Let $X$ be a normal projective variety and consider a blowup of $X$ $$Y \rightarrow X.$$ It is a theorem that global sections of $\mathcal{O}_Y = f^\star \mathcal{O}_X$ biject naturally (i.e. via pullback) with global sections of $\mathcal{O}_X$, see for example this question.

Let $\mathcal{F}$ be a coherent sheaf on $X$. Is it true that global sections of $\mathcal{F}$ coincide with global sections of $f^\star \mathcal{F}$ (by pulling back sections)?

My main reason to hope for the result is that it holds for the structure sheaf and I can find no counterexamples. (In fact I am interested in the case $Y\rightarrow X$ is a toric modification and mildly prefer to avoid projectivity hypothesis on $X$ so I would be delighted with the answer in this case).



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