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2.AEvery maximal Cohen-Macaulay module is torsion-free. Here a proof attempt for $M$$R$ a local CM ring with $K=R/m$, where $m$ is the maximal ideal of $R$.

Lemma: A noetherian module M is torsion-free if and only if $M$ is a first syzygy module $Hom_R(K,M)=0$.

(and this is also equivalentproof: The maps in $Hom_R(R/m,M)$ correspond to the conditionleft multiplications $L_z$ with $z \in Hom_R(R,M)=M$ such that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism)$z m=0$. )

But every maximal Cohen-Macaulay moduleNow assume $M$ is a first syzygy moduleMCM, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshinothen $depth(M) \geq 1$ and thus $Hom_R(K,M)=0$, thus $M$ is torsionfree.

(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)

2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)

2.Every maximal Cohen-Macaulay module is torsion-free. Here a proof attempt for $R$ a local CM ring with $K=R/m$, where $m$ is the maximal ideal of $R$.

Lemma: A noetherian module M is torsion-free if and only if $Hom_R(K,M)=0$.

(proof: The maps in $Hom_R(R/m,M)$ correspond to left multiplications $L_z$ with $z \in Hom_R(R,M)=M$ such that $z m=0$. )

Now assume $M$ is MCM, then $depth(M) \geq 1$ and thus $Hom_R(K,M)=0$, thus $M$ is torsionfree.

(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)

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2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

1(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)

2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.

2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

(1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.)

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A2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.

A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

2.A module $M$ is torsion-free if and only if $M$ is a first syzygy module (and this is also equivalent to the condition that the canonical map $M \rightarrow Hom_R(Hom_R(M,R),R)$ is a monomorphism).

But every maximal Cohen-Macaulay module is a first syzygy module, see for example chapter 1 in the book "Cohen-Macaulay modules over Cohen-Macaulay rings" of Yoshino.

1.I have not read the article, but assuming completeness often garantees the existence of a canonical module which is needed in some arguments.

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