auslander class and c–projective modules modulo exact...

$: Auslander class and C–projective modules modulo exact …math.ipm.ac.ir/conferences/2014/11th_Commalg/Slideshow/Amanzade… · Auslander class and C{projective modules modulo exact$
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Auslander class and C–projective modulesmodulo exact zero-divisors

Ensiyeh Amanzadeh

Kharazmi University

11th seminar on commutative algebra and related topicsSchool of Mathemaics, IPM, 2014

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Exact zero-divisors

Throughout, R is a commutative and noetherian ring and M anR–module.Exact zero–divisors introduced by Henriques and Sega, in 2009.

.Definition..

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An element x of R is called an exact zero–divisor on M if xM = 0,xM = M and there is y ∈ R such that the sequence ofmultiplication maps M

x−→ My−→ M

x−→ M is exact.In this case we say that x , y form a pair of exact zero–divisors onM.

.Example..

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Let a ∈ R be an R–regular element. For some integer r > 1, thesequence of multiplication maps

R/(ar )as−→ R/(ar )

ar−s

−→ R/(ar )as−→ R/(ar )

is exact for all 0 < s < r .Then as , ar−s form a pair of exact zero–divisors on R/(ar ).

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Exact zero-divisors

Throughout, R is a commutative and noetherian ring and M anR–module.Exact zero–divisors introduced by Henriques and Sega, in 2009..Definition..

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An element x of R is called an exact zero–divisor on M if xM = 0,xM = M and there is y ∈ R such that the sequence ofmultiplication maps M

x−→ My−→ M

x−→ M is exact.In this case we say that x , y form a pair of exact zero–divisors onM.

.Example..

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Let a ∈ R be an R–regular element. For some integer r > 1, thesequence of multiplication maps

R/(ar )as−→ R/(ar )

ar−s

−→ R/(ar )as−→ R/(ar )

is exact for all 0 < s < r .Then as , ar−s form a pair of exact zero–divisors on R/(ar ).

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Exact zero-divisors

.Basic properties..

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Let R be a local ring, x an exact zero–divisor on R.

• dim(R) = dim(R/xR) and depth(R) = depth(R/xR).(Avramov, Henriques and Sega, in 2010)

• R is Cohen-Macaulay if and only if R/xR is Cohen-Macaulay.(Avramov, Henriques and Sega, in 2010)

• R is Gorenstein if and only if R/xR is Gorenstein ring.(Henriques and Sega, in 2009)

• pdR(R/xR) = ∞.

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Exact zero-divisors

.Proposition (Dibaei and Gheibi)..

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Assume that x , y form a pair of exact zero–divisors on R. Let n bea non-negative integer. Consider the following statements.

(i) x , y form a pair of exact zero–divisors on M.

(ii) ExtiR(R/xR,M) = 0 for all i > n.

(iii) TorRi (R/xR,M) = 0 for all i > n.

Then (i)⇒(ii)⇔(iii). If one of the following conditions holds true,then the statements (i), (ii) and (iii) are equivalent.(a) xM = 0 and xM = M.(b) R is local and M is finite.

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Exact zero-divisors

We observed that, if x , y form a pair of exact zero–divisors on Rand if M

x−→ M is neither zero nor epimorphism, then x , y formalso a pair of exact zero–divisors on M whenever one of theconditions id(M) < ∞, pd(M) < ∞, or fd(M) < ∞ holds true.

.Proposition (Dibaei and Gheibi)..

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Assume that x , y form a pair of exact zero–divisors on both R andM and that N is an R/xR–module. Then the following statementshold true for all i ⩾ 0.

(i) ExtiR(N,M) ∼= ExtiR/xR(N,M/xM).

(ii) ExtiR(M,N) ∼= ExtiR/xR(M/xM,N).

(iii) TorRi (M,N) ∼= TorR/xRi (M/xM,N).

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Exact zero-divisors

We observed that, if x , y form a pair of exact zero–divisors on Rand if M

x−→ M is neither zero nor epimorphism, then x , y formalso a pair of exact zero–divisors on M whenever one of theconditions id(M) < ∞, pd(M) < ∞, or fd(M) < ∞ holds true..Proposition (Dibaei and Gheibi)..

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Assume that x , y form a pair of exact zero–divisors on both R andM and that N is an R/xR–module. Then the following statementshold true for all i ⩾ 0.

(i) ExtiR(N,M) ∼= ExtiR/xR(N,M/xM).

(ii) ExtiR(M,N) ∼= ExtiR/xR(M/xM,N).

(iii) TorRi (M,N) ∼= TorR/xRi (M/xM,N).

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Semidualizing modules

.Definition..

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An R–module C is called semidualizing, if

• C is finitely generated

• The natural homothety map χRC : R −→ HomR(C ,C ) is an

isomorphism

• For all i > 0, ExtiR(C ,C ) = 0

.Example..

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Examples of semidualizing modules include• R• The dualizing module of R if it exists (dualizing module is a

semidualizing module with finite injective dimension).

These are called the trivial semidualizing modules.

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Throughout C assumed to be a semidualizing R–module.

.Basic properties..

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• AnnR(C ) = 0 and SuppR(C ) = Spec(R).

• dimR(C ) = dim(R) and AssR(C ) = AssR(R).

• An element a ∈ R is R–regular if and only if it is C–regular.

• If a ∈ R is R–regular, then C/aC is semidualizingR/aR–module.

• If R is local, then depthR(C ) = depth(R).

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Throughout C assumed to be a semidualizing R–module..Basic properties..

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• AnnR(C ) = 0 and SuppR(C ) = Spec(R).

• dimR(C ) = dim(R) and AssR(C ) = AssR(R).

• An element a ∈ R is R–regular if and only if it is C–regular.

• If a ∈ R is R–regular, then C/aC is semidualizingR/aR–module.

• If R is local, then depthR(C ) = depth(R).

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Semidualizing and exact zero-divisor

Note that if x ∈ R is non–zero, then xC = 0. By Nakayama’slemma, xC = C if and only if (x) = R.Let x , y form a pair of exact zero–divisors on R.

• If pd(C ) < ∞, then Auslander-Buchsbaum formula impliesthat C is projective and so x , y form a pair of exactzero–divisors on C by definition.

• If R is Cohen-Macaulay local ring with dualizing module ω,then x , y form also a pair of exact zero–divisors on ω.

In general, we do not know whether a pair of exact zero–divisorson R is also a pair of exact zero–divisors on C .

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.Proposition (Dibaei, me)..

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Let (R,m, k) be a Cohen-Macaulay local ring that is notGorenstein, with dualizing module ω. Let f : R −→ S be a flatlocal ring homomorphism such that S/mS is not Gorenstein.Assume that x , y ∈ S form a pair of exact zero–divisors on S suchthat fdR(S/xS) < ∞. Then S ⊗R ω is a semidualizing S–modulewhich is not a dualizing S–module and pdS(S ⊗R ω) = ∞.Moreover, x , y form a pair of exact zero–divisors on S ⊗R ω.

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.Example (Dibaei, me)..

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Let R = k[X ,Y ]/(X ,Y )2, whenever k is a field. Then R is a localartinian ring that is not Gorenstein. As R is free k–module of rank3, ω = Homk(R, k) is dualizing R–module. SetS = R[U,V ,W ,Z ]/(U2,VW ,VZ ). Then S is free R–module andS/mS ∼= k[U,V ,W ,Z ]/(U2,VW ,VZ ) is not Cohen-Macaulay,where m is the maximal ideal of R. If u is the image of U in S ,then u, u form a pair of exact zero–divisors on S . We have anR–isomorphism S/uS ∼= R[V ,W ,Z ]/(VW ,VZ ) and so S/uS isfree R–module. Thus u, u form also a pair of exact zero–divisorson the semidualizing S–module S ⊗R ω.Note that S ⊗R ω is not a dualizing S–module withpdS(S ⊗R ω) = ∞.

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Semidualizing R/xR–modules


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Let B be a finite R–module. Assume that x , y form a pair of exactzero–divisors on both R and B. Then the following statements areequivalent.

(i) B is a semidualizing R–module.

(ii) B/xB and B/yB are semidualizing R/xR– andR/yR–modules, respectively.

.Corollary (Dibaei, me)..

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Let (R,m, k) be a Cohen-Macaulay local ring, D a finiteR–module. Assume that x , y form a pair of exact zero–divisors onboth R and D. If D/xD is a dualizing R/xR–module and D/yD isa semidualizing R/yR–module, then D is a dualizing R–module.

Note that the converse of the corollary was proved by Dibaei andGheibi, in 2011.

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The class GC (R)

.Definition..

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The class GC (R) consists of GC–projective R–modules, i.e. theclass of all finite R–modules M which satisfy the followingconditions.

• The natural homomorphismδCM : M −→ HomR(HomR(M,C ),C ) is an isomorphism.

• For all i > 0, ExtiR(M,C ) = 0 = ExtiR(HomR(M,C ),C ).

It is easy to see that if x , y form a pair of exact zero–divisors on R,then R/xR ∈ GR(R)..Proposition (Dibaei, me)..

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If x , y form a pair of exact zero–divisors on both R and C , then

R/xR ∈ GC (R).

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The class GC (R)

.Definition..

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It is easy to see that if x , y form a pair of exact zero–divisors on R,then R/xR ∈ GR(R).


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R/xR ∈ GC (R).

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The class GC (R)

.Definition..

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It is easy to see that if x , y form a pair of exact zero–divisors on R,then R/xR ∈ GR(R)..Proposition (Dibaei, me)..

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R/xR ∈ GC (R).

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Auslander class and Bass class

.Definition..

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The Auslander class AC (R) with respect to C is the class of allR–modules M satisfying the following conditions.• The natural map γCM : M −→ HomR(C ,C ⊗R M) is anisomorphism.• For all i > 0, TorRi (C ,M) = 0 = ExtiR(C ,C ⊗R M).

.Definition..

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The Bass class BC (R) with respect to C is the class of allR–modules M satisfying the following conditions.• The evaluation map ξCM : C ⊗R HomR(C ,M) −→ M is anisomorphism.• For all i > 0, ExtiR(C ,M) = 0 = TorRi (C ,HomR(C ,M)).

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The classes AC (R), BC (R) and GC (R)


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R/xR ∈ AC (R).

.Corollary (Dibaei, me)..

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Assume that x , y form a pair of exact zero–divisors on both R andC . If T is an R/xR–module, then the following statements holdtrue.

(i) If T is finite, then T ∈ GC (R) if and only ifT ∈ GC/xC (R/xR).

(ii) T ∈ AC (R) if and only if T ∈ AC/xC (R/xR).

(iii) T ∈ BC (R) if and only if T ∈ BC/xC (R/xR).

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Assume that x , y form a pair of exact zero–divisors on R, C andM.

• If M/xM ∈ AC/xC (R/xR) and M/yM ∈ AC/yC (R/yR), thenM ∈ AC (R).

• If M/xM ∈ BC/xC (R/xR) and M/yM ∈ BC/yC (R/yR), thenM ∈ BC (R).

• If M is finite, M/xM ∈ GC/xC (R/xR) andM/yM ∈ GC/yC (R/yR), then M ∈ GC (R).

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.Question (Dibaei, me)..

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Assume that x , y form a pair of exact zero–divisors on both R andM.

• If M ∈ GC (R), is M/xM ∈ GC (R)?

• If M ∈ BC (R), is M/xM ∈ BC (R)?

• If M ∈ AC (R), is M/xM ∈ AC (R)?

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Assume that x , y form a pair of exact zero–divisors on both R andM.

• If M ∈ GC (R), then M/xM ∈ GC (R) if and only if x , y formalso a pair of exact zero–divisors on HomR(M,C ).

• If M ∈ BC (R), then M/xM ∈ BC (R) if and only if x , y formalso a pair of exact zero–divisors on HomR(C ,M).

• If M ∈ AC (R) is finite , then M/xM ∈ AC (R) if and only ifx , y form also a pair of exact zero–divisors on C ⊗R M.

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The classes PC (R), FC (R) and IC (R).Definition..

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The classes of C–injective, C–projective and C–flat modules aredefined, respectively, as

IC (R) = { HomR(C , I ) | I is an injective R–module},PC (R) = { C ⊗R P | P is a projective R–module},FC (R) = { C ⊗R F | F is a flat R–module}.

They are the classes of injective, projective and flat R–modules,respectively, when C = R.

For any R–module M there exists an augmented properPC–projective resolution, that is, a complex

X+ = · · ·∂X2−→ C ⊗R P1

∂X1−→ C ⊗R P0

∂X0−→ M −→ 0

such that HomR(C ⊗R Q,X+) is exact for all projectiveR–module Q.

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The classes PC (R), FC (R) and IC (R)

The truncated complex

X = · · ·∂X2−→ C ⊗R P1

∂X1−→ C ⊗R P0 −→ 0

is called a proper PC–projective resolution of M.An augmented proper FC–projective resolution for M is definedsimilarly.Dually, for any R–module N there exists an augmented properIC–injective resolution, that is, a complex

Y+ = 0 −→ N −→ HomR(C , I 0)∂0Y−→ HomR(C , I 1)

∂1Y−→ · · ·

such that HomR(Y+,HomR(C , I )) is exact for all injective

R–module I .

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PC–projective dimension

.Definition..

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The PC–projective dimension of an R–module M is

PC −pd(M) = inf

{supX

∣∣∣ X is a proper PC − projectiveresolution of M

}where supX = sup{n | Xn = 0}. The modules of zeroPC–projective dimensions are the non-zero modules in PC (R); andwe set PC − pd(0) = −∞.The FC–projective dimension, denoted FC − pd(−), is definedsimilarly and the IC–injective dimension, denoted IC − id(−), isdefined dually.

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The class PC (R) modulo exact zero-divisors


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Let x , y form a pair of exact zero–divisors on both R and C .Assume that M is either in IC (R), PC (R), or FC (R) such thatM

x−→ M is neither zero nor epimorphism. Then x , y form a pairof exact zero-divisors on M.In any such case M/xM belongs to IC/xC (R/xR), PC/xC (R/xR),or FC/xC (R/xR), respectively.


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Assume that x , y form a pair of exact zero–divisors on both R andC . If PC–pd(M), FC–pd(M), or IC–id(M) is finit with xM = 0and xM = M, then x , y form a pair of exact zero–divisors on M.

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PC–projective dimension modulo exact zero-divisors


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Assume that x , y form a pair of exact zero–divisors on both R andC . Let xM = 0 and xM = M and set (−) = (−)⊗R R/xR. Thefollowing statements hold true.

(i) If PC − pd(M) < ∞, then PC − pd(M) ⩽ PC − pd(M).

(ii) If FC − pd(M) < ∞, then FC − pd(M) ⩽ FC − pd(M).

(iii) If M is finite with IC − id(M) < ∞, thenIC − id(M) ⩽ IC − id(M).

(iv) If R is local and M is finite, then equality holds in (i), (ii) and(iii).

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Thank You

auslander class and c–projective modules modulo exact...

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