Cousin complexes with applications to local cohomology and

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

Cousin complexes

Uniform local cohomological

annihilators

Attached primes of local

cohomologies

Cousin complexes

Uniform local cohomological

annihilators

Attached primes of local

cohomologies

Cohen-Macaulay locus

Cohen-Macaulay formal fibres

Generalized Cohen-Mcaulay

modules

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

Contents


- History

- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal





Contents


- History- Cohomology modules of Cousin complexes

- Uniform annihilators of local cohomology- Some partial characterizations- Height of an ideal





Contents


- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology

- Some partial characterizations- Height of an ideal





Contents


- History- Cohomology modules of Cousin complexes- Uniform annihilators of local cohomology- Some partial characterizations

- Height of an ideal





Contents







Contents







Contents




- Attached primes related to cohomologies of Cousin complexes

- Top local cohomological modules- Application to generalized Cohen-Macaulay modules



Contents




- Attached primes related to cohomologies of Cousin complexes- Top local cohomological modules

- Application to generalized Cohen-Macaulay modules



Contents







Contents







Contents






- Openness of Cohen-Macaulay locus

- Rings whose formal fibres are Cohen-Macaulay- Some comments

Contents






- Openness of Cohen-Macaulay locus- Rings whose formal fibres are Cohen-Macaulay

- Some comments

Contents







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).


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).


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.


Cousin Complexes


C(F ,M) : 0d−2

−→ M−1d−1

−→ M0 d0

−→ M1 d1

−→ · · · dn−1

−→ Mn dn

−→ Mn+1 −→ · · ·,


p∈Fn

(Coker dn−2)p.







Cousin Complexes


C(F ,M) : 0d−2

−→ M−1d−1

−→ M0 d0

−→ M1 d1

−→ · · · dn−1

−→ Mn dn

−→ Mn+1 −→ · · ·,


p∈Fn

(Coker dn−2)p.







Cousin Complexes


C(F ,M) : 0d−2

−→ M−1d−1

−→ M0 d0

−→ M1 d1

−→ · · · dn−1

−→ Mn dn

−→ Mn+1 −→ · · ·,


p∈Fn

(Coker dn−2)p.







Cousin Complexes

Throughout R is a commutative, noetherian ring with non–zeroidentity and M is a finitely generated R–module.


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?


History



Question



History



Question



CM


CM


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).


History





History





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.


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.
























































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.










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.



•••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.



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.



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.



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)).



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)).


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.




















•••Assume that M is of finite dimension and that CR(M) is finite.Then M has a u.l.c.a



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.




R has a u.l.c.a.



M has a u.l.c.a.





R has a u.l.c.a.



M has a u.l.c.a.





R has a u.l.c.a.



M has a u.l.c.a.





R has a u.l.c.a.



M has a u.l.c.a.





R has a u.l.c.a.



M has a u.l.c.a.





R has a u.l.c.a.



M has a u.l.c.a.




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.



Cor .


M has a u.l.c.a.

N has a u.l.c.a.

Cor .





Cor .


M has a u.l.c.a.

N has a u.l.c.a.

Cor .





Cor .


M has a u.l.c.a.

N has a u.l.c.a.

Cor .





Cor .


M has a u.l.c.a.

N has a u.l.c.a.

Cor .




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.



•••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.




M has a u.l.c.a.






M has a u.l.c.a.






M has a u.l.c.a.




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.







M has a u.l.c.a.







M has a u.l.c.a.







M has a u.l.c.a.


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.


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.


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?


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?


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.


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)).


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.





Att Htm(M) =


M : dim R/p = t − i − 1}.








Att Htm(M) =


M : dim R/p = t − i − 1}.






•••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.




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.


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)






a(K ) 6= 0}.

Rem.








a(K ) 6= 0}.

Rem.








a(K ) 6= 0}.

Rem.








a(K ) 6= 0}.

Rem.





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).



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).



Question


a (M) = T ?

Rem.




a (M).



Question


a (M) = T ?

Rem.




a (M).



•••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|.



•••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 .








r⋂i=1

Qi .








r⋂i=1

Qi .








r⋂i=1

Qi .


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 ).


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.



•••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.





M is a g.CM module.


R is g.CM.






M is a g.CM module.


R is g.CM.






M is a g.CM module.


R is g.CM.






M is a g.CM module.


R is g.CM.




•••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 .



•••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 .


gCM

CM



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.



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.



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.













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.



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.


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.



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.



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.



Rem.








Rem.








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.



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.


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}.


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.





M has a u.l.c.a.






M has a u.l.c.a.






M has a u.l.c.a.





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.








Rem.









Rem.









Rem.









Rem.









Rem.





V(∏d−1

i=−1(0 :R HiM)) =non–CM(M) = V(a(M)).



•••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)).
























•••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



•••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



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.



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.


gCM

CM

Formal fibres over minimal

primes are CM

Formal fibres over primes of CM locus are

CM

All formal fibres are

CM


gCM

CM

If all formal fibres of the base ring are CM


Some comments

Question

Assume that CR(M) is finite. Are the formal fibres of R over allprime ideals p ∈ Supp M, Cohen-Macaulay?


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)}.


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)}.


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


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Some comments

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Some comments

Problem


Rem.


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Some comments

Problem


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Cousin complexes with applications to local cohomology and

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