majid eghbali ipm november 30, 2011 8th seminar on...

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

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).

Why local cohomolgy

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.

Resent Work

HomR(Hdm(R), Hd

HomR(H ia(R), H i

HomR(Hda (R), Hd

Resent Work

HomR(Hdm(R), Hd

HomR(H ia(R), H i

HomR(Hda (R), Hd

Resent Work

HomR(Hdm(R), Hd

HomR(H ia(R), H i

HomR(Hda (R), Hd

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?

Questions

To Control top local cohomology

Express Hda (M) via Hd

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}.

Explanation

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.

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.

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.

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.

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.

S (M, S), E(R/m)), d := dim M,

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).

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).

Applications

(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.

a (R)) is a commutative semi-localNoetherian ring.

Applications

(4) AnnR̂(Hda (R)) = Qa(R̂).

a (R))

aR̂(R̂).

Applications

(4) AnnR̂(Hda (R)) = Qa(R̂).

a (R))

aR̂(R̂).

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.

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.

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.

Example

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.

Proposition

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.

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.

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,

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,

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.

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 .

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 .

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).

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).

THANK YOU VERY MUCH

majid eghbali ipm november 30, 2011 8th seminar on...

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