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As my comment up there says, not every minimal resolution is crepant. However, something like the reverse direction is true.

Suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_{Y/Z} > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_{Y/X}$ is also $> 0$, contradicting the crepant assumption.

EDIT: I just added the projective assumption on $Y \to X$, I don't it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

As my comment up there says, not every minimal resolution is crepant. However, something like the reverse direction is true.

Suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_{Y/Z} > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_{Y/X}$ is also $> 0$, contradicting the crepant assumption.

EDIT: I just added the projective assumption on $Y \to X$, I don't think it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

As my comment up there says, not every minimal resolution is crepant.

EDIT: I just added the projective assumption on $Y \to X$, I don't it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

On the other hand, suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_{Y/Z} > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_{Y/X}$ is also $> 0$, contradicting the crepant assumption.

As my comment up there says, not every minimal resolution is crepant. However, something like the reverse direction is true.

Suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_{Y/Z} > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_{Y/X}$ is also $> 0$, contradicting the crepant assumption.

EDIT: I just added the projective assumption on $Y \to X$, I don't it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

As my comment up there says, not every minimal resolution is crepant.

EDIT: I just added the projective assumption on $Y \to X$, I don't it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

On the other hand, suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_Y/Z > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_Y/X$ is also $> 0$, contradicting the crepant assumption.

As my comment up there says, not every minimal resolution is crepant.

EDIT: I just added the projective assumption on $Y \to X$, I don't it's necessary, but it makes $Y \to Z$ much clearer that it's a blow-up of something.

On the other hand, suppose you have a projective crepant resolution $Y \to X$. If you have a factorization $Y \to Z \to X$ where $Z$ is also a resolution of singularities, then $Z \to X$ is also necessarily crepant (the same computation on relative canonical divisors holds). Thus $Y \to Z$ is projective birational and $Z$ is smooth, so $Y \to Z$ can't be small.

Thus there are some new divisors on $Y$, and we know $K_{Y/Z} > 0$ again since $Z$ is smooth, but $K_Z$ is the same as the pullback of $K_X$ (since $Z \to X$ is crepant) so $K_{Y/X}$ is also $> 0$, contradicting the crepant assumption.