$\underline {Background}$: Let,$\mathcal F$ be a torsion free sheaf of rank $1$.

This means (according to the book by Huybrechts,Lehn (page $11$ definition $1.1.2$)) $$\frac {\alpha _d( \mathcal F)}{\alpha _d( \mathcal O_{X})}=1$$

i.e $\alpha _d$($\mathcal F$)=$\alpha _d$ ($\mathcal O_{X}$) ,

where the Hilbert polynomial $P$($\mathcal F$) can be uniquely written in the form

$$P(\mathcal F,m)=\sum_{i=0}^{dim(\mathcal F)} \alpha_{i} (\mathcal F) \frac {m^{i}}{i!}$$

$\underline {Question 1}$:rank of a coherent sheaf is defined in that book only when the sheaf is pure of dimension d.So does it make sense to apply the same definition in case of a torsion free sheaf?

i.e Is torsion free sheaf pure of dimension d?

$\underline {Question 2}$:How to show that $\mathcal F^{**}$ is a line bundle?

$\underline {Guess}$ :

For $\underline {Question 1}$ I think that it is the other way around,i.e being pure of dimension $d$ is more general than being torsion free

For $\underline {Question 2}$ I think it is better to have an equivalent definition of rank in terms of trivializing open sets or may be stalks(which I don't have)

Any help from anybody is welcome.


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