorphictop local cohomology modules of M with respect to all ideals of R
is equal to 2|AsshM|.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
•••Assume that R is complete, d ≥ 1 and setAssh M \ T = {q1, . . . , qr}. The following statements areequivalent.
There exists an ideal a of R such that Att Hda (M) = T .
For each i , 1 ≤ i ≤ r , there exists Qi ∈ Supp M withdim R/Qi = 1 such that⋂
p∈Tp * Qi and qi ⊆ Qi .
With Qi , 1 ≤ i ≤ r , as above, Att Hda (M) = T where a =
r⋂i=1
Qi .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
•••Assume that R is complete, d ≥ 1 and setAssh M \ T = {q1, . . . , qr}. The following statements areequivalent.
There exists an ideal a of R such that Att Hda (M) = T .
For each i , 1 ≤ i ≤ r , there exists Qi ∈ Supp M withdim R/Qi = 1 such that⋂
p∈Tp * Qi and qi ⊆ Qi .
With Qi , 1 ≤ i ≤ r , as above, Att Hda (M) = T where a =
r⋂i=1
Qi .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
•••Assume that R is complete, d ≥ 1 and setAssh M \ T = {q1, . . . , qr}. The following statements areequivalent.
There exists an ideal a of R such that Att Hda (M) = T .
For each i , 1 ≤ i ≤ r , there exists Qi ∈ Supp M withdim R/Qi = 1 such that⋂
p∈Tp * Qi and qi ⊆ Qi .
With Qi , 1 ≤ i ≤ r , as above, Att Hda (M) = T where a =
r⋂i=1
Qi .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Top local cohomology modules
•••Assume that R is complete, d ≥ 1 and setAssh M \ T = {q1, . . . , qr}. The following statements areequivalent.
There exists an ideal a of R such that Att Hda (M) = T .
For each i , 1 ≤ i ≤ r , there exists Qi ∈ Supp M withdim R/Qi = 1 such that⋂
p∈Tp * Qi and qi ⊆ Qi .
With Qi , 1 ≤ i ≤ r , as above, Att Hda (M) = T where a =
r⋂i=1
Qi .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Example
Set R = k[[X ,Y ,Z ,W ]], where k is a field and X ,Y ,Z ,W areindependent indeterminates. Let m = (X ,Y ,Z ,W ),
p1 = (X ,Y ) , p2 = (Z ,W ) , p3 = (Y ,Z ) , p4 = (X ,W )
and set M =R
p1p2p3p4as an R–module, Assh M = {p1, p2, p3, p4}.
We get {pi} = Att H2ai
(M), where
a1 = p2, a2 = p1, a3 = p4, a4 = p3, and {pi , pj} = Att H2aij
(M),where
a12 = (Y 2 + YZ ,Z 2 + YZ ,X 2 + XW ,W 2 + WX ),a34 = (Z 2 + ZW ,X 2 + YX ,Y 2 + YX ,W 2 + WZ ),a13 = (Z 2 + XZ ,W 2 + WY ,X 2 + XZ ),a14 = (W 2 + WY ,Z 2 + ZY ,Y 2 + YW ),a23 = (X 2 + XZ ,Y 2 + WY ,W 2 + ZW ),a24 = (X 2 + XZ ,Y 2 + WY ,Z 2 + ZW ).
Finally, we have {pi , pj , pk} = Att H2aijk
(M), wherea123 = (X ,W ,Y + Z ), a234 = (X ,Y ,W + Z ),a134 = (Z ,W ,Y + X ).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
Let (R,m) be a g.CM local ring. Then R/p has a u.l.c.a for allp ∈ Spec R. In particular, any equidimensional R–module M has au.l.c.a.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is a local ring such that R/p has a u.l.c.a forall p ∈ Spec R. Then the following statements are equivalent.
M is equidimensional R–module and for allp ∈ Supp M \ {m}, Mp is a Cohen-Macaulay Rp–module.
M is a g.CM module.
Cor.Assume that R is an equidimensional noetherian local ring. Thefollowing statements are equivalent.
R is g.CM.
For all p ∈ Spec R \ {m}, Rp is a Cohen-Macaulay ring andR/p has a uniform local cohomological annihilator.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is a local ring such that R/p has a u.l.c.a forall p ∈ Spec R. Then the following statements are equivalent.
M is equidimensional R–module and for allp ∈ Supp M \ {m}, Mp is a Cohen-Macaulay Rp–module.
M is a g.CM module.
Cor.Assume that R is an equidimensional noetherian local ring. Thefollowing statements are equivalent.
R is g.CM.
For all p ∈ Spec R \ {m}, Rp is a Cohen-Macaulay ring andR/p has a uniform local cohomological annihilator.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is a local ring such that R/p has a u.l.c.a forall p ∈ Spec R. Then the following statements are equivalent.
M is equidimensional R–module and for allp ∈ Supp M \ {m}, Mp is a Cohen-Macaulay Rp–module.
M is a g.CM module.
Cor.Assume that R is an equidimensional noetherian local ring. Thefollowing statements are equivalent.
R is g.CM.
For all p ∈ Spec R \ {m}, Rp is a Cohen-Macaulay ring andR/p has a uniform local cohomological annihilator.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is a local ring such that R/p has a u.l.c.a forall p ∈ Spec R. Then the following statements are equivalent.
M is equidimensional R–module and for allp ∈ Supp M \ {m}, Mp is a Cohen-Macaulay Rp–module.
M is a g.CM module.
Cor.Assume that R is an equidimensional noetherian local ring. Thefollowing statements are equivalent.
R is g.CM.
For all p ∈ Spec R \ {m}, Rp is a Cohen-Macaulay ring andR/p has a uniform local cohomological annihilator.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is a local ring such that R/p has a u.l.c.a forall p ∈ Spec R. Then the following statements are equivalent.
M is equidimensional R–module and for allp ∈ Supp M \ {m}, Mp is a Cohen-Macaulay Rp–module.
M is a g.CM module.
Cor.Assume that R is an equidimensional noetherian local ring. Thefollowing statements are equivalent.
R is g.CM.
For all p ∈ Spec R \ {m}, Rp is a Cohen-Macaulay ring andR/p has a uniform local cohomological annihilator.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is local.
I A finitely generated R–module M is g.CM if and only if allcohomology modules of CR(M) are of finite lengths.
I A finitely generated R–module M is quasi–Buchsbaummodule if and only if CR(M) is finite and mHi
M = 0 for all i .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Applications to generalized Cohen-Macaulay modules
•••Assume that (R,m) is local.
I A finitely generated R–module M is g.CM if and only if allcohomology modules of CR(M) are of finite lengths.
I A finitely generated R–module M is quasi–Buchsbaummodule if and only if CR(M) is finite and mHi
M = 0 for all i .
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
gCM
CM
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Throughout M is a finitely generated module of finite dimension dover a noetherian ring A. The Cohen-Macaulay locus of M isdenoted by
CM(M) := {p ∈ Spec A : Mp is Cohen-Macaulay as Ap–module}.
The topological property of the Cohen-Macaulay loci of modules isan important tool and have been studied by many authors.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Throughout M is a finitely generated module of finite dimension dover a noetherian ring A. The Cohen-Macaulay locus of M isdenoted by
CM(M) := {p ∈ Spec A : Mp is Cohen-Macaulay as Ap–module}.
The topological property of the Cohen-Macaulay loci of modules isan important tool and have been studied by many authors.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Grothendieck (1965) states that CM(M) is an open subset ofSpec R whenever R is an excellent ring.
Grothendieck (1966) shows that CM(R) is open when R possessesa dualizing complex.
Dibaei (2005) shows that CM(R) is open when R is a local ring,all of its formal fibres are Cohen-Macaulay and satisfying the Serrecondition (S2), the condition which is superfluous.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Grothendieck (1965) states that CM(M) is an open subset ofSpec R whenever R is an excellent ring.
Grothendieck (1966) shows that CM(R) is open when R possessesa dualizing complex.
Dibaei (2005) shows that CM(R) is open when R is a local ring,all of its formal fibres are Cohen-Macaulay and satisfying the Serrecondition (S2), the condition which is superfluous.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Grothendieck (1965) states that CM(M) is an open subset ofSpec R whenever R is an excellent ring.
Grothendieck (1966) shows that CM(R) is open when R possessesa dualizing complex.
Dibaei (2005) shows that CM(R) is open when R is a local ring,all of its formal fibres are Cohen-Macaulay and satisfying the Serrecondition (S2), the condition which is superfluous.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Rotthaus and Sega (2006) study the Cohen-Macaulay loci ofgraded modules over a noetherian homogeneous graded ringR =
⊕i∈N Ri considered as R0–modules.
Kawasaki (2008) shows that when the ring R is catenary, theopenness of CM(B) of any finite R–algebra B is a crucialassumption if one expects all equidimensional finite R–module Mhave finite Cousin complexes.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Cohen-Macaulay locus
Rotthaus and Sega (2006) study the Cohen-Macaulay loci ofgraded modules over a noetherian homogeneous graded ringR =
⊕i∈N Ri considered as R0–modules.
Kawasaki (2008) shows that when the ring R is catenary, theopenness of CM(B) of any finite R–algebra B is a crucialassumption if one expects all equidimensional finite R–module Mhave finite Cousin complexes.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Rem.
CM(M) = Spec R \ ∪i≥−1Supp R(Hi (CR(M))).
Rem.
If CR(M) is finite, then non-CM(M) = V(∏i
(0 :R HiM)) so that
CM(M) is a Zariski–open subsets of Spec R.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Rem.
CM(M) = Spec R \ ∪i≥−1Supp R(Hi (CR(M))).
Rem.
If CR(M) is finite, then non-CM(M) = V(∏i
(0 :R HiM)) so that
CM(M) is a Zariski–open subsets of Spec R.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Rem.
Assume that all formal fibres of R are Cohen-Macaulay. Then theCohen-Macaulay locus of M is a Zariski–open subset of Spec R.
•••Assume that (R,m) is a catenary local ring and that M isequidimensional R–module. Then
Min (non–CM(M)) ⊆ ∪0≤i≤dimM
Att Him(M)∪non–CM(R).
Cor.Assume that (R,m) is a catenary local ring and that thenon–CM(R) is a finite set. Then the Cohen-Macaulay locus of Mis open.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Rem.
Assume that all formal fibres of R are Cohen-Macaulay. Then theCohen-Macaulay locus of M is a Zariski–open subset of Spec R.
•••Assume that (R,m) is a catenary local ring and that M isequidimensional R–module. Then
Min (non–CM(M)) ⊆ ∪0≤i≤dimM
Att Him(M)∪non–CM(R).
Cor.Assume that (R,m) is a catenary local ring and that thenon–CM(R) is a finite set. Then the Cohen-Macaulay locus of Mis open.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Rem.
Assume that all formal fibres of R are Cohen-Macaulay. Then theCohen-Macaulay locus of M is a Zariski–open subset of Spec R.
•••Assume that (R,m) is a catenary local ring and that M isequidimensional R–module. Then
Min (non–CM(M)) ⊆ ∪0≤i≤dimM
Att Him(M)∪non–CM(R).
Cor.Assume that (R,m) is a catenary local ring and that thenon–CM(R) is a finite set. Then the Cohen-Macaulay locus of Mis open.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Example
Consider a local ring R satisfying Serre’s condition (Sd−2) suchthat CR(R) is finite.
Then HiR = 0 for i ≤ d − 4 and i ≥ d − 1, dimHd−3
R ≤ 1 and
dimHd−2R ≤ 0.
Thus non–CM(R) = SuppHd−2R ∪ SuppHd−1
R is a finite set.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Openness of Cohen-Macaulay locus
Example
Set S = k[[X ,Y ,Z ,U,V ]]/(X ) ∩ (Y ,Z ), where k is a field. It isclear that S is a local ring with Cohen-Macaulay formal fibres. ByRatliff’s weak existence theorem, there are infinitely many primeideals P of k[[X ,Y ,Z ,U,V ]], with(X ,Y ,Z ) ⊂ P ⊂ (X ,Y ,Z ,U,V ). For any such prime ideal P, SPis not equidimensional and so it is not Cohen-Macaulay. In otherwords, non–CM(S) is infinite.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Example
Ferrand and Raynaud (1970) show that there exists a local integraldomain (R,m) of dimension 2 such thatR = C[[X ,Y ,Z ]]/(Z 2, tZ ), where C is the field of complexnumbers and t = X + Y + Y 2s for some s ∈ C[[Y ]] \ C{Y }.As Ass R = {(Z ), (Z , t)}, R does not satisfy (S1). Thus H−1
R6= 0
while H−1R = 0. Now by a result of Petzl (1997), there exists aformal fibre of R which is not Cohen-Macaulay.As R is an integral local domain, we have non–CM(R) = {m}.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
M is equidimensional R–module and all formal fibres of Rover minimal members of Supp M are Cohen-Macaulay.
M has a u.l.c.a.
Cor.Assume that p is a prime ideal of Spec R such that R/p has au.l.c.a. Then the formal fibre of R over p is Cohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
M is equidimensional R–module and all formal fibres of Rover minimal members of Supp M are Cohen-Macaulay.
M has a u.l.c.a.
Cor.Assume that p is a prime ideal of Spec R such that R/p has au.l.c.a. Then the formal fibre of R over p is Cohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
M is equidimensional R–module and all formal fibres of Rover minimal members of Supp M are Cohen-Macaulay.
M has a u.l.c.a.
Cor.Assume that p is a prime ideal of Spec R such that R/p has au.l.c.a. Then the formal fibre of R over p is Cohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
M is equidimensional R–module and all formal fibres of Rover minimal members of Supp M are Cohen-Macaulay.
M has a u.l.c.a.
Cor.Assume that p is a prime ideal of Spec R such that R/p has au.l.c.a. Then the formal fibre of R over p is Cohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••The following statements are equivalent.
R is universally catenary ring and all of its formal fibres areCohen-Macaulay.
The Cousin complex CR(R/p) is finite for all p ∈ Spec R.
R/p has a u.l.c.a for all p ∈ Spec R.
Schenzel (1982) proves that the equality holds when M isequidimensional and R posses a dualizing complex.
Rem.
If a local R posses a dualizing complex, then it is a homomorphicimage of a Gorenstein local ring and so R is universally catenaryand all formal fibres of R are Cohen-Macaulay. Hence CR(M) isfinite for all equidimensional R–module M.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••Assume that CR(M) is finite. Then
V(∏d−1
i=−1(0 :R HiM)) =non–CM(M) = V(a(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••For an equidimensional R–module M, the following statements areequivalent.
R/q has a u.l.c.a for all q ∈ CM(M).
R/qR is equidimensional R–module and the formal fibre ring(Rq/qRq)⊗R R is Cohen-Macaulay for all q ∈ CM(M).
non–CM(M) = V(a(M)).
non–CM(M) ⊇ V(a(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••For an equidimensional R–module M, the following statements areequivalent.
R/q has a u.l.c.a for all q ∈ CM(M).
R/qR is equidimensional R–module and the formal fibre ring(Rq/qRq)⊗R R is Cohen-Macaulay for all q ∈ CM(M).
non–CM(M) = V(a(M)).
non–CM(M) ⊇ V(a(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••For an equidimensional R–module M, the following statements areequivalent.
R/q has a u.l.c.a for all q ∈ CM(M).
R/qR is equidimensional R–module and the formal fibre ring(Rq/qRq)⊗R R is Cohen-Macaulay for all q ∈ CM(M).
non–CM(M) = V(a(M)).
non–CM(M) ⊇ V(a(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••For an equidimensional R–module M, the following statements areequivalent.
R/q has a u.l.c.a for all q ∈ CM(M).
R/qR is equidimensional R–module and the formal fibre ring(Rq/qRq)⊗R R is Cohen-Macaulay for all q ∈ CM(M).
non–CM(M) = V(a(M)).
non–CM(M) ⊇ V(a(M)).
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••Assume that p ∈ Spec R. A necessary and sufficient condition forR/p to have a u.l.c.a. is that there exists an equidimensionalR–module M such that p ∈ Supp M \ V(a(M)).
Cor.Assume that M satisfies the condition (Sn). If CR(M) is finite,then the formal fibres of R over all prime ideals p ∈ Supp M withhtMp ≤ n are Cohen-Macaulay
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
•••Assume that p ∈ Spec R. A necessary and sufficient condition forR/p to have a u.l.c.a. is that there exists an equidimensionalR–module M such that p ∈ Supp M \ V(a(M)).
Cor.Assume that M satisfies the condition (Sn). If CR(M) is finite,then the formal fibres of R over all prime ideals p ∈ Supp M withhtMp ≤ n are Cohen-Macaulay
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
Cor.Assume that CR(M) is finite. Then the formal fibres of R over allprime ideals p ∈ Supp M with htMp ≤ 1 are Cohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Rings whose formal fibres are Cohen-Macaulay
Cor.Assume that CR(M) is finite. Then the formal fibres of R over allprime ideals p ∈ CM(M) ∪ {p ∈ Supp M : htMp = 1} areCohen-Macaulay.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
gCM
CM
Formal fibres over minimal
primes are CM
Formal fibres over primes of CM locus are
CM
All formal fibres are
CM
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
gCM
CM
If all formal fibres of the base ring are CM
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Question
Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Cor.Assume that CR(M) is finite and dim M ≤ 3. Then the formalfibres of R over all prime ideals p ∈ Supp M are Cohen-Macaulay.
Cor.Assume that CR(M) is finite and dim (non–CM(M)) ≤ 1. Then theformal fibres of R over all prime ideals p ∈ Supp M areCohen-Macaulay.
dim (non–CM(M)) = sup{dim R/p : p ∈ non–CM(M)}.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Cor.Assume that CR(M) is finite and dim M ≤ 3. Then the formalfibres of R over all prime ideals p ∈ Supp M are Cohen-Macaulay.
Cor.Assume that CR(M) is finite and dim (non–CM(M)) ≤ 1. Then theformal fibres of R over all prime ideals p ∈ Supp M areCohen-Macaulay.
dim (non–CM(M)) = sup{dim R/p : p ∈ non–CM(M)}.
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Problem
Assume that CR(M) is finite and x is a non–zero–divisor on M.Then CR(M/xM) is finite.
Rem.
Solving the above problem is equivalent to find a positive answerfor our question.
Question
Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?
THE END
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Problem
Assume that CR(M) is finite and x is a non–zero–divisor on M.Then CR(M/xM) is finite.
Rem.
Solving the above problem is equivalent to find a positive answerfor our question.
Question
Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?
THE END
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Problem
Assume that CR(M) is finite and x is a non–zero–divisor on M.Then CR(M/xM) is finite.
Rem.
Solving the above problem is equivalent to find a positive answerfor our question.
Question
Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?
THE END
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Finite Cousin complexes Attached primes of local cohomology Cohen-Macaulay locus
Some comments
Problem
Assume that CR(M) is finite and x is a non–zero–divisor on M.Then CR(M/xM) is finite.
Rem.
Solving the above problem is equivalent to find a positive answerfor our question.
Question
Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?
THE END