# How to prove $L$ is an invertible sheaf?

Suppose $$R$$ is a reduced noetherian ring, and $$L$$ is a finitely generated flat $$R-$$module such that $$L_p\simeq R_p$$ for every prime. Then how to prove $$L$$ is an invertible sheaf of $$\operatorname{Spec} R$$?(means we need to extend the isomorphism of stalks to a neighbourhood)

• Here is how you could try to tackle this: Take generators $g_i$ for $L$, and look at their images in $R_p$. Then find an open affine neighbourhood $D(f)$ of $p$, where those images are defined, which gives a map $R_f^n \to R_f$. By localizing further, make sure this factors over $L_f \to R_f$, and gives an isomorphism. Sep 14 at 13:13
• Check out 13.7.F & 13.7.K in Vakil: For nice enough settings, a module which is free at a point is free at a neighbourhood around that point (13.7.F) - and a module which has constant rank is locally free (13.7.K). Sep 14 at 13:57

There is always an evaluation homomorphism $$ev_L:L\otimes L^{*}\to R, \;\;\; (a,b)\mapsto b(a).$$ This evaluation is natural in the sense that if $$f:M\to N$$ is a module isomorphism, then there is a module isomorphism $$ev(f):M\otimes M^*\to N\otimes N^*,$$ which forms a commutative triangle with both evaluations (i.e. $$ev_M\circ ev(f)=ev_N)$$. We also note that the evaluation map $$R_p\otimes R^*_p\to R_p$$ is a isomorphism.
In particular, we see that after localization at $$p$$, the maps $$(L\otimes L^*)_p=L_p\otimes L^*_p\to R_p$$ is the composition of isomporphisms $$L_p\otimes L^*_p\to R_p\otimes R_p^*\to R_p$$ and so in particualr $$ev_L$$ is an isomorphism after any localization. As any homorphism that is an isomorphism after any localization is an isomorphism, we see that $$ev_L$$ is an isomorphism, so that $$L$$ is invertable.
• In the last step, how can we know $L$ is invertible? The definition I know is: $L$ is locally free of rank 1. But how to prove that(using $ev_L$ is an isomorphism)? Sep 15 at 1:53