majid eghbali ipm november 30, 2011 8th seminar on...
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On an endomorphism ring of local cohomology
Majid Eghbali
IPMNovember 30, 2011
It is based on a joint work with Peter Schenzel8th Seminar on Commutative Algebra and Related Topics
Majid Eghbali On an endomorphism ring of local cohomology
![Page 2: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/2.jpg)
Why local cohomolgy
Local cohomology was invented by Grothendieck in 1960s toprove some theorems in algebraic geometry. It has manyapplications in topology, geometry, combinatorics, andcomputational subjects.
DefinitionLet M be an R-module and a an ideal of R, then we define i-thlocal cohomology module as H i
a(M) = lim−→ExtiR(R/an, M).
Majid Eghbali On an endomorphism ring of local cohomology
![Page 3: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/3.jpg)
Why local cohomolgy
Local cohomology was invented by Grothendieck in 1960s toprove some theorems in algebraic geometry. It has manyapplications in topology, geometry, combinatorics, andcomputational subjects.
DefinitionLet M be an R-module and a an ideal of R, then we define i-thlocal cohomology module as H i
a(M) = lim−→ExtiR(R/an, M).
Majid Eghbali On an endomorphism ring of local cohomology
![Page 4: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/4.jpg)
Why local cohomolgy
Local cohomology was invented by Grothendieck in 1960s toprove some theorems in algebraic geometry. It has manyapplications in topology, geometry, combinatorics, andcomputational subjects.
DefinitionLet M be an R-module and a an ideal of R, then we define i-thlocal cohomology module as H i
a(M) = lim−→ExtiR(R/an, M).
Majid Eghbali On an endomorphism ring of local cohomology
![Page 5: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/5.jpg)
Resent Work
Hochster-Huneke (1994)
HomR(Hdm(R), Hd
m(R)), where d := dim R and (R,m) local ring.
Hellus-Stückrad (2007), Hellus-Schenzel(2008),Schenzel(2009)
HomR(H ia(R), H i
a(R)), where i := height of a.
Eghbali- Schenzel(2011)
HomR(Hda (R), Hd
a (R)), where d := dim R.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 6: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/6.jpg)
Resent Work
Hochster-Huneke (1994)
HomR(Hdm(R), Hd
m(R)), where d := dim R and (R,m) local ring.
Hellus-Stückrad (2007), Hellus-Schenzel(2008),Schenzel(2009)
HomR(H ia(R), H i
a(R)), where i := height of a.
Eghbali- Schenzel(2011)
HomR(Hda (R), Hd
a (R)), where d := dim R.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 7: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/7.jpg)
Resent Work
Hochster-Huneke (1994)
HomR(Hdm(R), Hd
m(R)), where d := dim R and (R,m) local ring.
Hellus-Stückrad (2007), Hellus-Schenzel(2008),Schenzel(2009)
HomR(H ia(R), H i
a(R)), where i := height of a.
Eghbali- Schenzel(2011)
HomR(Hda (R), Hd
a (R)), where d := dim R.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 8: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/8.jpg)
Resent Work
Hochster-Huneke (1994)
HomR(Hdm(R), Hd
m(R)), where d := dim R and (R,m) local ring.
Hellus-Stückrad (2007), Hellus-Schenzel(2008),Schenzel(2009)
HomR(H ia(R), H i
a(R)), where i := height of a.
Eghbali- Schenzel(2011)
HomR(Hda (R), Hd
a (R)), where d := dim R.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 9: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/9.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 10: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/10.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 11: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/11.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 12: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/12.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 13: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/13.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 14: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/14.jpg)
Questions
Some questions in local cohomology
Let a be an ideal of a local ring (R,m) and ER(R/m) be theinjective hull of the residue field R/m and M a finitely generatedR-module of dimension d:
1 How one can express Hda (M) via Hd
m(M).2 What are the properties of HomR(Hdim R
a (R), ER(R/m)).3 What are the properties of HomR̂(Hdim R
a (R), Hdim Ra (R)).
4 What are some applications of the above questions?
Majid Eghbali On an endomorphism ring of local cohomology
![Page 15: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/15.jpg)
To Control top local cohomology
Express Hda (M) via Hd
m(M).
Explanation
Put d := dim M. When Hda (M) 6= 0 one of the most important
views concerning this is to express Hda (M) via Hd
m(M). Moreprecisely the kernel of the natural epimorphismHdim Mm (M)→ Hdim M
a (M) has been calculated explicitly.
NoteFor an R-module M let 0 = ∩n
i=1Qi(M) denote a minimalprimary decomposition of the zero submodule of M. That isM/Qi(M), i = 1, ..., n, is a pi -primary R-module. ClearlyAssR M = {p1, ..., pn}.
Majid Eghbali On an endomorphism ring of local cohomology
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To Control top local cohomology
Express Hda (M) via Hd
m(M).
Explanation
Put d := dim M. When Hda (M) 6= 0 one of the most important
views concerning this is to express Hda (M) via Hd
m(M). Moreprecisely the kernel of the natural epimorphismHdim Mm (M)→ Hdim M
a (M) has been calculated explicitly.
NoteFor an R-module M let 0 = ∩n
i=1Qi(M) denote a minimalprimary decomposition of the zero submodule of M. That isM/Qi(M), i = 1, ..., n, is a pi -primary R-module. ClearlyAssR M = {p1, ..., pn}.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 17: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/17.jpg)
To Control top local cohomology
Express Hda (M) via Hd
m(M).
Explanation
Put d := dim M. When Hda (M) 6= 0 one of the most important
views concerning this is to express Hda (M) via Hd
m(M). Moreprecisely the kernel of the natural epimorphismHdim Mm (M)→ Hdim M
a (M) has been calculated explicitly.
NoteFor an R-module M let 0 = ∩n
i=1Qi(M) denote a minimalprimary decomposition of the zero submodule of M. That isM/Qi(M), i = 1, ..., n, is a pi -primary R-module. ClearlyAssR M = {p1, ..., pn}.
Majid Eghbali On an endomorphism ring of local cohomology
![Page 18: Majid Eghbali IPM November 30, 2011 8th Seminar on ...math.ipm.ac.ir/conferences/2011/commalg2011/talks/Eghbali.pdf · Majid Eghbali IPM November 30, 2011 It is based on a joint work](https://reader033.vdocuments.us/reader033/viewer/2022050611/5fb25a31e1b8f054b107a53e/html5/thumbnails/18.jpg)
Some Definitions
DefinitionLet a ⊂ R denote an ideal of R. We define two disjoint subsetsU, V of AssR M related to a
(a) U = {p ∈ AssR M|dim R/p = d and dim R/a + p = 0}.(b) V = {p ∈ AssR M|dim R/p < d or dim R/p =
d and dim R/a + p > 0}.Finally we define Qa(M) = ∩pi∈UQi(M). In the case U = ∅, putQa(M) = M.
DefinitionLet M denote a finitely generated module over the local ring(R,m). Let a ⊂ R denote an ideal. Then define Pa(M) as theintersection of all the primary components of AnnR M such thatdim R/p = dim M and dim R/a + p = 0. Clearly Pa(M) is thepre-image of QaR/ AnnR M(R/ AnnR M) in R.
Majid Eghbali On an endomorphism ring of local cohomology
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Some Definitions
DefinitionLet a ⊂ R denote an ideal of R. We define two disjoint subsetsU, V of AssR M related to a
(a) U = {p ∈ AssR M|dim R/p = d and dim R/a + p = 0}.(b) V = {p ∈ AssR M|dim R/p < d or dim R/p =
d and dim R/a + p > 0}.Finally we define Qa(M) = ∩pi∈UQi(M). In the case U = ∅, putQa(M) = M.
DefinitionLet M denote a finitely generated module over the local ring(R,m). Let a ⊂ R denote an ideal. Then define Pa(M) as theintersection of all the primary components of AnnR M such thatdim R/p = dim M and dim R/a + p = 0. Clearly Pa(M) is thepre-image of QaR/ AnnR M(R/ AnnR M) in R.
Majid Eghbali On an endomorphism ring of local cohomology
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Main Theorem
TheoremLet a denote an ideal of a local ring (R,m). Let M be a finitelygenerated R-module and d = dim M. Then there is a naturalisomorphism
Hda (M) ∼= Hd
mR̂(M̂/Q
aR̂(M̂)) ∼= HdmR̂
(M̂/Pa(M̂)M̂).
ProofUsing the short exact sequence
0→ Qa(M)→ M → M/Qa(M)→ 0
and applying local cohomology module to it we prove the claim.to this end note that AssR Qa(M) = V , AssR M/Qa(M) = U andU ∪ V = AssR M.
Majid Eghbali On an endomorphism ring of local cohomology
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Main Theorem
TheoremLet a denote an ideal of a local ring (R,m). Let M be a finitelygenerated R-module and d = dim M. Then there is a naturalisomorphism
Hda (M) ∼= Hd
mR̂(M̂/Q
aR̂(M̂)) ∼= HdmR̂
(M̂/Pa(M̂)M̂).
ProofUsing the short exact sequence
0→ Qa(M)→ M → M/Qa(M)→ 0
and applying local cohomology module to it we prove the claim.to this end note that AssR Qa(M) = V , AssR M/Qa(M) = U andU ∪ V = AssR M.
Majid Eghbali On an endomorphism ring of local cohomology
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Homological Properties of HomR(Hda (R), ER(R/m))
NotationFor an ideal a ⊂ R with dim R/a = d we will denote by ad theintersection of those primary components in a minimal reducedprimary decomposition of a which are of dimension d .
Notation and DefinitionFor a local ring (R,m) which is a factor ring of a Gorenstein ring(S, n) with r = dim S. Then there are functorial isomorphisms
Hdm(M) ∼= HomR(Extr−d
S (M, S), E(R/m)), d := dim M,
where M is a finitely generated R-module. The moduleKM = Extr−d
S (M, S) is called the canonical module of M.
Majid Eghbali On an endomorphism ring of local cohomology
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Homological Properties of HomR(Hda (R), ER(R/m))
NotationFor an ideal a ⊂ R with dim R/a = d we will denote by ad theintersection of those primary components in a minimal reducedprimary decomposition of a which are of dimension d .
Notation and DefinitionFor a local ring (R,m) which is a factor ring of a Gorenstein ring(S, n) with r = dim S. Then there are functorial isomorphisms
Hdm(M) ∼= HomR(Extr−d
S (M, S), E(R/m)), d := dim M,
where M is a finitely generated R-module. The moduleKM = Extr−d
S (M, S) is called the canonical module of M.
Majid Eghbali On an endomorphism ring of local cohomology
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Homological Properties of HomR(Hda (R), ER(R/m))
NotationFor an ideal a ⊂ R with dim R/a = d we will denote by ad theintersection of those primary components in a minimal reducedprimary decomposition of a which are of dimension d .
Notation and DefinitionFor a local ring (R,m) which is a factor ring of a Gorenstein ring(S, n) with r = dim S. Then there are functorial isomorphisms
Hdm(M) ∼= HomR(Extr−d
S (M, S), E(R/m)), d := dim M,
where M is a finitely generated R-module. The moduleKM = Extr−d
S (M, S) is called the canonical module of M.
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
LemmaLet a denote an ideal in a d-dimensional local ring (R,m). Then(1) Ta(R) = HomR(Hd
a (R), ER(R/m)) is a finitely generatedR̂-module.
(2) AssR̂ Ta(R) = {p ∈ Ass R̂|dim R̂/p =
dim R and dim R̂/aR̂ + p = 0}.(3) KR̂(R̂/Qa(R̂)) ∼= Ta(R). In particular, It satisfies the S2
condition. Furthermore when R̂/Qa(R̂) is Cohen-Macaulaythen so is Ta(R).
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
LemmaLet a denote an ideal in a d-dimensional local ring (R,m). Then(1) Ta(R) = HomR(Hd
a (R), ER(R/m)) is a finitely generatedR̂-module.
(2) AssR̂ Ta(R) = {p ∈ Ass R̂|dim R̂/p =
dim R and dim R̂/aR̂ + p = 0}.(3) KR̂(R̂/Qa(R̂)) ∼= Ta(R). In particular, It satisfies the S2
condition. Furthermore when R̂/Qa(R̂) is Cohen-Macaulaythen so is Ta(R).
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
Lemma
(4) AnnR̂(Hda (R)) = Qa(R̂).
TheoremLet a denote an ideal in a local ring (R,m). Let
Φ : R̂ → HomR̂(Hda (R), Hd
a (R))
the natural homomorphism. Then(1) ker Φ = Q
aR̂(R̂).
(2) Φ is surjective if and only if R̂/QaR̂(R̂) satisfies S2.
(3) HomR̂(Hda (R), Hd
a (R)) is a finitely generated R̂-module.
(4) HomR̂(Hda (R), Hd
a (R)) is a commutative semi-localNoetherian ring.
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
Lemma
(4) AnnR̂(Hda (R)) = Qa(R̂).
TheoremLet a denote an ideal in a local ring (R,m). Let
Φ : R̂ → HomR̂(Hda (R), Hd
a (R))
the natural homomorphism. Then(1) ker Φ = Q
aR̂(R̂).
(2) Φ is surjective if and only if R̂/QaR̂(R̂) satisfies S2.
(3) HomR̂(Hda (R), Hd
a (R)) is a finitely generated R̂-module.
(4) HomR̂(Hda (R), Hd
a (R)) is a commutative semi-localNoetherian ring.
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Applications
Lemma
(4) AnnR̂(Hda (R)) = Qa(R̂).
TheoremLet a denote an ideal in a local ring (R,m). Let
Φ : R̂ → HomR̂(Hda (R), Hd
a (R))
the natural homomorphism. Then(1) ker Φ = Q
aR̂(R̂).
(2) Φ is surjective if and only if R̂/QaR̂(R̂) satisfies S2.
(3) HomR̂(Hda (R), Hd
a (R)) is a finitely generated R̂-module.
(4) HomR̂(Hda (R), Hd
a (R)) is a commutative semi-localNoetherian ring.
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
TheoremLet a be an ideal of a complete local ring (R,m). For an integerr ≥ 2 we have the following statements:(1) Suppose R/Qa(R) has S2. Then Ta(R) satisfies the
condition Sr if and only if H im(R/Qa(R)) = 0 for
d − r + 2 ≤ i < d .
(2) R/Qa(R) satisfies the condition Sr if and only ifH im(Ta(R)) = 0 for d − r + 2 ≤ i < d and
R/Qa(R) ∼= HomR(Hda (R), Hd
a (R)).
In particular, if R/Qa(R) has S2 it is a Cohen-Macaulay ring ifand only if the module Ta(R) is Cohen-Macaulay.
Majid Eghbali On an endomorphism ring of local cohomology
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Applications
TheoremLet a be an ideal of a complete local ring (R,m). For an integerr ≥ 2 we have the following statements:(1) Suppose R/Qa(R) has S2. Then Ta(R) satisfies the
condition Sr if and only if H im(R/Qa(R)) = 0 for
d − r + 2 ≤ i < d .
(2) R/Qa(R) satisfies the condition Sr if and only ifH im(Ta(R)) = 0 for d − r + 2 ≤ i < d and
R/Qa(R) ∼= HomR(Hda (R), Hd
a (R)).
In particular, if R/Qa(R) has S2 it is a Cohen-Macaulay ring ifand only if the module Ta(R) is Cohen-Macaulay.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
HartshorneLet (R,m) denote a local ring such that depth R ≥ 2. ThenSpec R \ {m} is connected in Zariski topology.
Example
Let R := k [x , y , u, v ]/((x , y) ∩ (u, v)). Then R is a twodimensional ring such that Spec R \ {m} is disconnected. ThenR can not be Cohen-Macaulay ring.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
HartshorneLet (R,m) denote a local ring such that depth R ≥ 2. ThenSpec R \ {m} is connected in Zariski topology.
Example
Let R := k [x , y , u, v ]/((x , y) ∩ (u, v)). Then R is a twodimensional ring such that Spec R \ {m} is disconnected. ThenR can not be Cohen-Macaulay ring.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
HartshorneLet (R,m) denote a local ring such that depth R ≥ 2. ThenSpec R \ {m} is connected in Zariski topology.
Example
Let R := k [x , y , u, v ]/((x , y) ∩ (u, v)). Then R is a twodimensional ring such that Spec R \ {m} is disconnected. ThenR can not be Cohen-Macaulay ring.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
DefinitionLet (R,m) denote a local ring. We denote by G(R) theundirected graph whose vertices are primes p ∈ Spec R suchthat dim R = dim R/p, and two distinct vertices p, q are joinedby an edge if and only if (p, q) is an ideal of height one.
Proposition
Let (R,m) denote a local ring and d = dim R. Then thefollowing conditions are equivalent:(1) The graph G(R) is connected.(2) Spec R/0d is connected in codimension one.(3) For every ideal JR/0d of height at least two,
Spec(R/0d ) \ V (JR/0d ) is connected.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
DefinitionLet (R,m) denote a local ring. We denote by G(R) theundirected graph whose vertices are primes p ∈ Spec R suchthat dim R = dim R/p, and two distinct vertices p, q are joinedby an edge if and only if (p, q) is an ideal of height one.
Proposition
Let (R,m) denote a local ring and d = dim R. Then thefollowing conditions are equivalent:(1) The graph G(R) is connected.(2) Spec R/0d is connected in codimension one.(3) For every ideal JR/0d of height at least two,
Spec(R/0d ) \ V (JR/0d ) is connected.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
DefinitionLet (R,m) denote a local ring. We denote by G(R) theundirected graph whose vertices are primes p ∈ Spec R suchthat dim R = dim R/p, and two distinct vertices p, q are joinedby an edge if and only if (p, q) is an ideal of height one.
Proposition
Let (R,m) denote a local ring and d = dim R. Then thefollowing conditions are equivalent:(1) The graph G(R) is connected.(2) Spec R/0d is connected in codimension one.(3) For every ideal JR/0d of height at least two,
Spec(R/0d ) \ V (JR/0d ) is connected.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
Hochster-HunekeLet (R,m) be a complete local equidimensional ring andd = dim R. Then the following conditions are equivalent:(1) Hd
m(R) is indecomposable.(2) KR, the canonical module of R is indecomposable.(3) The ring HomR(KR, KR) is local.(4) For every ideal J of height at least two, Spec(R) \ V (J) is
connected.(5) The graph G(R) is connected.
Majid Eghbali On an endomorphism ring of local cohomology
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Some connectedness results
Hochster-HunekeLet (R,m) be a complete local equidimensional ring andd = dim R. Then the following conditions are equivalent:(1) Hd
m(R) is indecomposable.(2) KR, the canonical module of R is indecomposable.(3) The ring HomR(KR, KR) is local.(4) For every ideal J of height at least two, Spec(R) \ V (J) is
connected.(5) The graph G(R) is connected.
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Some connectedness results
The extension of Hochster-Huneke TheoremLet (R,m) denote a complete local ring and d = dim R. For anideal a ⊂ R the following conditions are equivalent:(1) Hd
a (R) is indecomposable.(2) HomR(Hd
a (R), E(R/m)) is indecomposable.(3) The endomorphism ring of Hd
a (R) is a local ring.(4) The graph G(R/Qa(R)) is connected,
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Some connectedness results
The extension of Hochster-Huneke TheoremLet (R,m) denote a complete local ring and d = dim R. For anideal a ⊂ R the following conditions are equivalent:(1) Hd
a (R) is indecomposable.(2) HomR(Hd
a (R), E(R/m)) is indecomposable.(3) The endomorphism ring of Hd
a (R) is a local ring.(4) The graph G(R/Qa(R)) is connected,
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
NotationWe describe t , the number of connected components ofG(R/Qa(R)).
DefinitionA connected component of an undirected graph is a subgraphin which any two vertices are connected to each other by paths,and which is connected to no additional vertices.
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
NotationWe describe t , the number of connected components ofG(R/Qa(R)).
DefinitionA connected component of an undirected graph is a subgraphin which any two vertices are connected to each other by paths,and which is connected to no additional vertices.
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
NotationWe describe t , the number of connected components ofG(R/Qa(R)).
DefinitionA connected component of an undirected graph is a subgraphin which any two vertices are connected to each other by paths,and which is connected to no additional vertices.
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
DefinitionLet a be an ideal in a local ring (R,m). Suppose that Q = Qa(R)is a proper ideal. Let Gi , i = 1, . . . , t , denote the connectedcomponents of G(R/Q). Let Qi , i = 1, . . . , t , denote theintersection of all p-primary components of a reduced minimalprimary decomposition of Q such that p ∈ Gi . Then Q = ∩t
i=1Qiand G(R/Qi) = Gi , i = 1, . . . , t , is connected. Moreover, letai , i = 1, . . . , t , denote the image of the ideal a in R/Qi .
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
DefinitionLet a be an ideal in a local ring (R,m). Suppose that Q = Qa(R)is a proper ideal. Let Gi , i = 1, . . . , t , denote the connectedcomponents of G(R/Q). Let Qi , i = 1, . . . , t , denote theintersection of all p-primary components of a reduced minimalprimary decomposition of Q such that p ∈ Gi . Then Q = ∩t
i=1Qiand G(R/Qi) = Gi , i = 1, . . . , t , is connected. Moreover, letai , i = 1, . . . , t , denote the image of the ideal a in R/Qi .
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
TheoremLet a denote an ideal of a complete local ring (R,m) withd = dim R ≥ 2. Then
End Hda (R) ' End Hd
a1(R/Q1)× . . .× End Hd
at(R/Qt )
is a semi-local ring, End Hdai
(R/Qi), i = 1, . . . , t , is a local ringand therefore t is equal to the number of maximal ideals ofEnd Hd
a (R).
Majid Eghbali On an endomorphism ring of local cohomology
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Number of connected components
TheoremLet a denote an ideal of a complete local ring (R,m) withd = dim R ≥ 2. Then
End Hda (R) ' End Hd
a1(R/Q1)× . . .× End Hd
at(R/Qt )
is a semi-local ring, End Hdai
(R/Qi), i = 1, . . . , t , is a local ringand therefore t is equal to the number of maximal ideals ofEnd Hd
a (R).
Majid Eghbali On an endomorphism ring of local cohomology
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THANK YOU VERY MUCH
Majid Eghbali On an endomorphism ring of local cohomology