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Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Let $A=\mathbb{C}[x,y,u,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Let $A=\mathbb{C}[x,y,u,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

My questions is: how to compute $M$ concretely?(say as a quotient of sum of $A$)

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.

Thanks for the help!

Let $A=\mathbb{C}[x,y,u,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ is CM but not Gorenstein. Then the canonical sheaf comes from a reflexive module $M$ of rank 1 over $A$ which is not locally free.

Question. How to compute $M$ concretely (say as a quotient of sum of $A$)?

I know that if $U\subset A$ is the regular locus, i.e. $A$ cut off the unique singular point, then $\tilde{M}$ is the push forward of the canonical bundle of $U$. However I could not gain much from this description.