# Is the pull-back of canonical sheaf invertible (modulo torsion)?

Let $$X$$ be a $$\mathbb{Q}$$-Gorenstein (isolated) singularity of dimension at least $$2$$ and $$f:Y \to X$$ be a resolution of singularities. In this case the canonical sheaf $$K_X$$ is not necessarily invertible, it is only reflexive.

Question. Is the pull-back $$f^*K_X$$ invertible? If not, can we say that $$f^*K_X/\mathrm{tors}$$ is invertible?

I think that $$f^*K_X$$ is not invertible in general. For instance, take as $$X$$ a quotient surface singularity of type $$\frac{1}{4}(1, \, 1)$$. Then straightforward computations give $$K_Y=f^*K_X - \frac{1}{2}E,$$ where $$E$$ is the exceptional divisor. We infer that $$f^*K_X = K_Y + \frac{1}{2}E$$ is not an invertible sheaf on $$Y$$. In fact, the exceptional divisor $$E$$ is not $$2$$-divisible in $$\operatorname{Pic}(Y)$$. The way I see this is that $$X$$ is the singularity given by a cone over a rational normal curve of degree $$4$$ in $$\mathbb{P}^4$$, and $$Y$$ is the blow-up at the vertex. Then $$E$$ is a section of a (rational) fibration on $$Y$$, hence it cannot be divisible.
• In my example, it seems to me that $f^*K_X$ is torsion-free, so the answer should be again no. Am I missing something? Commented Oct 13, 2021 at 11:58
• I mean, a torsion sheaf is supported on a proper subvariety. But $2f^*K_X=2K_Y+E$ is a line bundle, hence its torsion part is zero. So also the torsion part of $f^*K_X$ is zero. Commented Oct 13, 2021 at 12:05
• My problem is, I cannot imagine a torsion-free sheaf of rank 1 over a smooth variety which is not invertible but if we tensor it twice it becomes invertible. I am thinking using the natural short exact sequence that you get from a torsion-free sheaf to its reflexive hull with non-trivial cokernel. I cannot imagine this cokernel vanishing if we tensor it twice. I find the language used in the literatures a bit ambiguous. In particular, a lot of the times by $f^*K_X$ they mean actually the reflexive hull of this sheaf, which is of course invertible. Commented Oct 14, 2021 at 21:16