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Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2+yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2+yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$. I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.

Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.