# Gorenstein-Projective Modules over a Class of Morita Rings

Authors: Dadi Asefa
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetΔ0,0=AANBBMAB be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a Δ0,0-module X,Y,f,g to be Gorenstein-projective. As an application, we give sufficient conditions when the algebras A and B inherit the strongly CM-freeness of Δ0,0.


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## Body

## 1. Introduction

Gorenstein algebra and Gorenstein-projective modules are important topics of research in Gorenstein homological algebra. A fundamental problem in Gorenstein homological algebra is determining all the Gorenstein-projectiveA-modules for a given algebra A. The class of Gorenstein-projective modules is a key component of relative homological algebra and has received a great deal of attention in the study of representation theory (e.g., [1–6, 8–13, 16–18, 20, 23–27]).For algebrasA and B, bimodules BMA and ANB, and a B-B-bimodule map ϕ:M⊗AN⟶B, and an A-A-bimodule map ψ:N⊗BM⟶A satisfying some special conditions. Bass [7] introduced Morita algebra Δϕ,ψ=AANBBMAB, where the special conditions for ϕ and ψ are to guarantee that the multiplication of Δϕ,ψ has the associativity. Morita algebras Δϕ,ψ give a very large class of algebras, and many important algebras can be realized as Morita algebras. For example, the 2×2 matrix algebra M2A=AAAA over A, the algebra Δ0,0=AAAA, the upper triangular matrix algebra AANB0B, the algebras defined by finite quivers and relations. Thus, researching Morita rings is pivotal.Asefa [1] obtained sufficient conditions for Gorenstein-projective module X,Y,f,g over Δϕ,ψ, implying that X is a Gorenstein-projective A-module and Y is a Gorenstein-projective B-module. Gao and Psaroudakis [13] constructed Gorenstein-projective modules over a Morita ring Δ0,0. They stated that [13], [Theorem 3.10] does not give sufficient conditions for a module X,Y,f,g Gorenstein-projective ([13], Remark 3.13). As a result, it is natural to ask, “When is a module X,Y,f,g Gorenstein-projective?”. This paper is motivated to answer this question. In the following main result, we give sufficient conditions for X,Y,f,g to be a Gorenstein-projective module over a Morita ring Δ0,0.Theorem 1.
LetΔ0,0 be a Morita ring. Assume that(i)
MA and NB have finite flat dimensions.(ii)
BM and AN have finite projective dimensions.
Then, if each of the following conditions holds, aΔ0,0-module X,Y,f,g is Gorenstein-projective.(1)
Cokerg is Gorenstein-projective A-module;(2)
Cokerf is Gorenstein-projective B-module; and(3)
M⊗ACokerg≅Imf and N⊗BCokerf≅Img.
Lastly, we give sufficient conditions when the algebrasA and B inherit the strongly CM-freeness of Δ0,0.


## 2. Preliminaries

This section discusses some basic definitions and facts that will be used throughout the paper.Throughout, rings mean a ring with unity and anR-module mean a left R-module. Let R be a ring. Let M be an R-module, then the projective(injective and flat) dimension of M will be denoted by projdimM (injdimM and flatdimM). The class of modules isomorphic to direct summands of direct sums of copies of M is denoted by AddM.AnR-module M is Gorenstein-projective if there exists an exact sequence of projective R-modules(1)P•≔⋯⟶P−1⟶P0⟶d0P1⟶⋯.such thatHomRP•,Q is exact for an arbitrary projective R-module Q and that M≅Kerd0. The class of Gorenstein-projective R-modules will be denoted by GProjR.LetA and B be rings, ANB and BMA bimodules, and ϕ:M⊗AN⟶B and ψ:N⊗BM⟶A bimodules homomorphism. This paper focuses on the case of ϕ=0=ψ. Then,(2)Δ0,0≔AANBBMAB=anmb|a∈A,b∈B,m∈M,n∈N,is a Morita ring, where the addition is that of a matrix, and multiplication of this Morita ring is given as follows:(3)anmb.a′n′m′b′=aa′an′+nb′ma′+bm′bb′.The caseϕ=0=ψ is a subclass of the general Morita rings(e.g., [7, 13–15, 21].)


### 2.1. Modules overΔ0.0

A left module over Δ(0.0) is given asX,Y,f,g, where X is an A-module, Y is a B-module, and(4)f:M⊗AX⟶Y,g:N⊗BY⟶X,where g is an A-module map and f is a B-map.AΔ0,0-module morphism is given by a,b:X,Y,f,g⟶X′,Y′,f′,g′, where a:X⟶X′ is a homomorphism in A−Mod and b:Y⟶Y′ is a homomorphism in B−Mod such that the following diagrams are commutative.(5)Lemma 1 (see [13]).
LetΔϕ,ψ=AANBBMABbe a Morita ring.(1)
A sequence0⟶X′′,Y′′,f′′,g′′⟶X,Y,f,g⟶X′,Y′,f′,g′⟶0is exact inΔ-Mod if and only if the sequence0⟶X′′⟶X⟶X′⟶0is exact inA-Modand the sequence0⟶Y′′⟶Y⟶Y′⟶0isB-Mod.(2)
Letα,β:X,Y,f,g⟶X1,Y1,f1,g1a morphism inΔ-module and consider the mapsσ:Kerα⟶Xandγ:Kerβ⟶Y. Then,Kerα,βis given byKerα,Kerβ,t,swhere the mapstandsare induced from the commutative diagrams given below.(6)Similarly, the Cokernel ofa,bcan be described.


### 2.2. We Now Recall Functors Given in [16]

(1)
The functorTA:A-Mod ⟶Δ0,0-Mod is given by TAX≔X,M⊗AX,1,0 for any object X in A-Mod.(2)
The functorTB:B-Mod ⟶Δ0,0-Mod is given by TBY≔N⊗BY,Y,0,1 for any object Y in B-Mod.(3)
The functorUA:Δ0,0-Mod ⟶A-Mod is given by UAX,Y,f,g:=X for any object X,Y,f,g in Δ0,0-Mod.(4)
The functorUB:Δ0,0-Mod ⟶B-Mod is given by UBX,Y,f,g:=Y for any object X,Y,f,g in Δ0,0-Mod.(5)
LetX∈A be any object in Mod, then we denote by ϵX:N⊗BHomAN,X⟶X the map A-module given by involution. The functor HA:A-Mod ⟶Δ0,0-Mod is given by HAX≔X,HomAN,X,0,ϵX for any object X in A-Mod.(6)
LetY be any object in B-Mod, the we denote by ϵY:M⊗AHomBM,Y⟶Y the map B-module given by involution. The functor HB:B-Mod ⟶Δ0,0-Mod is given by HBY≔HomBM,Y,Y,ϵY,0 for any object Y in B-Mod.(7)
The functorZA:A-Mod ⟶Δ0,0-Mod is defied by ZAX:=X,0,0,0 for any object X in A-Mod. The functor ZB:B-Mod ⟶Δ0,0-Mod can be similarly defined.More information about the functors given above can be found in the following result.Proposition 1.
([16], Proposition 2.4]), LetΔ0,0be Morita ring. Then,(1)
The functorsHA,HB,TA, andTB, are fully faithful.(2)
The pairsUA,HA,UB,HB,TA,UA, andTB,UBare adjoint pairs.(3)
The functorsUAandUBare exact.Lemma 2.
LetΔ0,0be Morita ring.(1)
[19], [Theorem 7.3] A leftΔ0,0-moduleP,Q,f,gis projective if and only ifP,Q,f,g=TAX⊕TBY=X,M⊗AX,1,0⊕Y,N⊗BY,Y,0,1for some projective leftA-moduleXand projective leftB-moduleY.(2)
[22], [Corollary 2.2] A leftΔ0,0-moduleI,J,f,gis injective if and only ifI,J,f,g=HAX⊕HBY=X,HomAN,X,0,ϵX⊕HomBM,Y,Y,ϵY,0for some injective leftA-moduleXand injective leftB-moduleY.


## 2.1. Modules overΔ0.0

A left module over Δ(0.0) is given asX,Y,f,g, where X is an A-module, Y is a B-module, and(4)f:M⊗AX⟶Y,g:N⊗BY⟶X,where g is an A-module map and f is a B-map.AΔ0,0-module morphism is given by a,b:X,Y,f,g⟶X′,Y′,f′,g′, where a:X⟶X′ is a homomorphism in A−Mod and b:Y⟶Y′ is a homomorphism in B−Mod such that the following diagrams are commutative.(5)Lemma 1 (see [13]).
LetΔϕ,ψ=AANBBMABbe a Morita ring.(1)
A sequence0⟶X′′,Y′′,f′′,g′′⟶X,Y,f,g⟶X′,Y′,f′,g′⟶0is exact inΔ-Mod if and only if the sequence0⟶X′′⟶X⟶X′⟶0is exact inA-Modand the sequence0⟶Y′′⟶Y⟶Y′⟶0isB-Mod.(2)
Letα,β:X,Y,f,g⟶X1,Y1,f1,g1a morphism inΔ-module and consider the mapsσ:Kerα⟶Xandγ:Kerβ⟶Y. Then,Kerα,βis given byKerα,Kerβ,t,swhere the mapstandsare induced from the commutative diagrams given below.(6)Similarly, the Cokernel ofa,bcan be described.


## 2.2. We Now Recall Functors Given in [16]

(1)
The functorTA:A-Mod ⟶Δ0,0-Mod is given by TAX≔X,M⊗AX,1,0 for any object X in A-Mod.(2)
The functorTB:B-Mod ⟶Δ0,0-Mod is given by TBY≔N⊗BY,Y,0,1 for any object Y in B-Mod.(3)
The functorUA:Δ0,0-Mod ⟶A-Mod is given by UAX,Y,f,g:=X for any object X,Y,f,g in Δ0,0-Mod.(4)
The functorUB:Δ0,0-Mod ⟶B-Mod is given by UBX,Y,f,g:=Y for any object X,Y,f,g in Δ0,0-Mod.(5)
LetX∈A be any object in Mod, then we denote by ϵX:N⊗BHomAN,X⟶X the map A-module given by involution. The functor HA:A-Mod ⟶Δ0,0-Mod is given by HAX≔X,HomAN,X,0,ϵX for any object X in A-Mod.(6)
LetY be any object in B-Mod, the we denote by ϵY:M⊗AHomBM,Y⟶Y the map B-module given by involution. The functor HB:B-Mod ⟶Δ0,0-Mod is given by HBY≔HomBM,Y,Y,ϵY,0 for any object Y in B-Mod.(7)
The functorZA:A-Mod ⟶Δ0,0-Mod is defied by ZAX:=X,0,0,0 for any object X in A-Mod. The functor ZB:B-Mod ⟶Δ0,0-Mod can be similarly defined.More information about the functors given above can be found in the following result.Proposition 1.
([16], Proposition 2.4]), LetΔ0,0be Morita ring. Then,(1)
The functorsHA,HB,TA, andTB, are fully faithful.(2)
The pairsUA,HA,UB,HB,TA,UA, andTB,UBare adjoint pairs.(3)
The functorsUAandUBare exact.Lemma 2.
LetΔ0,0be Morita ring.(1)
[19], [Theorem 7.3] A leftΔ0,0-moduleP,Q,f,gis projective if and only ifP,Q,f,g=TAX⊕TBY=X,M⊗AX,1,0⊕Y,N⊗BY,Y,0,1for some projective leftA-moduleXand projective leftB-moduleY.(2)
[22], [Corollary 2.2] A leftΔ0,0-moduleI,J,f,gis injective if and only ifI,J,f,g=HAX⊕HBY=X,HomAN,X,0,ϵX⊕HomBM,Y,Y,ϵY,0for some injective leftA-moduleXand injective leftB-moduleY.


## 3. Gorenstein-Projective Modules overΔ0,0

This section aims to construct Gorenstein-projective modules overΔ0,0.The following lemmas are required in order to prove the main theorems of this paper.Lemma 3.
LetAbe a ring andMaB-A-bimodule with finite flat dimension. If a complex of flatA-modulesℱ•is exact, then, the sequenceM⊗Aℱ•is also exact.Proof.
Assume thatℱ• is an exact complex of flat A-modules. Because M has a finite flat dimension, we have the following flat resolution of M.(7)0⟶Fn⟶Fn−1⟶⋯⟶F0⟶MA⟶0.
We obtain the following exact sequence of complexes because all terms in the complexℱ• are flat.(8)0⟶Fn⊗Aℱ•⟶Fn−1⊗Aℱ•⟶⋯⟶F0⊗Aℱ•⟶MA⊗Aℱ•⟶0.
Since the complexesFi⊗Aℱ• are exact for all i, so is M⊗Aℱ•. □Lemma 4.
LetBbe a ring. If aB-moduleNhas finite injective dimension and the complex of projectiveB-modules,(9)Q•≔⋯⟶Qn−1⟶Qn⟶Qn+1⟶⋯,is exact, then so isHomBQ•,N.Lemma 5.
LetΔ0,0be a Morita ring with zero bimodule homomorphisms. Then(1)
[13], [Lemma 3.8] For eachX∈A-Mod and eachY∈B-Mod we have the following exact sequences inΔ0,0-Mod.(10)0⟶ZBM⊗AX⟶TAX⟶ZAX⟶0.and(11)0⟶ZAN⊗BY⟶TBY⟶ZBY⟶0.(2)
[13], [Lemma 3.9] For allX,X′∈A-Mod andY,Y′∈B-Mod, we have the following isomorphisms:(12)HomΔ0,0TAX⊕TBY,ZAX′≅HomAX,X′.and(13)HomΔ0,0TAX⊕TBY,ZBY′≅HomBY,Y′.The following result provides sufficient conditions for the functorTA:A−Mod⟶Δ0,0−Mod and the functor TB:B−Mod⟶Δ0,0−Modto preserve Gorenstein-projective modules.Proposition 2.
(1)
Assume thatMA has a finite flat dimension and that AN has a finite projective dimension. TAX is a Gorenstein-projective Δ0,0-module if X is a Gorenstein-projective A-module.(2)
Assume thatNB has a finite flat dimension and that BM has a finite projective dimension. TBY is a Gorenstein-projective Δ0,0-module if Y is a Gorenstein-projective B-module.Proof.
We show (1) and (2) can be proved in a similar manner. Since an A-module X is a Gorenstein-projective, there is an exact sequence of projective A-modules,(14)P•:⋯⟶P−1⟶P0⟶d0P1⟶⋯,such that X≅Kerd0, and HomAP•,Q exact for any projective A-module Q. Lemma 3 states that the assumption that MA has finite flat dimension implies the sequence M⊗AP• is exact. Hence we get the exact sequence of projective Δ0,0-modules,(15)TAP•:⋯⟶TAP−1⟶TAP0⟶d0,1⊗d0TAP1⟶⋯,such that TAX≅Kerd0,1⊗d0. Now, it is left to show that HomΔ0,0TAP•,X′,Y′,f′,g′ is exact for any projective Δ0,0-module X′,Y′,f′,g′. By Lemma 2, this can be proved by showing the exactness of HomΔ0,0TAP•,TAP and HomΔ0,0TAP•,TBQ for any projective A-module P, and any projective B-module Q. By Proposition 1 the functor TA is fully faithful. Thus, HomΔ0,0TAP•,TAP≅HomAP•,P. Hence HomΔ0,0TAP•,TAP because HomAP•,P is exact. Since TA,UA are adjoint pairs, we have the following equation:(16)HomΔ0,0TAP•,TBQ≅HomAP•,N⊗BQ.
A moduleN⊗BQ has finite projective dimension because it is isomorphic to a direct summand of direct sums of copies of N. Since P• is a complete A-projective resolution, the complex HomAP•,N⊗BQ is exact(see [18], [Proposition 2]). Thus, HomΔ0,0TAP•,TBQ is exact. Hence HomΔ0,0TAP•,X′,Y′,f′,g′ is exact for any projective Δ0,0-module X′,Y′,f′,g′. Therefore, TAX is a Gorenstein-projective Δ0,0-module.In the following result, we give sufficient conditions for aΔ0,0-module X,Y,f,g to be Gorenstein-projective.Theorem 2.
LetΔ0,0be a Morita ring. Assume that(i)
MAandNBhave finite flat dimensions.(ii)
BMandANhave finite projective dimensions.
Then, if each of the following conditions holds, aΔ0,0-moduleX,Y,f,gis Gorenstein-projective.(1)
Cokergis a Gorenstein-projectiveA-module;(2)
Cokerfis a Gorenstein-projectiveB-module; and(3)
M⊗ACokerg≅ImfandN⊗BCokerf≅Img.Proof.
Suppose that conditions (1)–(3) are true. Since Cokerf is a Gorenstein-projective B-module, there exists an exact complex of projective B-modules,(17)Q•:⋯⟶Q−1⟶Q0⟶d′0Q1⟶⋯.
such thatCokerf≅Kerd′0 and HomBQ•,Q is exact for each projective B-module Q. Thus, we get the following exact sequence,(18)0⟶N⊗BCokerf⟶N⊗BQ0⟶Id⊗d′0N⊗BQ1⟶⋯,
becauseNB has a finite flat dimension. Since Cokerg is a Gorenstein-projective A-module, there exists a complete projective resolutions,(19)P•:⋯⟶P−1⟶P0⟶d0P1⟶⋯,of A-modules such that Cokerg≅Kerd0.
Letπ1:X⟶Cokerg and π2:Y⟶Cokerf. Consider the following commutative diagram of A-modules.(20)
Since ψ = 0, the above equation implies that there exists anA-map i1:N⊗BCokerf⟶X that is unique and g=i1°IdN⊗π2. Thus, from Img≅N⊗BCokerf it follows that i1 is an injective A-map. Thus, we get the exact sequence as follows:(21)0⟶N⊗BCokerf⟶i1X⟶π1Cokerg⟶0
Similarly, the sequence(22)0⟶M⊗ACokerg⟶i2Y⟶π2Cokerf⟶0,is exact.
Since eachN⊗AQi has finite projective dimension, and since each Kerdi is a Gorenstein-projective A-module, we have that ExtA1Kerdi,N⊗BQi=0,∀i≥0. Applying generalized Horseshoe Lemma([26], Lemma 1.6 ii) to the exact sequences (18) and (21), we obtain an exact sequence as follows:(23)0⟶X⟶P0⊕N⊗BQ0⟶α0P1⊕N⊗BQ1⟶α1⋯.with αi=di0γiIdN⊗d′i, γi:Pi⟶N⊗BQi+1, ∀i∈ℤ, such that the following diagram(24)is commutative. The dual argument obtains the commutative diagram with exact rows shown below.(25)
When we combine (24) and (25), we get the exact sequence shown below.(26)⋯⟶P−2⊕N⊗BQ−2⟶α−2P−1⊕N⊗BQ−1−⟶α−1P0⊕N⊗BQ0⟶α0P1⊕N⊗BQ1⋯α1.with Kerα0=X.
We now construct an exact sequence similar to (26) for a left B-module Y. Since each M⊗APi has finite projective dimension as B-module by assumption on M, and Kerd′i is a Gorenstein-projective B-module, it follows that ExtB1Kerd′i,M⊗APi=0. Thus, by ([26], Lemma 1.6 ii) again, we obtain the exact sequence as follows:(27)0⟶Y⟶M⊗AP0⊕Q0⟶β0M⊗AP1⊕Q1⟶β1⋯.with βi=IdM⊗diσi0d′i, and σi:Qi⟶M⊗APi+1, ∀i∈ℤ, such that the diagram(28)is commutative. The dual argument gives the commutative diagram with the exact rows(29)
As a result, combining (28) and (29) yields the following exact sequence, which is similar to the following equation:(30)⋯⟶M⊗AP−2⊕Q−2⟶β−2M⊗AP−1⊕Q−1−⟶β−1M⊗AP0⊕Q0⟶β0M⊗AP1⊕Q1⋯β1with Kerβ0=Y.
Glue together the exact sequences (26) and (30) to obtain the following sequence:(31)T•:⋯⟶TAP−1⊕TBQ−1−⟶α−1β−1TAP0⊕TBQ0⟶α0β0⋯with Kerα0β0=X,Y,f,g.
The morphismαiβi∀i∈ℤ, is a Δ0,0-map because(32)and(33)
are commutative diagrams.
Since the complexes (26) and (30) are exact, it follows from Lemma 1 (1) that the sequence T• is exact. The object X,Y,f,g arises as the kernel of the morphism α0β0, and we see from Lemma 1 (2) that f=i2°IdM⊗π1 and g=i1°IdN⊗π2. However, based on the commutative diagram of A-modules shown below,(34)
We know thatg is uniquely determined by i1°IdN⊗π2. Similarly, f is uniquely determined by i1°IdM⊗π1.
We are now left with showing thatHomΔ0,0T•,X′,Y′,f′,g′ is exact for each projective Δ0,0-module X′,Y′,f′,g′. We can deduce from Lemma 2 that it is enough to show that HomΔ0,0T•,TAP and HomΔ0,0T•,TBQ are exact for each projective A-module P and for each projective B-module Q. By Lemma 5 (1) the sequence 0⟶ZBM⊗AP⟶TAP⟶ZAP⟶0 is exact. Since each term in the complex T• is a projective Δ0,0-module, the sequence(35)0⟶HomΔ0,0T•,ZBM⊗AP⟶HomΔ0,0T•,TAP⟶HomΔ0,0T•,ZAP⟶0,is exact. By Lemma 5 (2) we have the following equations,(36)HomΔ0,0T•,ZAP≅HomAP•,P.
The complexHomAP•,P is exact because P• is a complete projective resolution. Thus, the complex HomΔ0,0T•,ZAP is exact. Also, by Lemma 5 (2), we have(37)HomΔ0,0T•,ZBM⊗AP≅HomBQ•,M⊗AP.
To show the exactness ofHomBQ•,M⊗AP, we know that a B-module M⊗AP has finite projective dimension, since M⊗AP is isomorphic to direct summand of a direct sum of copies of M. Thus, HomBQ•,M⊗AP is exact by [18], [Proposition 2.3], which implies HomΔ0,0T•,ZBM⊗AP is exact. Hence from the exact sequence of complexes in (35) it follows that the complex HomΔ0,0T•,TAP is exact. Similarly, the complex(38)HomΔ0,0T•,TBQ,is exact. Thus, HomΔ0,0T•,X′,Y′,f′,g′ is exact for each projective Δ0,0-module X′,Y′,f′,g′. Therefore, a Δ0,0-module X,Y,f,g is a Gorenstein-projective.If the converse of Theorem2 holds, then Gorenstein-projective modules over Δ0,0 will be fully determined. However, whether the converse is true or not is an open problem.Corollary 1.
LetΔ0,0=AAAAbe Morita ring, andX,Y,f,gΔ0,0-module. IfCokerg,Cokerfare Gorenstein-projectiveA-modules such thatCokerg≅ImfandCokerf≅Img, thenX,Y,f,gis a Gorenstein-projectiveΔ0,0-module.


## 4. Application

In this section, we study when the class of all Gorenstein-projectiveA-modules and B-modules coincides with the class of projective A-modules and B-modules, respectively.If each finitely generated projective leftR-module is projective, then a ring R is said to be left CM-free. And R is said to be strongly left CM-free if each Gorenstein-projective left module is projective(see [12]).The results that follow provide sufficient conditions for the algebrasA and B to inherit the stronglyCM-freeness of Δ0,0.Proposition 3.
LetΔ0,0=AANBBMABbe Morita ring.(1)
Assume thatMAhas finite flat dimension,ANis projective, andΔ0,0is a strongly CM-free, thenAis a strongly CM-free.(2)
Assume thatNBhas finite flat dimension,BMis projective, andΔ0,0is a strongly CM-free, thenBis a strongly CM-free.Proof.
(1)
AssumeAN is projective and Δ0,0 is a strongly CM-free. Let X be a Gorenstein-projective A-module. Because MA has a finite projective dimension, Proposition 2 (1) asserts that TAX=X,M⊗AX,IdM⊗AX,0 is a Gorenstein-projective Δ0,0-module. The assumption that Δ0,0 is a strongly CM-free implies that TAX is a projective Δ0,0-module. By Lemma 2 (1), TAX=TAP for some projective A-module PorTAX=TBQ=N⊗BQ,Q,0,1 for some projective B-module Q. Hence X=P, or X=N⊗BQ. An A-module N⊗BQ is projective because it is isomorphic to a direct summand of a direct sum of copies of AN and AN is projective. Thus, X is a projective A-module. Therefore, A is a strongly CM-free.(2)
AssumeBM is projective and Δ0,0 is a strongly CM-free. Let Y be a Gorenstein-projective B-module. By similar argument as in(1), Y is a projective B-module. Therefore, B is a strongly CM-free.As a consequence we have the following corollary.Corollary 2.
LetΔ0,0=AAAAbe Morita ring. IfΔ0,0is a strongly CM-free, thenAis a strongly CM-free.



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