On a smooth algebraic variety X, every coherent sheaf F has a finite resolution by
locally free sheaves. Using such resolution, we can define the determinant
of F, det F, which is a line bundle on X.


My question is :


why if the support of F is of codimension greater or equal to 2
is the determinant of F trivial ?


It is mentionned without proof on the book "The geometry of moduli spaces of
sheaves", D. Huybrechts, M. Lehn.
I have verified this result on some explicit examples for which I know some
explicit locally free resolutions but I don't see how to do the general case.



triviality of determinant sheaf
Outside the support of $F$, the resolution is an exact sequence, so the alternating tensor product of the determinants is trivial. On a smooth scheme, a line bundle trivial outside a codimension $2$ subset is trivial.


