# On rank$1$ torsion free sheaves

$$\underline {Background}$$: 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.